ROTATION IN THE PLEIADES WITH K2. III. SPECULATIONS ON ORIGINS AND EVOLUTION

At constant mass, the K dwarfs in the Pleiades show a range of ∼40 in their observed rotation periods (i.e., from ∼0.25 days to ∼10 days), and hence a similar range in their inferred total angular momentum assuming solid-body rotation. Understanding why this very large dispersion in rotational histories exists has obvious importance for understanding both star and planet formation. Theoretical models for angular momentum evolution (e.g., Gallet & Bouvier 2013, 2015) can account for this large range in rotation at a given mass by adjusting parameters such as the disk-locking timescale, core-envelope coupling timescale, and wind saturation velocities. However, it is possible that other mechanisms may also contribute. Binarity is one obvious potential mechanism for affecting the angular momenta of ZAMS-age stars because binary companions have at least the potential to truncate circumstellar disks and reduce the efficacy of PMS disk locking (Patience et al. 2002; Meibom et al. 2007). In order to test this idea, we have identified a sample of photometric binaries among the K2 Pleiades sample using the V versus $V-{I}_{{\rm{C}}}$ and V versus ${(V-{K}_{{\rm{s}}})}_{0}$ color–magnitude diagrams (CMDs). In the former (which uses photometry from Stauffer et al. 1987 and references therein, and from Kamai et al. 2014), we identify stars as photometric binaries if they lie more than 0.3 mag above a single-star main-sequence locus. For the latter, using the estimates of ${(V-{K}_{{\rm{s}}})}_{0}$ as described in Paper I, we adopt ΔV = 0.45 mag for ${(V-{K}_{{\rm{s}}})}_{0}$ < 5 and ΔV = 0.55 mag for ${(V-{K}_{{\rm{s}}})}_{0}$ > 5 to identify probable photometric binaries. The required ΔV is larger for the ${(V-{K}_{{\rm{s}}})}_{0}$ CMD because of the lower accuracy of the (less homogeneous) ${(V-{K}_{{\rm{s}}})}_{0}$ data; the limit changes at ${(V-{K}_{{\rm{s}}})}_{0}$ = 5 because the slope of the single-star locus increases at about that color. Stars selected as binaries from the ${(V-{K}_{{\rm{s}}})}_{0}$ CMD are illustrated in Figure 4. The advantage of this purely photometric selection technique is that we can apply it for every star for which we have K2 rotation period data.

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We replot the period–color data for the Pleiades FGK stars (roughly stars with 1.1 < ${(V-{K}_{{\rm{s}}})}_{0}$ < 3.7) in Figure 5, where the photometric binaries we have identified are now shown as diamonds. This diagram shows that members of the slow sequence are infrequently identified as photometric binaries (9 photometric binaries out of 105 total), whereas the stars with intermediate rotation periods are photometric binaries with significantly higher frequency. In Paper I (Section 3.2, Figure 11), we show that more than a dozen stars in the slowly rotating sequence have been identified as spectroscopic binaries. Nearly all are single-line spectroscopic binaries, consistent with many of them not having been identified by us as photometric binaries. Our association of being a photometric binary and having somewhat more rapid rotation could be reconciled with these additional spectroscopic binaries in the slow sequence if the physical mechanism leading to relatively rapid rotation operates most efficiently for nearly equal mass systems having relatively large semimajor axes.

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In order to illustrate this same point in a more quantitative way, we have calculated "vertical" displacements of each star from the single-star V versus ${(V-{I}_{{\rm{c}}})}_{0}$ locus (dV) and normalized period displacements from the slow-sequence locus curves (${dP}/{P}_{{\rm{seq}}}$)—where ${P}_{{\rm{seq}}}$ is the period predicted by the polynomial fits—and plot those two indices in Figure 6. We use ${(V-{I}_{{\rm{c}}})}_{0}$ rather than ${(V-{K}_{{\rm{s}}})}_{0}$ for this purpose because the ${(V-{I}_{{\rm{c}}})}_{0}$ photometry is more accurate, it is less affected by spottedness (Stauffer et al. 2003), and in this magnitude range we have published ${(V-{I}_{{\rm{c}}})}_{0}$ photometry for more than three-quarters of the Pleiades members (Stauffer et al. 1987 and references therein; Kamai et al. 2014). Figure 6 shows that the stars with intermediate rotation rates are much more frequently displaced significantly above the single-star CMD locus than either of the other two groups. The fraction of stars more than 0.3 mag above the single-star locus is 10% for the slow sequence, 41% for the intermediate sequence, and 14% for the fast sequence. Collapsing the distribution in the other axis, 62% (18 of 29) of the stars identified as photometric binaries versus 22% (26/120) of the single stars are members of the intermediate rotation group. Just from binomial statistics, the probability that so many of the objects in the intermediate rotation group are photometric binaries is less than 1 part in 1000, while the probability that so few of the objects in the slowly rotating group are photometric binaries is of order 1 in 100. We therefore conclude that there is very likely a real link between rotation rate at Pleiades age in this mass range and having a comparable mass companion. A plausible but not unique physical explanation for this correlation is that binary stars with some range of parameters (mass ratio and separation presumably being most important) inhibit disk-dependent PMS angular momentum regulation mechanisms.¹⁷

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If the correlation between binarity and rotation rate arises due to the effect of binarity on the lifetime or amount of dust in the inner disk during PMS evolution, it is possible that this effect could be reflected in the presence or strength of debris disk signatures at later epochs. A number of models predict that the rotation period of low-mass PMS stars will be locked to the Keplerian rotation period of their inner circumstellar disk (Königl 1991; Bouvier et al. 2007, p. 479). This suggests the possibility that the stars with the longest-lived PMS circumstellar disks might arrive on the main sequence as slow rotators (e.g., Bouvier et al. 1997). If those long-lived disks generate more planets and planetessimals, then it is possible that their signature at later epochs could be more dust from planetessimal collisions, which would be detected in the mid- to far-infrared as debris disks. We might then expect a correlation where the stars in the slowly rotating sequence are also frequently identified as having detectable debris disks. Such a correlation was noted by Sierchio et al. (2010), but the statistical significance was meager, in part because only $v\sin i$ data were available for many of the stars at that time. In Figure 7, we again replot the K2 period–color data for the Pleiades, this time highlighting stars with detected debris disks as red circles. Unfortunately, in the color range where there are detections (basically spectral type F and early G), there is no clear separation between a slowly and rapidly rotating sequence, and therefore (in agreement with Sierchio et al.) we cannot draw any conclusions concerning a possible correlation of debris disks and rotation. The stars of interest are too faint for WISE; a test of the hypothesized correlation between debris disk IR excess and rotation must await a more sensitive set of far-IR data, extending into the late G and K spectral types in the Pleiades.

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High-resolution spectra obtained more than 50 yr ago (Anderson et al. 1966) showed that, on average, rotational velocities for Pleiades members more massive than the Sun decrease rapidly going from early to late F. Improved $v\sin i$ for Pleiades F stars were later obtained by Soderblom et al. (1993a) and Queloz et al. (1998). Those data showed mean rotation rates decreasing from about 50 km s⁻¹ at F2 to about 15 km s⁻¹ at G0. Therefore, the most rapidly rotating Pleiades F dwarfs are at the earliest subtypes. However, because early F dwarfs are not expected to have outer convective envelopes, they should not have magnetic dynamos or solar-like starspots. Without spots, these stars should not have rotationally driven photometric variability, and hence they should not appear in our K2 period catalog. Outer convective envelopes of sufficient depth to generate significant energy from a magnetic dynamo are believed to first appear around spectral type F5 (Wilson 1966), so it is near that spectral type where one might expect to identify spot-induced rotational variability. Our K2 data should allow us to accurately determine that transition point. Unfortunately, ZAMS F stars also occupy a ${T}_{{\rm{eff}}}$ range where pulsation is expected to generate periodic photometric variability (Dziembowski 1977; Dupret et al. 2004). In the late A to early F spectral range, one expects to find δ Sct pulsators; for somewhat cooler stars—spectral types generally about F0–F3—one expects γ Dor type pulsation (Balona et al. 2011). From the perspective of this paper, it is fortunate that the δ Sct pulsators have typical periods of 0.03–0.3 days (Balona et al. 2015), allowing their pulsational variability to be easily distinguished from the rotational modulation of F stars, where one expects periods greater than 0.5 days (also see Paper II).

γ Dor variables, on the other hand, are expected to have dominant periods of order 0.4–3.0 days (Kaye et al. 1999), placing their pulsation range directly in the range of rotation periods expected for ZAMS-age F dwarfs. Indeed, while Balona et al. (2011) identify several hundred stars with light curves in the main Kepler field as probable γ Dor variables, they also acknowledge that the variability of many of those stars may in fact be due to rotational modulation and starspots. For our purposes, we choose to avoid this issue by relying on the amplitude of variability (as determined in Paper I) to define the boundary redward of which we are reasonably certain that spots dominate the variability (and the periods we find are the star's rotation period). Figure 8 shows the variation of the amplitude of photometric variability for stars where we determine periods versus ${(V-{K}_{{\rm{s}}})}_{0}$ color. There is a fairly sharp boundary at ${(V-{K}_{{\rm{s}}})}_{0}$ ∼ 1.1, redward of which the amplitudes become much larger; the spectral type corresponding to that color (from Pecaut & Mamajek 2013, Table 5) is F5, the same boundary where one begins to see significant chromospheric emission. Immediately blueward of that boundary, we believe it is entirely possible that much of the variability seen in our K2 light curves is rotational modulation due to small (possibly transient) magnetic spots, but separating that variability from pulsational signatures is not simple and should be the subject of a dedicated paper. From analysis of Q3 Kepler data for active stars, Reinhold et al. (2013) also concluded that spectral type F5 marked a dividing line, blueward of which it was much more difficult to separate rotational variability from pulsational variability.¹⁸

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There is a clearly defined slow-rotation sequence in our K2 Pleiades data, running from ${(V-{K}_{{\rm{s}}})}_{0}$ ∼ 1.1, or spectral type F5 (see discussion in the preceding section), all the way to ${(V-{K}_{{\rm{s}}})}_{0}$ ∼ 3.7, or spectral type K8. The mean rotation period increases from $P\sim 2$ days at the blue edge to $P\sim 11$ days at the red edge. Is the relatively sharp cutoff to the slow sequence at ${(V-{K}_{{\rm{s}}})}_{0}$ ∼ 3.7 simply a direct and readily measureable indicator of the cluster's age, or is there some other physical mechanism that contributes to this sharp end to the slowly rotating sequence at this particular age? We suspect that the former answer is mostly correct, and we provide one figure that strongly supports this view; however, we also offer one (possibly coincidental) piece of evidence in favor of the latter idea.

Direct evidence that the location of the red edge for the slowly rotating sequence is a signpost of the age for a coeval population of stars comes from comparison of the Pleiades K2 rotation period data to similarly processed and analyzed K2 Field 5 data for the much older (∼600–800 Myr) Praesepe cluster. We have conducted a quick-look analysis of the publicly available K2 Field 5 light curves for Praesepe for the sole purpose of making the comparison to the Pleiades data. Figure 9 overplots the period data for Praesepe versus that for the Pleiades; the solid curve is a polynomial fit to the well-defined Praesepe slow sequence.¹⁹ Looking at the slow sequences for the two clusters near ${(V-{K}_{{\rm{s}}})}_{0}$ ∼ 3.5, it is obvious that the slow sequence extends much further to the red at ∼700 Myr than at 125 Myr, and therefore the location of this edge is at least partially a measure of the age of the cluster. A well-defined slow sequence at 500–800 Myr extending to early M is also quite evident in the ground-based rotation studies of Praesepe (Agueros et al. 2011), as well as M37 (Hartman et al. 2009) and the Hyades (Delorme et al. 2011).

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While Figure 9 is quite persuasive, it is nevertheless true that the red edge in the Pleiades occurs almost precisely at the boundary between spectral types K and M. That boundary corresponds approximately to the point where molecules become prevalent in the photospheres of low-mass main-sequence stars, and hence at the point where free electrons become less prevalent (Mould 1976). This point is illustrated in a J − H versus $H-{K}_{s}$ color–color diagram on the K2 Pleiades stars (Figure 10), where we have highlighted the stars in the slow sequence. It is apparent that the slow sequence in the Pleiades ends almost exactly where the dwarf sequence colors turn over in the J − H versus $H-{K}_{s}$ diagram (i.e., where J − H stops getting redder as ${T}_{{\rm{eff}}}$ decreases and instead slowly becomes bluer with decreasing ${T}_{{\rm{eff}}}$). It could be that it is simply coincidental that the red limit to the slow sequence happens to occur at Pleiades age at exactly this color. Or, it may be that there is a (possibly small) decrease in the average angular momentum loss rate at this color as a result of the reduced ionization fraction in the photosphere, such that the red limit to the slow sequence stalls at this color for a time rather than increasing more or less linearly with age. We will return to this topic at the end of the next section.

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We noted a kink at ${(V-{K}_{{\rm{s}}})}_{0}$ ∼ 2.6 in the slowly rotating Pleiades sequence in Section 2. We highlight that kink here in Figure 11 with an expanded plot of just the relevant portion of the $\mathrm{log}P$ versus ${(V-{K}_{{\rm{s}}})}_{0}$ diagram, which again compares stars in the Pleiades and Praesepe. The curves are the second-order polynomial fits to the Pleiades slow-sequence stars described in Section 2. There is obviously no kink in the Praesepe slow sequence near ${(V-{K}_{{\rm{s}}})}_{0}$ = 2.6, but the kink seems fairly evident in the Pleiades.²⁰ This kink in the period–color distribution could simply be illusory and attributable to small number statistics. However, data exist both in other clusters and within our K2 Pleiades observations that suggest that the kink may be real; we describe these data now.

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The presence of a kink in the slow sequence at Pleiades age (∼125 Myr) but not at Praesepe age (∼600–800 Myr) suggests that this could be an age-related phenomenon. If so, other clusters of similar age to the Pleiades should also show the kink. Large samples of rotation periods have now been published for many open clusters (e.g., Irwin et al. 2007, 2008, 2009; Messina et al. 2010; Barnes et al. 2015). However, in many cases the exposure times and areal coverage for those surveys are such that M dwarfs are well sampled in the data but relatively few K dwarfs are included, making those data unsuitable for our purposes. Two relatively young clusters for which rotation periods have been published with good coverage of the K dwarf regime do exist, though; these two clusters are M35 (age ∼ 160 Myr) and M34 (age ∼ 220 Myr). Figure 12 compares the rotation period distribution for those two clusters to Praesepe's rotation data, where the rotation data for these clusters come from ground-based campaigns reported in Meibom et al. (2009, 2011). Close examination of those plots shows the following:

• 1.  
  The M35 slow sequence is very similar to the Pleiades slow sequence (as previously noted by Hartman et al. 2010). The main difference relative to the Pleiades is that the feature that could be interpreted as a kink occurs at a color significantly redder than for the Pleiades, at ${(V-{K}_{{\rm{s}}})}_{0}$ ∼ 2.9.
• 2.  
  The M34 slow sequence is exactly as expected, with a locus that is between that of the younger Pleiades and older Praesepe. Its slow sequence has what may be a similar kink to the Pleiades, but at a redder color than either the Pleiades or M34 (${(V-{K}_{{\rm{s}}})}_{0}$ ∼ 3.1), with a few stars redder than those which loosely follow the lower branch of the Pleiades slow locus curve.

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A logarithmic scale can sometimes be misleading and can underemphasize real but small effects. In Figure 13, we replot a portion of the period data for the young clusters with a linear y-axis. The top left panel of Figure 13 shows the Pleiades and Praesepe stars, zooming in on the region around ${(V-{K}_{{\rm{s}}})}_{0}$ = 2.6. For the other three panels of Figure 13, for each cluster we plot the difference in period between a star's observed period and the period appropriate to that color for a star in Praesepe (using the polynomial fit to the Praesepe period data shown in the first panel). These plots emphasize both the reality of a change in the period distribution at the colors we have advocated in the previous paragraphs and the fact that the location of this effect progresses to redder colors for older clusters.

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What is somewhat concerning is that the red branch of the Pleiades slow sequence (the stars with red circles and ${(V-{K}_{{\rm{s}}})}_{0}$ > 2.9 in the first panel, or correspondingly the stars with ΔP ∼ 4 in the second panel) seems much less populated in M34 and M35. This could be a selection bias—these stars would be precisely the ones most difficult for ground surveys to detect their periodicity because of their expected long periods, relatively small amplitudes, and likely strong spot evolution/migration (as found in the Pleiades—see next few paragraphs). Alternatively, the Pleiades may be unusual for a young cluster in having the red portion of the slow sequence so well populated. In either event, this could argue against designating the red edge of the slow sequence (${(V-{K}_{{\rm{s}}})}_{0}$ = 3.7 for the Pleiades) as an empirical signpost of age for young clusters. Instead, the color of the kink in the distribution (${(V-{K}_{{\rm{s}}})}_{0}$ = 2.6, 2.9, and 3.1 for ages of 125, 160, and 220 Myr, respectively) could be a better age indicator. M35 was observed by K2 in Field 0, and so it may eventually be possible to use those data to derive periods for more late K dwarfs in that cluster and possibly determine better the shape of its slow sequence and thereby allow a more detailed comparison to the Pleiades rotational data.

If the kink in the Pleiades slow-sequence period–color relation marks a change in magnetic field properties or wind properties, it is reasonable to believe that this change could be reflected in other measureable properties of the K2 light curves. As noted in Section 2 (based on the histograms shown in Figure 3), the fraction of slowly rotating stars does change abruptly at this color, going from ∼60% for ${(V-{K}_{{\rm{s}}})}_{0}$ < 2.6 to of order 30% for ${(V-{K}_{{\rm{s}}})}_{0}$ > 2.6. There is also a marked change in the morphology of the K2 light curves at that color. Figure 14 again shows the period versus color plot for the Pleiades, but this time marking stars found in Section 3.4 of Paper II to have two resolved, close peaks in their LS periodogram suggestive of either latitudinal differential rotation or spot evolution (blue diamonds in the figure) and those stars with single-peaked LS periodograms but with evolving light-curve shape (red circles in the figure). A total of 78% of the FGK stars in the slow, intermediate, and rapidly rotating sequences in Figure 2 have one or both of these signatures, versus a negligible fraction for the M dwarfs. This alone illustrates that the magnetic field morphology differs between the GK dwarfs and the M dwarfs. Blueward of ${(V-{K}_{{\rm{s}}})}_{0}$ = 2.6, most (54%) of the symbols are blue, meaning that stars with two close, resolved peaks in their periodograms predominate; redward of ${(V-{K}_{{\rm{s}}})}_{0}$ = 2.6, most (76%) of the symbols are red, corresponding to single-peaked periodograms and evolving light-curve shape. The driver of the morphology changes in either case could be spot evolution or latitudinal differential rotation or a combination of both—here we only note that there is an empirical change in the frequency of the two morphologies at this ${(V-{K}_{{\rm{s}}})}_{0}$ ∼ 2.6. While these effects are not definitive, they hint at a change in the average magnetic field topology at this boundary. Because this boundary appears to move to lower mass with increasing age, it must be driven by some internal process that depends on those two properties.

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Analysis of ground-based rotation period studies of two nearby, young open clusters (NGC 2547, age ∼ 40 Myr, Irwin et al. 2008; NGC 2516, age ∼ 150 Myr, Irwin et al. 2007) has been used to argue that the rotation period distribution for ZAMS-age M dwarfs follows a $J\propto M$ (or, equivalently, a $P\propto {M}^{2}$) relation. Those authors note that stars in this same mass range, 0.1 < M/${M}_{\odot }$ < 0.4, in star-forming regions with ages of a few megayears also follow a $J\propto M$ relation. The Pleiades K2 data provide a larger sample of well-determined rotation periods in this mass range than are available for any other cluster; we use those data here in order to determine how those periods depend on mass (and how this compares to the stars in NGC 2547 and NGC 2516). We use the Baraffe et al. (2015, hereafter BHAC15) 120 Myr isochrone to estimate masses for the Pleiades stars, as described in Appendix A. We also choose to reexamine how the Pleiades (ZAMS-age) period–mass relation evolves with time by comparing it to period data for NGC 2264 (we describe the NGC 2264 period data and mass determinations in Appendix B). We have compiled our own list of NGC 2264 periods, determined ${(V-{K}_{{\rm{s}}})}_{0}$ colors for those stars, estimated masses using the same BHAC15 models as for the Pleiades (except now for 3 Myr), and culled the sample to remove stars with strong IR excesses, large reddening, or that fall below the main locus of NGC 2264 members in a CMD).

Figure 15 shows the rotational velocity distribution of late M dwarfs in the Pleiades and NGC 2264, in histogram format. Comparison of those diagrams shows that the Pleiades distribution is similar in shape, but somewhat narrower, than the distribution in NGC 2264. That is, the distributions are unimodal (as opposed to the bimodal distribution seen at higher mass), and the 10%–90% width of the distribution when plotted in terms of log P is about 1 dex (but broader for NGC 2264). The Pleiades stars obviously have spun up considerably relative to their younger cousins; the median rotational period at 0.3 ${M}_{\odot }$ in NGC 2264 is about 3.8 days, while it is about 0.68 days in the Pleiades, a change by about a factor of six assuming that the Pleiades and NGC 2264 have the same parent distribution. According to the BHAC15 models, the moment of inertia for a 0.3 ${M}_{\odot }$ star decreases by about a factor of 12 between 3 and 125 Myr. PMS angular momentum loss mechanisms therefore, on average, drain about half of the angular momentum from 0.3 ${M}_{\odot }$ M dwarfs during this time interval. The similarity in the shape, but somewhat broader distribution in NGC 2264, suggests that angular momentum loss rates scale only weakly with rotation period in this mass and age regime.

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In order to determine the dependence of period on mass for the Pleiades M dwarfs, we have divided the stars into six color bins of 0.4 mag width, beginning at ${(V-{K}_{{\rm{s}}})}_{0}$ = 4.1 (approximately 0.6 ${M}_{\odot }$) and ending at ${(V-{K}_{{\rm{s}}})}_{0}$ = 6.5 (approximately 0.1 ${M}_{\odot }$), and determined the median period in each color bin. Figure 16 shows these median periods and several possible period–mass power-law relations. For stars somewhat higher in mass than the Sun, Kraft proposed an initial angular momentum distribution of the form $J/M\propto {M}^{2/3}$ (Kraft 1970). With $J\propto {{MR}}^{2}{\rm{\Omega }}$, and given that at Pleiades age in this mass range it is reasonable to adopt the approximation that $R\propto M$, the Kraft law reduces to $P\propto {M}^{4/3}$. For the Pleiades M dwarfs (Figure 16), the Kraft relation seems a bit too shallow, whereas a $P\propto {M}^{2}$ relation appears to be too steep. The $P\propto {M}^{1.5}$ (corresponding to $J\propto {M}^{1.5}$) relation appears to be a reasonably good fit. In Appendix B, Figure 26 shows a similar plot for NGC 2264. In this case, the best fit is for $P\propto {M}^{1.15}$. Given the uncertainties in estimating masses at young ages, it is not certain that these two results are significantly different. At 3 Myr, R scales about as ${M}^{0.5}$ (rather than $R\sim M$ on the ZAMS), so our period–mass relation at 3 Myr corresponds approximately to $J\propto M$). These period and angular momentum power-law relationships are in approximate accord with the previous determinations by Irwin et al. (2008, 2009).

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To illustrate more directly the similarity of the rotational velocity distributions for the M dwarfs in the two clusters, Figure 17 plots the median periods and best polynomial fits for both NGC 2264 and the Pleiades as a function of mass (as estimated from the BHAC15 isochrones). In this figure, we have divided all the NGC 2264 periods by 5, in order to approximately account for spin-up between 3 and 125 Myr. This scaling succeeds remarkably well in aligning the median periods for the two clusters, indicating that the angular momentum loss mechanism in this mass and age range preserves the functional dependence of rotation period on mass.

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The 1 dex width for the late M rotation distribution in the two clusters is somewhat misleading because much of that width simply reflects the strong correlation between period and mass. The median period at ${(V-{K}_{{\rm{s}}})}_{0}$ = 5.0 (median mass = 0.39 ${M}_{\odot }$) is 1.10 days, while at ${(V-{K}_{{\rm{s}}})}_{0}$ = 5.8 (median mass = 0.21 ${M}_{\odot }$) it is 0.37 days. To get a better measure of the true range in P at a given mass for the late M dwarfs in the Pleiades, we have subtracted an appropriately normalized power law ($P=4.482\times {M}^{1.5}$) from the observed periods and zero-point-shifted the periods to the median period at ${(V-{K}_{{\rm{s}}})}_{0}$ = 5.0. The logarithmic distribution of these normalized periods is shown in Figure 18, where it can be seen that the width of the distribution is now only about half of that deduced without correction for the strong correlation of P with mass.

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For the Pleiades FGK stars, in Paper II we have associated the presence of close, resolved peaks (ΔP/P < 0.45) in the LS periodogram with latitudinal differential rotation or spot evolution on a single star, and distant, resolved peaks (ΔP/P > 0.45) with the rotation periods of the two stars in a binary. Theory predicts that fully convective stars should rotate as solid bodies; therefore, we might have simply assumed that all resolved peaks in the LS periodogram for mid- to late M dwarfs were indicative of binary stars. However, we prefer to adopt an empirical approach and allow the data to determine the physical mechanism for multiple peaks in the LS periodogram of late M dwarfs. We believe that the Pleiades data do require a binary star interpretation, but also that those data have interesting implications for the initial angular momentum distribution of binary M dwarfs.

For most of the subsequent analysis, we limit ourselves to mid- to late M dwarfs (5.0 < ${(V-{K}_{{\rm{s}}})}_{0}$ < 6.0), with the blue boundary being set at where one expects stars to become fully convective and the red boundary being set where our period data become significantly incomplete due to the faintness of the stars.

The top left panel of Figure 19 shows where M stars identified in Paper II as having two close peaks in the LS periodogram fall in a period–color diagram. About 10% of the stars with periods in this color range are identified with this signature. The top right panel of Figure 19 provides the V versus ${(V-{K}_{{\rm{s}}})}_{0}$ diagram for these same stars, as well as for stars identified in Paper II as having distant resolved peaks in the LS periodogram. Nearly all of the stars in both categories fall well above the single-star locus in the CMD, most easily interpreted as indicating that they are photometric binaries. This diagram alone is prima facie evidence that resolved peaks of any form in this color range should be interpreted as the periods of both components of a binary because we can think of no reason why latitudinal differential rotation should only be detectable on photometric binaries. The only possible escape we can think of for this interpretation would be if the stars with two resolved peaks were more spotted than single-peak stars and if heavily spotted late M Pleiades members were significantly redder in ${(V-{K}_{{\rm{s}}})}_{0}$ than otherwise similar Pleiads. The bottom left panel of Figure 19 shows that this escape route is not viable, because the resolved close-peak stars are just as strongly displaced above the single-star locus in a g versus g − i CMD, and one expects g − i to not be significantly affected by spottedness (Stauffer et al. 2003).²¹ Rappaport et al. (2014) also concluded that rapidly rotating, field M dwarfs with two periods detected in their LS periodogram are almost always best interpreted as young binary stars.

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There is at least one apparent problem with interpreting all of the resolved peak stars as binaries; this problem is illustrated in the bottom right panel of Figure 19, where we now show only fully convective stars (5.0 < ${(V-{K}_{{\rm{s}}})}_{0}$ < 6.0) and we plot both periods for each star. Comparing the top left and bottom right panels of Figure 19, it seems that the two identified periods for a large fraction of the stars with two resolved LS peaks are too close to each other to have been drawn randomly from the overall distribution. We have run a Monte Carlo simulation to test this assertion, where we created 1000 sets of 45 binaries, with periods chosen randomly from the observed distribution, selecting stars differing in ${(V-{K}_{{\rm{s}}})}_{0}$ by less than 0.15. Figure 20 provides two visualizations of the output from this simulation. The top panel of Figure 20 compares the median normalized period difference to the median first period for the simulations and the observed population of two resolved peak Pleiades M dwarfs with 5.0 < ${(V-{K}_{{\rm{s}}})}_{0}$ < 6.0. None of the simulated groups have normalized period differences as small as found for the late M stars with two resolved peaks, indicating P < 0.1% for a random selection process. The stars in the resolved peak group also appear to be relatively rapid rotators, with their median period being shorter than the median period of all of the simulation runs. The bottom panel of Figure 20 compares a histogram of the observed normalized period differences to that from the average of the simulated groups. There is a large excess of systems with period differences less than 30% of the mean period.

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A hypothesis that would seem to fit our data is that at the end of the PMS accretion phase, very low mass (VLM) binaries composed of comparable-mass stars have periods that are shorter and more similar to each other than they would be if drawn randomly from all the stars of that mass in their birth group. Because there is little dependence of angular momentum loss rate on either mass or period from that point to the age of the Pleiades (see previous section), this rotation bias is still retained at Pleiades age. At much older ages, where mass- or period-dependent loss rates become important, the similarity of rotation periods may become less apparent—thereby explaining why there has been little evidence for this effect up to now (though from a spectroscopic study of 11 field VLM binary stars, Konopacky et al. [2012] concluded that "the components of VLM binaries are not randomly paired in $v\sin i$"). The tendency for late M photometric binary stars to rotate faster than single stars of the same color is in the same sense as we found for G and K dwarf binaries in Section 3.1.

There are suggestive correlations between the distribution of rotation periods in the Pleiades and the measurements of surface magnetic field topologies being made in recent years with ZDI and other techniques (Donati & Landstreet 2009; Linsky & Scholler 2015). In particular, those data show a sharp transition in magnetic field topology at about spectral type M4, corresponding to about ${(V-{K}_{{\rm{s}}})}_{0}$ = 5.0. Cooler than that boundary, M dwarfs generally have primarily poloidal, axisymmetric fields and less complex small-scale magnetic structures. They also generally show little latitudinal differential rotation. There is also a step up in the large-scale magnetic energy going from early M to mid-M (Morin et al. 2008, 2010). Early M dwarfs have strong toroidal fields and weak poloidal fields, have more small-scale magnetic structure, and often have fairly strong latitudinal differential rotation. They also have more rapid evolution in their magnetic topology, perhaps due to the stronger shearing induced by the differential rotation (Donati et al. 2008). The sharp transition in magnetic properties plausibly occurs at or near the mass where M dwarfs become fully convective (M ∼ 0.35 ${M}_{\odot }$; spectral type ∼M3.5).

We have already shown in Figure 14 that the K2 data corroborate that the GK dwarfs very often show signatures of spot evolution or latitudinal differential rotation. There are other morphological features of the K2 light curves that may shed light on the magnetic topologies of Pleiades-age M dwarfs. The top panel of Figure 21 highlights a different facet of light-curve morphology; the stars circled in red in this figure have LS periodograms with just one, narrow peak and phased light curves that show no significant evolution with time and which have sinusoidal or nearly sinusoidal shapes. Light curves of this type are almost absent in the slowly rotating FGK sequence (10% frequency, for 1.1 < ${(V-{K}_{{\rm{s}}})}_{0}$ < 3.7), whereas among the rapid rotators in the same color range (the blue circled points in Figure 2), 45% have light curves of this type. Among the early M dwarfs with 3.7 < ${(V-{K}_{{\rm{s}}})}_{0}$ < 4.6, 53% (48/91) have this signature, whereas for later M dwarfs with 4.6 < ${(V-{K}_{{\rm{s}}})}_{0}$ < 6.0, the fraction rises to 73% (300/410). The latter number should be regarded as a lower limit because some of the late M dwarfs are very faint and/or have very low amplitude variability, making it difficult to characterize the morphological status of the light curve. These stable, sinusoidal light curves imply long-lived spot groups at relatively high latitudes.

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The bottom panel of Figure 21 highlights a different morphological signature, stars with two or more peaks or minima in their phased light curves and where there is little or no evolution in the light-curve shape over the time period of the full K2 campaign.²² That signature is best explained as due to the presence of spot groups on opposite hemispheres of the star (see, e.g., Figure 5 of McQuillan et al. 2013 or Figure 4 of Davenport et al. 2015), with little or no differential rotation (and little or no change in spot size or shape).

Our K2 data fit well into the magnetic field topology story line for M dwarfs:

• 1.  
  There is a striking transition in the period distribution between the early and mid- to late M dwarfs, at apparently the same place where the B field topologies change. The early M dwarfs show a broad spread in rotation, showing neither a well-defined slow sequence (as present for the FGK dwarfs) nor a well-defined rapidly rotating sequence (as present for the late M dwarfs).
• 2.  
  A very large fraction of the late M dwarfs show primarily short periods and stable light curves suggestive of solid-body rotation, long active region lifetimes, and primarily axisymmetric, poloidal fields. Evolving light-curve shapes due to either spot evolution or latitudinal differential rotation are present for about one-quarter of the early M dwarfs (see Figure 13 of Paper II), in agreement with the ZDI measurement of rapid evolution in magnetic topology for some early M dwarfs.

Figure 22 includes four F dwarfs that seem to be "lost in space"—they all have ${(V-{K}_{{\rm{s}}})}_{0}$ ∼ 1.1, spectral type ∼F4, and periods near 7 days, as opposed to an expected period of order 2 days if they were located on the Pleiades slowly rotating sequence. Table 1 lists the properties of these stars. All four stars appear to be very abnormally slowly rotating. Why is this the case?

Table 1.  F Stars with Abnormally Slow Rotation Periods

EPIC	Name	Spectral Type	${(B-V)}_{o}$	${(V-{K}_{{\rm{s}}})}_{0}$	Period	$v\sin i$
			(mag)	(mag)	(days)	(km s−1)
211121734	HII 605	F3	0.40	1.05	7.40	...
210996505	HII 1132	F5	0.45	1.16	6.85	>40
211072160	HII 1338	F3	0.42	1.10	7.60	...
211138217	HII 1766	F4	0.43	1.16	6.72	22.7

Note. Spectral types from Mendoza (1956), except for HII 1132, which is from Kraft (1967b); photometry and periods from Paper I; $v\sin i$ values from Queloz et al. (1998). We have a Keck HIRES spectrum for HII 1132, from which we estimate $v\sin i$ ∼ 50 km s⁻¹.

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We note that the two simplest explanations do not work. First, membership in the Pleiades is not an issue; all four stars are essentially certain members of the cluster. Second, their measured properties as listed in Table 1, and in particular their periods, are well determined.

For two of these stars, HII 1132 and HII 1766, the most probable solution to the conundrum is that the star is a binary, and the K2 period measures the rotation period of the secondary star and not the primary. In fact, for these stars, the measured period cannot be the rotation period for the primary because the $v\sin i$ values for these stars are much larger than their inferred rotational velocity if one adopts the ∼7 day period and a radius of order 1.4 ${R}_{\odot }$ appropriate for their spectral type. Examination of DSS and 2MASS finding charts for all four stars shows no nearby line-of-sight companion that would likely contaminate the K2 photometry.

For HII 1132 and HII 1766, one can perform a thought experiment. If the K2 period derives from a companion star that lies on the slowly rotating FGK sequence polynomial relation, what would its magnitude be in the Kepler band and what amplitude would its variability have to be in order to explain the observed light-curve amplitude (where the K2 data are assumed to include a constant signal from the F star and the variable light curve from the secondary)? Doing that experiment for HII 1132, we find ${(V-{K}_{{\rm{s}}})}_{0}$ ∼ 2.3, ${K}_{{\rm{kep}}}\sim 12.0$, and an inferred photometric amplitude of close to 7% for the secondary. That is a large amplitude for a Pleiades K dwarf, but within the range found for slow rotators (see Figure 12 in Paper I). A similar value would be inferred for HII 1766. This situation only arises in the Pleiades for F star primaries, where any intrinsic variability of the F star has very low amplitude (Figure 8) and where plausible, slowly rotating secondary stars a few magnitudes fainter could have relatively large photometric amplitudes. HII 1132 and HII 1766 are both displaced above the single-star V versus ${(V-{I}_{{\rm{c}}})}_{0}$ main sequence by about 0.3 mag, making it possible but not certain that they are binaries with relatively low mass companions.

For HII 1338, instead, we believe there is a different explanation. HII 1338 is a known double-line spectroscopic binary, with a measured orbital period of 7.76 days (Raboud & Mermilliod 1998). The stars in the system likely have their rotation periods tidally locked (or nearly so) to the orbital period, and that is the physical explanation for their slow rotation rate.

HII 605 is also a known spectroscopic binary (Pearce & Hill 1975; Liu et al. 1991). The radial velocity amplitude is at least 40 km s⁻¹, but no orbital parameters have been measured. It is possible therefore that it also has an orbital period short enough to induce tidal locking at the observed rotation period; alternatively, if the orbital period is long, then the explanation proposed for HII 1132 and HII 1776 may apply.

Finally, we note that HII 1132 has by far the largest IR excess of any known Pleiades member (Rhee et al. 2008). It also has a brown dwarf companion (Rodriguez et al. 2012). While one could invent possible connections between these observables and the very slow rotation rate for HII 1132, with only one object in the sample the connection would be tenuous.

Even in a nearby cluster like the Pleiades, membership studies are imperfect, particularly as one pushes to relatively faint magnitudes. As discussed in Paper I, our original list of potential Pleiades members with K2 light curves included of order 1000 stars. We conducted our own literature search for these stars in order to verify membership, and we ended up eliminating about 150 stars as likely non-members (see Section 2.5 of Paper I). Many of these stars either had no periodic signature in the K2 data or had quite long periods for their spectral type if they were Pleiades members (see Appendix C of Paper I). However, even after that culling, several M dwarfs remain that seem to have rotation periods that are abnormally long; these are the stars marked in red with P > 10.0 days in Figure 22. Are these stars simply non-members that have slipped past our vetting process, or are they stars that have arrived at Pleiades age with rotation rates that fall outside the norm for their mass?

Table 2 provides basic data for these comparatively slowly rotating M dwarfs, including membership probabilities from the most recent survey efforts (Lodieu et al. 2012; Bouy et al. 2015) and proper motions from the URAT all sky survey (Zacharias et al. 2015). All of the stars in this table are photometric and proper-motion members of the cluster based on these papers and catalogs, and based on our own photometry. There are essentially no other published data for these stars that provide additional useful information on their cluster membership. To help interpret their periods, we have obtained Keck HIRES spectra for four of these stars: HHJ 353, SK 17, HII 370, and DH 668. The membership of DH 668 is uncertain—our measured radial velocity is slightly discrepant from the mean Pleiades motion, and it has Hα strongly in absorption, more consistent with it being a field star. The other three stars have radial velocities within ∼1 km s⁻¹ of that expected for Pleiades members. In addition, all three have Hα equivalent widths suggestive of youth (that is, they have either weak Hα emission or "filled-in" Hα). In Figure 23, we show the Hα profiles for these three stars along with a slowly rotating, late K dwarf Pleiades member (HII 3030) for which we have a WIYN-Hydra echelle spectrum (published in Terndrup et al. 2000). We do not have a K2 light curve for HII 3030; we include it here because of the resemblance of its Hα profile to that for HII 370 and SK 17 and because its $v\sin i$ indicates that it is a relatively slow rotator. HII 3030, HII 370, and SK 17 all have similar, and slightly unusual, Hα emission profiles consisting of two narrow-emission peaks located in what would have been the wings of their Hα absorption profile if they were not chromospherically active; HHJ 353 has the strongest Hα emission and has a typical Hα profile for a slowly rotating dMe star. The Hα profiles for HII 3030, HII 370, and SK 17 are essentially as predicted by models for a weakly active dMe star (Cram & Mullan 1979). The Hα equivalent widths for these four stars are similar to or slightly less than other members for their color. Given that the three stars from Table 2 having these new Hα data are more slowly rotating than any other members of their color, it is probably appropriate that they also have somewhat weaker Hα equivalent widths. We therefore conclude that at least these three—and probably the majority of the others in Table 2—are actual Pleiades members, and they are indeed rotating abnormally slowly. Something either in their initial birth environment or in their angular momentum loss between birth and Pleiades age has allowed them to spin down more than all the other stars of their mass.

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Table 2.  M Stars with Abnormally Slow Rotation Periods

EPIC	${K}_{{\rm{s}}}$	Period	${(V-{K}_{{\rm{s}}})}_{0}$	Bouy	Lodieu	μR.A.	μDecl.	Name
	(mag)	(days)	(mag)
211074500	13.18	10.06	5.65	0.99	0.84	27	−48	BPL 88
210865020	11.24	13.05	4.21	0.88	⋯	16	−43	UGCS J03411+205117
211007577	11.67	13.70	4.57	0.99	0.54	23	−49	HHJ 353
211007344	11.08	14.58	4.11	0.98	0.72	16	−44	SK 17
211056297	10.45	17.28	3.88	0.99	⋯	10	−59	HII 370
210855272	10.94	17.60	3.64	0.72	0.11	11	−38	DH 668
210975876	12.53	22.14	4.91	0.54	⋯	13	−41	UGCS J040550+223553

Note. Bouy & Lodieu columns are membership probabilities from Bouy et al. (2015) and Lodieu et al. (2012), respectively. The columns providing proper motions in R.A. (μR.A.) and decl. (μdecl.) are in units of mas yr⁻¹, as provided by the URAT proper-motion survey (Zacharias et al. 2015). The mean cluster motion is μR.A. = 19.5 mas yr⁻¹ and μdecl. = −45.5 mas yr⁻¹ (Lodieu et al. 2012).

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