Rotation of Late-type Stars in Praesepe with K2

Praesepe members and candidate members were observed in K2 Campaign 5, which lasted for 75 days between 2015 April and October. Figure 1 shows the distribution of objects with K2 LCs on the sky; note the gaps between detectors. All of the stars shown were observed in the long-cadence (∼30 minute exposure) mode. Some of these stars were additionally observed in fast cadence (∼1 minute exposure), but those data are beyond the scope of the present paper. There are 984 unique K2 long-cadence LCs corresponding to members or candidate members of Praesepe (see Section 2.5.4 below). The tidal radius of Praesepe is 12.1 pc (Holland et al. 2000), which at a distance of 184 pc (see Section 2.5 below) is ∼38 across, which is easily covered by K2. (Stars that are more than 5° away from the cluster center are highlighted in Appendix H.)

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Kepler pixel sizes are relatively large, $3\buildrel{\prime\prime}\over{.} 98\times 3\buildrel{\prime\prime}\over{.} 98$, and the 95% encircled energy diameter ranges from 3.1 to 7.5 pixels with a median value of 4.2 pixels. During the K2 portion of the mission, because only two reaction wheels can be used, the whole spacecraft slowly drifts and then repositions regularly every 0.245 days. This drift is ∼01 per hour (Cody et al. 2017).

Since these data were reduced at the same time as we reduced our Pleiades data (Rebull et al. 2016b), we have the same sets of LCs available to us: (1) the pre-search data conditioning (PDC) version generated by the Kepler project and obtained from MAST, the Mikulski Archive for Space Telescopes, (2) a version with moving apertures obtained following A. M. Cody et al. (2017, in preparation), (3) the version using a semiparametric Gaussian process model used by Aigrain et al. (2015, 2016), and (4) the "self-flat-fielding" approach used by Vanderburg & Johnson (2014) as obtained from MAST. To this, late in the process, we added (5) the LCs from the EVEREST pipeline (Luger et al. 2016), which uses pixel level decorrelation. We removed any data points corresponding to thruster firings and any others with bad data flags set in the corresponding data product.

Again, following the same approach as in the Pleiades, we inspected LCs from each reduction approach, and we selected the visually "best" LC from among the original four LC versions. Since the EVEREST LCs became available late in our analysis process, for the most part we used the EVEREST LCs to break ties or clarify what the LC was doing. About 7% had such severe artifacts that no "best" LC could be identified; as for the Pleiades, this is often a result of instrumental (non-astrophysical) artifacts (because the star is too bright or too faint or adversely affected by nearby stars, etc.). As for the Pleiades, though, the periods we report here are not generally ambiguous, and are detected in all the LC versions; they just look cosmetically better in one version or another. Our approach of comparing several different LC versions also minimizes the likelihood that any of these LC reductions removes stellar signal.

Out of the 984 LCs, only one pair of stars for which a K2 LC was requested were within 4'' of each other (within a Kepler pixel). These objects are discussed further in Appendix A.

Our approach for finding periods was identical to what we used in the Pleiades in Rebull et al. (2016b). In summary, we used the Lomb–Scargle (LS; Scargle 1982) approach as implemented by the NASA Exoplanet Archive Periodogram Service¹² (Akeson et al. 2013). We looked for periods between 0.05 and 35 days, with the upper limit being set by roughly half the campaign length.

Figure 2 shows LCs, periodograms, and phased LCs for some Praesepe stars with single, unambiguous periods. As we found in the Pleiades, the overwhelming majority of them have zero, or effectively zero, false alarm probability (FAP); a significant fraction have more than one peak with 0 FAP—see Section 4 below.

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For stars of the mass range considered here, the phased LCs are mostly sinsuoidal and therefore best attributed to star spot-modulated rotation. We find periods for 828/984 K2 LCs, or 84% of all K2 LCs of candidate or confirmed Praesepe members (see Section 2.5.4). There is no significant trend with color (as a proxy for mass) for the fraction of periodic stars.

As for the Pleiades, we removed from this distribution any periods that are likely to be eclipsing binaries (EBs; see Appendix B) or those whose waveforms did not seem to be rotation periods (see Appendix C).

There are a few multi-periodic stars that we suspect to be pulsating variables (see Appendix D); many of them are also reported in the literature as δ Scuti-type variables. For some pulsators, the first period is still likely to be similar to the rotation period (see discussion in Rebull et al. 2016a). For the remaining multi-periodic stars (∼25% of the sample; see Section 4 below), for the most part, we took the period corresponding to the strongest peak in the periodogram as the rotation period to be used for subsequent analysis. In a few cases (e.g., double-dip stars; see Section 5), a secondary peak is the right ${P}_{\mathrm{rot}}$ to use.

We have chosen four of the most recent surveys looking for rotation rates in Praesepe for a detailed comparison of our results; they are summarized in Figure 3. There are 220 Praesepe stars in the literature with at least one estimate of ${P}_{\mathrm{rot}};$ 60 of those do not have K2 LCs.

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Scholz et al. (2011) used the Isaac Newton Telescope to monitor very low mass members, reporting 49 periods. We have periods for 40 of the stars in common between the two studies; the remaining Scholz et al. stars were not monitored (either because they are too faint for K2 or because they are in the gaps between chips). The periods derived for these 40 stars are shown in Figure 3. There is generally very good agreement; the median fractional difference ($| {P}_{\mathrm{Scholz}+}-{P}_{\mathrm{here}}| /{P}_{\mathrm{here}}$) is 2.4%. There are eight stars for which their period and ours do not match, and most of them are harmonics, because they are off by factors of 2 (too small or large). In four of these cases, the period we derive is very close to 1 day, which is hard to recover with a ground-based telescope. We believe our periods to be correct in all cases for the epochs in which we observed this cluster. Two stars are of special note. EPIC 211951438/HSHJ396 has a period of 0.593 days in Scholz et al., but we cannot recover any period for this star, so we do not retain it as periodic. EPIC 211984058/2MASSJ08384128+1959471 was not a star for which we initially found a period, because all five LC versions are different, and it was not clear which LC version is the "best" (see discussion above in Section 2). However, upon examination of the K2 thumbnail and POSS images of the region, several of the LC versions appear to have been dragged off to a nearby bright star. The Aigrain et al. version is the only one that both stays on the target star and has a periodic signal. The period we derive from this is identical to that from Scholz et al. (2011), so we retained this star as periodic with that period.

Delorme et al. (2011) used SuperWASP to monitor 71 cluster members, looking for periods from 1.1 to 20 days, determining that 52 were periodic. Of these periodic stars, we have 39 in common (the remaining 13 fall in the gaps between K2 chips, or are otherwise off the K2 FOV). The periods derived for these 39 are shown in Figure 3. There is generally very good agreement; the median fractional difference ($| {P}_{\mathrm{SuperWASP}}-{P}_{\mathrm{here}}| /{P}_{\mathrm{here}}$) is 2.7%. There are three stars for which our periods do not agree. One is EPIC 212013132/JS379 = 2MASSJ08404426+2028187, where it is likely that one of the surveys detected an harmonic. SuperWASP obtains a period of 4.27 days; we obtain three periods, including one that is close to 4.4 days, but the primary period we obtained for this star is 2.129 days. The second star is EPIC 211950227/2MASSJ08402554+1928328, where the K2 LC is very, very messy. SuperWASP obtains 13.15 days. The period we adopted for this one is 0.8984 days, and comes from only part of the LC, with most of the LC having been corrupted by instrumental effects. The third star is EPIC 211995288/KW30 = 2MASSJ08372222+2010373, where we find a period of 7.8 days, a factor of 2 larger than the 3.9 days found in SuperWASP, so another harmonic. Having tested these periods by phasing our LCs at these alternate periods, we believe our periods to be correct in all cases for the epochs in which we observed these stars. There are no stars in common where SuperWASP has a period and we do not have a period.

Kovács et al. (2014) used HATNet (Hungarian Automated Telescope Networks) to monitor 381 members, finding 180 rotation periods, all ranging between 2.5 and 15 days. They identified 10 more stars as having what we would call "timescales," that is, repeated patterns that may or may not correspond to periods. There are 152 stars (150 out of their rotation periods and 2 more out of their "timescales") in common between the surveys, shown in Figure 3; most of the remaining stars fall in the gaps between chips, while a few could have been but simply were not observed. For the stars that we have in common, the median fractional difference ($| {P}_{\mathrm{HATNet}}-{P}_{\mathrm{here}}| /{P}_{\mathrm{here}}$) is 2.2%. There are 13 stars for which the periods do not match, 7 of which are likely harmonics; we believe our period to be correct. One of the two from their "timescales" category is EPIC 211918335/HAT-269-0000582/KW244 = TX Cnc, which Kovács et al. identify as an EB and Whelan et al. (1973) identifies it as a W UMa-type; we drop this ${P}_{\mathrm{orb}}$. The other is 211947631/HAT-269-0000465/BD+19d2087 (which Kovács et al. simply call "miscellaneous"); we retain P = 4.74 days as a rotation period. There are no stars for which we both have LCs but HATNet has a period and we do not.

Agüeros et al. (2011) and Douglas et al. (2014) are a two-part study on the Palomar Transient Factory (PTF) observations of Praesepe. They used PTF to monitor 534 members, 40 of which were periodic between 0.52 and 35.85 days. There are 30 stars in common between the studies (see Figure 3), with the remaining 10 falling in the gaps between chips. The derived periods are generally in very good agreement; the median fractional difference ($| {P}_{\mathrm{PTF}}-{P}_{\mathrm{here}}| /{P}_{\mathrm{here}}$) is just 0.36%. There is one star (EPIC 211989299/AD2552 = 2MASSJ08392244+2004548) for which one of us is likely to have found an harmonic, although the LC is messy (we obtain 12.791 and they obtain 25.36). We believe our periods to be correct for the epochs in which we observed these stars. For another one (EPIC 211886612/JS525), they found a period of 22.6 days; we retrieved an LS peak at 17.2 days for this object, but relegated it to "timescales" (in Appendix C; this is why only 29 stars appear in Figure 3). For the LC we have, it just does not look as nice as the rest of the clear ${P}_{\mathrm{rot}}$ here.

We note here that, using K2, we find periods for a far higher number and higher fraction of Praesepe members than have been accomplished before. Moving to space enables higher precision and continuous photometry, resulting in a far higher fraction of detectable rotation periods.

We started with the list of Praesepe members that we had proposed for this campaign and added targets from other related programs. From this list, we then amassed supporting data and assessed membership for each star. For some objects, we obtained additional Keck/HIRES spectroscopy; see Appendix I.

Table 1 includes the supporting photometric data we discuss in this section, plus the periods we derive here (in Section 2.2), for members of Praesepe. For completeness, non-members (NMs) appear in Appendix E.

Table 1.  Periods and Supporting Data for Praesepe Members with K2 Light Curves

Label	Contents
EPIC	Number in the Ecliptic Plane Input Catalog (EPIC) for K2
coord	R.A. and decl. (J2000) for target
othername	Alternate name for target
Vmag	V magnitude (in Vega mags), if observed
Kmag	${K}_{{\rm{s}}}$ magnitude (in Vega mags), if observed
vmk	$(V-{K}_{{\rm{s}}})$, as directly observed (if V and ${K}_{{\rm{s}}}$ exist), or as inferred (see text)
P1	Primary period, in days (taken to be rotation period)
P2	Secondary period, in days
P3	Tertiary period, in days
P4	Quaternary period, in days
LC	LC used as "best"a
single/multi-P	Indicator of whether single or multi-period star; if object has a "timescale" (see Appendix C), that is indicated
dd	Indicator of whether or not it is a double-dip LC
ddmoving	Indicator of whether or not it is a moving double-dip LC
shch	Indicator of whether or not it is a shape changer
beat	Indicator of whether or not the full LC has beating visible
cpeak	Indicator of whether or not the power spectrum has a complex, structured peak and/or has a wide peak
resclose	Indicator of whether or not there are resolved close periods in the power spectrum
resdist	Indicator of whether or not there are resolved distant periods in the power spectrum
pulsator	Indicator of whether or not the power spectrum and period suggest that this is a pulsator

Note.

^aLC1—PDC, from MAST; LC2—version following A. M. Cody et al. (2017, in preparation); LC3—version following Aigrain et al. (2015); LC4—version reduced by Vanderburg & Johnson (2014) and downloaded from MAST; LC5—version reduced by EVEREST code (Luger et al. 2016) and downloaded from MAST.

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.

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2.5.1. Metallicity, Reddening, Age

2.5.2. Target List

2.5.3. Literature Photometry

2.5.4. Membership

2.5.5. Obtaining $(V-{K}_{{\rm{s}}})$

Since we want to compare to the results we obtained in the Pleiades, we wanted to use $(V-{K}_{{\rm{s}}})$ as a mass proxy in the same fashion as we did in the Pleiades.

We originally approached this the same way we did for the Pleiades, and used data from URAT. However, Gaia data are now available. Since V − K_s is available (with V and K_s directly observed) for ∼250 Praesepe stars with K2 LCs, we derived a formula to convert G − K_s colors to V − K_s colors by comparing the V − K_s to the G − K_s. However, there are no observed V photometry of known members for colors redder than about V − K_s = 5.5. To extend the calibration of G − K_s, we have used faint Pleiades and Hyades stars (where V photometry was obtained for very low-mass stellar and brown dwarf candidates in Pleiades and Hyades, respectively, by Stauffer et al. (1989a, 1994) and Bryja et al. (1992, 1994)). To these, we added GJ512B, a field star. These stars are listed in Table 2. The final relationship between V − K_s and G − K_s is

$$\begin{eqnarray}(V-{K}_{{\rm{s}}}) & = & 0.27354+0.7336\times (G-{K}_{{\rm{s}}})+0.1646\\ & & \times {(G-{K}_{{\rm{s}}})}^{2}-0.000922\times {(G-{K}_{{\rm{s}}})}^{3}.\end{eqnarray} \tag{ 1 }$$

We used this to interpolate $(V-{K}_{{\rm{s}}})$ for those Praesepe stars lacking $(V-{K}_{{\rm{s}}})$, but having a Gaia measurement. For about a dozen more stars, there is no Gaia measurement but there is a URAT measurement; we used the relationship we derived for the Pleiades to obtain a $(V-{K}_{{\rm{s}}})$ estimate from the URAT data for these remaining stars.

Table 2.  Red Stars from the Pleiades and Hyades Used to Extend the Relation Between V − K_s and G − K_s

Star	V (mag)	G (mag)	Ks (mag)	Source of V
HHJ3	22.14	18.78	14.13	Stauffer et al. (1998)
HHJ5	20.71	18.45	13.93	Stauffer et al. (1998)
HHJ8	20.60	18.38	14.23	Stauffer et al. (1998)
HHJ10	20.74	18.39	13.77	Stauffer et al. (1998)
PPL1	22.08	19.49	14.31	Stauffer et al. (1989a)
PPL2	22.12	19.32	14.37	Stauffer et al. (1989a)
PPL14	21.56	19.12	14.48	Stauffer et al. (1994)
Bry804	19.27	16.83	12.37	Bryja et al. (1994)
Bry816	18.83	16.61	12.20	Bryja et al. (1994)
GJ512B	13.70	12.10	8.30	field star

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All of the stars with a measured period have a measurement or estimate of $(V-{K}_{{\rm{s}}})$; five of the non-periodic stars are missing a $(V-{K}_{{\rm{s}}})$. For $(V-{K}_{{\rm{s}}})$ < 2.5, median uncertainties (based on the scatter in the Gaia calibration above) are ∼0.017 mag; for 2.5 < $(V-{K}_{{\rm{s}}})$ > 5.5, median uncertainties are ∼0.085 mag.

In the Pleiades, we discarded stars with ${K}_{{\rm{s}}}$ ≲ 6 and ${K}_{{\rm{s}}}$ ≳ 14.5 as being too bright and faint, respectively, for the K2 LCs to be reliable. Here in Praesepe, the appropriate limits are less obvious; we have dropped the brightest (${K}_{{\rm{s}}}$ ≲ 6) and retained the rest.

There are two stars with ${K}_{{\rm{s}}}$ < 6, one of which we determined to be periodic, and both of which are listed in Appendix F. Both of them are discarded from our sample as too bright.

There are 21 stars with 6 < ${K}_{{\rm{s}}}$ < 8; ${K}_{{\rm{s}}}$ = 8 is roughly an F5 spectral type. At least 11 of them are likely pulsators (with 6 more likely pulsators that have fainter ${K}_{{\rm{s}}}$); see Appendix D. We have left these in the sample to allow for comparison to our Pleiades work (which also included likely pulsators), but have identified those pulsators where necessary in the remaining discussion.

In the Pleiades, there were very few sources with K2 LCs in the optical CMD below ${K}_{{\rm{s}}}$ = 14.5; as we will see below in Figure 4, such a cutoff is not as obvious here. There are many targets with 14.5 < ${K}_{{\rm{s}}}$ < 15.5 that have clear periods, and we have left those in the sample.

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Final star counts are as follows. From an initial sample of 984 candidate members, we find 941 members that are also not too bright. Limiting it further, there are 809 stars that we find to be periodic in these K2 data.

Figure 4 shows color–magnitude diagrams for the stars detected as periodic and not detected as periodic. The periodic stars for the most part follow the expected main-sequence relation for Praesepe. The stars we do not detect as periodic appear to have a less well-defined main-sequence relation, which would be consistent with those stars more likely to be non-members, despite satisfying the membership criteria described in Section 2.5.4 above.

In the Pleiades, we determined 92% of the sample to be periodic (Rebull et al. 2016b); here, we obtain 809/941 = 86% of the member sample to be periodic.¹⁴ If we have more non-members inadvertently included in the sample for Praesepe than we did for the Pleiades (despite very similar selection methods), a lower fraction of periodic stars might be expected. However, Praesepe is considerably older than the Pleiades, so the stars are expected to have fewer spots (hence lower amplitude signals) and rotate more slowly. Both of these factors would contribute to a lower fraction of detectably periodic members in Praesepe.

The distribution of periods we found is shown in Figure 5; note that this excludes the stars with periods that were determined to be non-members (see Section 2.5.4 and Appendix E). The stars are rotating more slowly, on average, than the analogous figure from the Pleiades (see Figure 3 in Rebull et al. 2016b or Figure 9 in Rebull et al. 2016a). Whereas the Pleiades is strongly peaked at <1 day, only ∼20% of the Praesepe stars with rotation periods rotate faster than a day, with ∼42% rotating between 1 and 10 days, and ∼37% rotating between 10 and 35 days.

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There seems to be a bimodal distribution of periods in Praesepe, with one peak near ∼1 day and another peak near ∼10 days. Assuming a Skumanich law (Skumanich 1972), ${v}_{\mathrm{rot}}\propto {t}^{-0.5}$, and that the Pleiades is 125 Myr and Praesepe is 800 Myr, we would expect the peak at ∼0.3 days from the Pleiades to become the peak at ∼1 day in Praesepe. This is the case for the M stars, which compose most of the Pleiades peak at ∼0.3 days and most of the Praesepe peak at ∼1 day.

There is a substantial number of Praesepe stars, however, with periods near 10–15 days, about 35% of the distribution. The stars contributing to this peak are for the most part more massive than those stars composing the other, faster rotating peak. Using the Skumanich law, these stars should correspond to Pleiades stars with periods of 4–6 days. Indeed, about 20% of the Pleiades stars with K2 rotation rates have periods greater than about 4 days; presumably, these go on to populate the slower peak in older clusters. However, there are fractionally more stars in the slower peak in Praesepe, which suggests that this peak includes stars with a larger range of masses than we have assumed in the rough Skumanich calculation.

Note that the 17 stars we have identified as likely pulsators are still in this sample shown in Figure 5; their periods do not make a significant difference to the histograms of P₁ (first period, e.g., the ${P}_{\mathrm{rot}}$) or P₂ (secondary period), just because so many stars are represented in these plots. However, they do make a significant difference to the histograms of P₃ and P₄ (tertiary and quaternary periods), where they are largely responsible for the periods in these histograms.

Qualitatively, the morphology of the period versus color diagram for Praesepe, and the evolution of that distribution from young ages to Praesepe age, has been well-documented in the literature. The fact that by Praesepe's age G and K dwarfs have a very narrow distribution in period at a given mass was first shown for Hyades stars in the 1980s (Duncan et al. 1984; Radick et al. 1987). That F and early G stars arrive on the main sequence with higher rotational velocities and subsequently spin down on the main sequence was first shown by Robert Kraft via spectroscopic rotational velocities for stars in a number of open clusters (Kraft 1967 and references therein). Later spectroscopic studies of those same clusters with more modern spectrographs and detectors showed that low-mass stars of all masses arrive on the main sequence with a wide range in rotational velocities and that angular momentum loss on the main sequence causes those stars to converge over time to a much narrower range in rotation at a given mass, with the convergence time being longer for lower masses (Stauffer & Hartmann 1986; Stauffer et al. 1987, 1989b). The subsequent development of wide-format CCDs made it possible to obtain rotation periods for large samples of stars in open clusters, allowing the distribution of rotation rate as a function of mass to be determined for many of Kraft's open clusters (and other clusters), largely confirming the spectroscopic results but with better precision and larger samples of stars; see Gallet & Bouvier (2015) and Coker et al. (2016) for a review of the rotation period data and theoretical models of angular momentum loss which attempt to explain the data. The LCs for open clusters and star-forming regions from K2 builds on this heritage, but also adds the benefits of photometric stability and signal-to-noise provided by space-based observations. For the open clusters observed with K2, these data for the first time allow the rotation periods of both stars in most binary systems to be determined, and for all cluster members observed, the LCs allow an assessment of the shape of the phased LC, which provides insight into the size and location of star spots on their surfaces.

Figure 6 shows the relationship between P and $(V-{K}_{{\rm{s}}})$ for Praesepe using K2 data. For comparison, Figure 6 also shows the P versus $(V-{K}_{{\rm{s}}})$ plot for the Pleiades K2 data. There have clearly been significant changes in these distributions between the ages of the Pleiades and Praesepe. As we noted above, Praesepe stars are on the whole rotating more slowly. However, the M stars in both clusters are primarily rapid rotators.

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Figure 7 has the P versus $(V-{K}_{{\rm{s}}})$ plots again for both Praesepe and the Pleiades, but with color coding to aid in this discussion. In Praesepe, there are three relatively well-defined and apparently linear sequences. At the blue end (omitting the pulsators), for $(V-K)\lt 1.3$ (≲F8), $\mathrm{log}P=1.748\,\times (V-K)-1.653$. P is changing rapidly over a very small range of $(V-{K}_{{\rm{s}}})$. There is an abrupt transition between the reddest end of this blue sequence and the bluest end of the next sequence; this is the Kraft break (Kraft 1967), where magnetic braking becomes less efficient for bluer stars. In the middle (yellow portion) of Figure 7, $1.3\leqslant (V-K)\lt 4.5$ (∼F8 to ∼M3), $\mathrm{log}P=0.138\times (V-K)+0.692$. In this regime, the points are tightly clumped around this relationship. For $4.5\leqslant (V-K)\lt 6.5$, ∼M3 to ∼M6 (which encompasses both the green and red points in Figure 7), $\mathrm{log}P\,=-1.303\times (V-K)+7.360$. Here again, P is changing rapidly with $(V-{K}_{{\rm{s}}})$. This end of the distribution contributes substantially to the bimodal nature of the P distribution (Figure 5); there are many stars with $4.5\leqslant (V-K)\lt 6.5$ and relatively few stars with $2\lt P\lt 10$. The transition between the blue and yellow regions in Figure 7 is abrupt. The transition between the yellow and green regions is less obvious; the green points, at least the ones with $P\gt 10$, could justifiably be included in the linear fit of the yellow points and the bulk of these points are already consistent with that fit. The green points, however, are also consistent with the relationship delineated by the red points (or even a slightly steeper relation). The end of the slow sequence (yellow through the green points consistent with the relationship) in Praesepe is $(V-{K}_{{\rm{s}}})$ ∼ 5.2, or M4.

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Figure 7 also includes the Pleiades (Rebull et al. 2016b) for context. The color coding in Figure 7 follows Stauffer et al. (2016), Figure 24, and is not meant to trace exactly the same populations as seen in Praesepe, but simply illustrates different sections of the diagram that we call out in the text. In the Pleiades, we discussed the "slowly rotating sequence" for 1.1 ≲ $(V-{K}_{{\rm{s}}})$ ≲ 3.7 (∼F5 to ∼K9¹⁵ , 2 ≲ P ≲ 11 days), which is the blue and yellow points together in Figure 7. The transition of blue to yellow is placed at the location of the "kink" in the slowly rotating sequence (∼K3; see Stauffer et al. 2016). The green points delineate a region in which there seems to be a "disorganized relationship" between P and $(V-{K}_{{\rm{s}}})$ between 3.7 ≲ $(V-{K}_{{\rm{s}}})$ ≲ 5.0 (∼K9 to ∼M3, 0.2 ≲ P ≲ 15 days). Finally, the red points represent the "fast sequence," with $(V-{K}_{{\rm{s}}})$ ≳ 5.0 (≳M3, 0.1 ≲ P ≲ 2 days).

Now between the Pleiades (∼120 Myr) and Praesepe (∼800 Myr), parts of this diagram see tremendous change (earlier types), and parts show more subtle changes (later types). All of the G and K (and even early M) stars (yellow in the Praesepe plot) have spun down into a well-defined, linear relationship from the curved relationship (blue and yellow in the Pleiades plot). The "disorganized region" (green in the Pleiades plot, and even to some extent the yellow) is no longer quite so disorganized by Praesepe, where many of the earlier M stars have spun down into a relationship consistent with the G and K stars. The Praesepe early-M stars are on average slow rotators, and the later M stars are on average more rapid rotators, but there is clear overlap between the rotation rates of early and late-M stars; one cannot divide the stars by mass and have them also divided by period. In the Pleiades, there is less obviously a bimodal distribution in M star periods, and (in contrast to Praesepe) the division between fast and slow rotators is also roughly a division in mass. The overall M star relationship (the red points in both panels in Figure 7) sees less obvious changes compared to the large changes for the more massive stars, but there are differences for the M stars, too. The slope of the M star relationship between period and color is overall much steeper in Praesepe than in the Pleiades; this is most easily seen from the black curved line in Figure 7, which is the relationship for Pleiades M stars derived in Stauffer et al. (2016).

In the Pleiades, some of the M stars are still contracting (still spinning up); this is not the case in Praesepe. The M3 and M4 stars in Praesepe have spun down considerably, and they have a shorter contraction time than the M5 and later stars. The distribution of M5 stars has not changed much between the Pleiades and Praesepe. In the older cluster, angular momentum loss via wind braking has had more time to counteract the contraction in the M3–4 stars, and the balance between contraction/spin up and angular momentum loss must be different in the M5 (and later) stars.

The existence of a well-defined slow sequence of late F and even early G stars (blue points in Figure 7) in Praesepe presumably points to their having at least some amount of angular momentum loss. If they had no angular momentum loss at all, there would be a scatter in rotation, reflecting a range in initial angular momentum and a range in disk lifetimes. However, presumably their angular momentum loss rate is quite small since they have so little outer convective envelope. As a speculation, it is possible that the F dwarfs in Praesepe still have rapidly rotating radiative cores, with their observed rotation periods representing a balance between the angular momentum feeding up from below with the angular momentum lost from their winds. The G/K/M stars have much larger angular momentum loss rates, and have had time to spin down their cores. So, the two sequences reflect that dichotomy: core/envelope still decoupled for the F dwarfs, core/envelope coupled for the G, K, and early Ms.

The bimodal period distribution of M stars in Praesepe (but not the Pleiades) is interesting because field M stars are found to have a bimodal P distribution in the Kepler field (e.g., McQuillan et al. 2013; Davenport 2016) and in nearby M field stars (e.g., Kado-Fong et al. 2016; Newton et al. 2016). These field stars are older, on average, than Praesepe, and the locations of the two peaks are slower (∼19 and ∼33 days) compared to Praesepe with ∼1 and ∼17 days for the ∼500 Praesepe stars with $(V-{K}_{{\rm{s}}})$ ≥ 3.79, the color corresponding to M0. Assuming Skumanich evolution, these two peaks cannot both evolve together in lockstep. However, it is interesting that both distributions are bimodal. It may be that the mechanism that causes some M stars to rapidly spin down (e.g., Brown 2014; Newton et al. 2016) has started to operate in some of the Praesepe stars. We note, of course, that our data cannot constrain the behavior of M stars later than about M5 or M6.

Although the relationships delineated by the majority of the Praesepe stars in Figures 6 and 7 are striking, it is worth looking at some of the outliers in this distribution. Notes on specific stars appear in Appendix G, as well as optical color–magnitude and P versus $(V-{K}_{{\rm{s}}})$ diagrams with these objects highlighted.

The G and K stars in young open clusters like the Pleiades have bimodal rotational velocity distributions, with a majority of stars on a slowly rotating branch and a minority in a rapidly rotating branch. The latter stars are generally believed to descend from pre-main-sequence stars that lost their circumstellar disks (and hence their ability to rapidly shed angular momentum) very early. Our P versus color plot also shows several rapidly rotating G/K stars. Are these the descendants of the rapidly rotating G/K stars in the Pleiades? For most or all of these stars, we believe not. Instead, they are best explained as tidally locked short-period binaries. Most of them are known short-period binaries; many of the remaining ones have little to no spectroscopic information, but tidally locked binaries are a logical explanation for stars that are rotating too quickly in comparison to other stars of similar colors.

The stars with periods much longer than average for their $(V-{K}_{{\rm{s}}})$ color are also curious. For a few of them, tidal synchronization (or pseudo-synchronization) in a short-period binary may be the explanation, as has been advocated for similar outliers in M35 (Meibom et al. 2006) and other clusters. Because of the steep dependence of the synchronization time on the binary–star separation, for the longer period outliers ($P\gt \sim 20\,\mathrm{days}$), this becomes a less viable possibility. If these stars are all Praesepe members, and if we are correct that all of them are rotation periods, then something about these stars made them spin down more than most other stars in the cluster. For these longest-period stars, interpretation of their LCs as rotation periods is more fraught than other shorter-period stars, as there is more variation from cycle to cycle, and it becomes more of a judgement call as to whether or not the star has a ${P}_{\mathrm{rot}}$, or just a repeated pattern that may or may not be tied to rotation. Slowly rotating stars may undergo more significant spot evolution, so this may be a real astrophysical effect. Having fewer complete cycles within the K2 campaign, coupled with significant changes every cycle, makes it hard to assess. Many of the long-P G and K stars from the bulk of the distribution have more obviously sinusoidal LCs, and many of the longest P M stars (ones that are still with the bulk of the distribution) share LC characteristics with these long-P outlier G and K stars; there is a continuum of LC properties such that it is not always easy to draw a line between ${P}_{\mathrm{rot}}$ and just a repeated pattern.

Specifically because of this ambiguity, we obtained follow-up Keck/HIRES spectroscopy of many of these very slowly rotating stars (see Appendix I). Of the ones we observed, about half of them have RVs inconsistent with cluster membership. We identified these as non-members (Section 2.5.4 and Appendix E). Four of the longest P objects (for which we have yet to obtain spectra) appear just below the single-star main sequence, suggesting that they may also be non-members; given the uncertainties in $(V-{K}_{{\rm{s}}})$, we have provisionally left them in the list of members (see Appendix G).

In Rebull et al. (2016a), we presented a set of empirical structures in the K2 Pleiades LCs and periodograms, finding via visual inspection 11 different categories. These categories are discussed in detail in Rebull et al. (2016a), and we do not repeat that discussion here. All but one of the categories of LCs can be found in Praesepe as well; see Table 3 for a list. Many of the categories have more than one significant period. Figure 8 shows some examples of these classes in Praesepe, two of which have more than one measured P; these examples span a range of brightnesses, periods, and categories. (For examples of each of the categories in the Pleiades, see Rebull et al. 2016b.)

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Before we can compare the total counts of objects in the various periodogram categories, we need to be sure that we are sampling the same range of masses in the two clusters. Figure 9 shows the sample fraction for the periodic member samples as a function of $(V-{K}_{{\rm{s}}})$ as a proxy for mass. The two samples are comparable over most of the range of $(V-{K}_{{\rm{s}}})$; there are sample completeness effects at the reddest and bluest bins, which are the most poorly populated. The most notable differences are at the bluest end ($(V-{K}_{{\rm{s}}})$ ≲ 1), which will affect primarily stars whose measured K2 periods are most likely to be pulsation rather than rotation.

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Now with the knowledge that the samples are of comparable mass ranges, we can compare sample fractions of LC/periodogram categories. Table 3 summarizes the total counts of objects in each LC/periodogram category for Praesepe and the Pleiades (in both cases, only cluster members are included). There are some significant differences between the clusters. Overall, the fraction of stars that are singly periodic and multiply periodic are roughly comparable in the two clusters (∼75%–80% and ∼20%–25%, respectively). However, there is a slightly higher fraction of singly periodic sources (and a slightly lower fraction of multi-periodic sources) in Praesepe.

In the context of this work, we identified an inconsistency in the way that we were identifying close and distant resolved peaks in our earlier Pleiades work. As stated there, we calculated ${\rm{\Delta }}P/{P}_{1}$ (see also Section 6 below) and used that value to identify close (${\rm{\Delta }}P/{P}_{1}\lt 0.45$) and distant (${\rm{\Delta }}P/{P}_{1}\gt 0.45$) peaks. However, we went on to say (without explanation) that some objects could be identified as both close and distant peaks. We now more consistently identify an object as both close and distant peaks if the difference in any two periods divided by the ${P}_{\mathrm{rot}}$ is <0.45 for two of at least three periods, and >0.45 for (a different) two of at least three periods (unless the star is already identified as a pulsator, in which case it is not identified as either close or distant peaks). As a result, three objects from the Pleiades that were already identified as resolved close peaks should also have been tagged as resolved distant peaks (EPIC 210877423, 211112974, and 211128979). This has been corrected in the statistics in Table 3.

Among the LC/periodogram categories, there is a higher fraction of moving double-dip and shape changers in Praesepe than there are in the Pleiades; there are more than twice the fraction of shape changers as in the Pleiades (∼40% versus ∼15%). In Rebull et al. (2016a) and Stauffer et al. (2016), we suggested that the shape changers and moving double dips are due to latitudinal differential rotation and/or spot/spot group evolution. These LC types were found primarily in the slower rotators in the Pleiades; with more slower rotators in Praesepe, it may not be surprising that we have more shape changers, because this may reflect a real difference in the incidence rate of differential rotation.

Because there is a slightly lower fraction of multi-period sources in Praesepe, there is a lower fraction of nearly all the multi-period subcategories in Praesepe. However, the most discrepant fractions are found in the resolved close peaks category, where the fraction is half what it was in the Pleiades. We postulated in Rebull et al. (2016a) and Stauffer et al. (2016) that the resolved peaks categories could be a result of latitudinal differential rotation and/or spot/spot group evolution among the G and K stars.

The shape changers, moving double dips, and close resolved peaks (in the G and K stars) may all be subject to an observational bias in the following sense. We noted in Rebull et al. (2016a) that, particularly for the shape changers/moving double dips, it was possible that if there had been much more data, ≫70 days, encompassing more cycles, then it might have been possible to differentiate the two (or more?) periods contributing to the changing shape. In Praesepe, since the periods are on average longer than in the Pleiades, consequently there are (on average) fewer complete cycles encompassed in the K2 campaign. Thus, perhaps the higher occurrence rate of shape changers/moving double dips and the lower occurrence rate of resolved close peaks (for G and K stars) in Praesepe may be attributable to the number of complete cycles available for stars in Praesepe (as compared to the Pleiades).

For the M stars in the Pleiades, we postulated that the close resolved peaks were binaries, since most of them appeared above the single-star main sequence. The distant resolved peaks we thought were most likely to be binaries. The same basic result is true in Praesepe as well—the M stars with resolved peaks are above the single-star main sequence (see Section 5.3 below). From theory, we expect stronger differential rotation in earlier stars (e.g., Kitchatinov & Olemskoy 2012) and effectively solid-body rotation in the M stars. Praesepe has a comparable if not slightly greater fraction of resolved distant peaks, though there are many more instances where the difference between the peaks is very large indeed (>6 days; see Section 6 below).

The phased LCs of most spotted stars are simple, showing a morphology that is more or less sinusoidal or one that has one or two broad "humps" or dips. That is as expected, because spots located at most positions on a stellar surface and when viewed from most vantage points will be visible for half or more of the rotation period. That means that most spots will have a contribution to the phased LC that spans 180° or more in phase. Flux dips or other structures that cover less than 90° in phase are very hard to produce with spots. The only way they can be produced is by placing the spot very near the upper or lower limb of the star. Because of limb darkening and geometric foreshortening, light from such a location contributes little to the total integrated brightness of the star; therefore, flux dips from such spots cannot yield LC features that are very deep.

In the Pleiades (Rebull et al. 2016a), we nevertheless identified six stars with short-duration flux dips (full width at zero intensity, FWZI, <0.2 in phase); one star had three such dips, another had two, and the remaining four had just one flux dip. The dips were to first order constant in shape and depth over the duration of the K2 campaign. All six stars were mid-M dwarfs with very short ($P\lt 0.7$ days) periods. We identified another 19 rapidly rotating, mid-to-late-M dwarfs in the 8 Myr old Upper Sco association whose K2 phased LCs seemed to show more structure than could be explained by spots (Stauffer et al. 2017). We have attributed the LC structure to warm clouds of coronal gas orbiting the stars near their Keplerian co-rotation radius, following models by Jardine & van Ballegooijen (2005) and Townsend & Owocki (2005).

We have found no Praesepe stars with the same properties (short-duration flux dips or other structure, rapid rotation, mid-to-late-M dwarf spectral type). However, there are three slowly rotating Praesepe M dwarfs ($(V-{K}_{{\rm{s}}})$ ∼ 5) that do have more structured LCs than we expect; these are shown in Figure 10: 212011416/2MASSJ08330845+2026372, 211915940/JS208, and 211931651/AD3196 = CP Cnc. One (EPIC 212011416/2MASSJ08330845+2026372) has a nearby saturated column in the K2 data, but multiple LC versions obtain the same shape and (large!) depth of the features. Note that in the first two cases, the highest peak in the periodogram results in a phased LC that has more scatter than when phased at three times the peak value; this is similar to the double dips (see discussion in Rebull et al. 2016a). These stars have periods ≳10× longer than the stars in Pleiades or USco.

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We do not believe the physical mechanism producing these LC morphologies is the same as for the Pleiades and Upper Sco. Because the flux dips are broader than in the Pleiades and are generally low amplitude, it is (barely) possible that spots could be responsible. However, the full amplitude of one of them (EPIC 212011416) is about 4%, which would be very difficult to produce using spots located near the limb of the visible hemisphere.

In order to help constrain the nature of these stars, we obtained single-epoch spectroscopy of these sources with Keck/HIRES (see Appendix I). All are consistent with being RV members of Praesepe and all are narrow-lined (and single-lined).

The phased LC morphologies for these three stars are much more complex than most of the rest of the LCs in Praesepe. Because they represent a tiny minority of the Praesepe sample, we do not believe that our failure to understand their properties should affect any of the other conclusions in the paper. We plan a future paper (Hebb et al.) that will discuss modeling of these LCs.

Kovács et al. (2014) also identified LC/power spectrum classes in their study of Praesepe. They had (a) monoperiodic sinusoids (13% of their periodic sample); (b) single but unstable sinusoid (39%); (c) two peaks interpreted as latitudinal differential rotation (7%); (d) power moved into first harmonic (41%).

(a) The monoperiodic sinusoids of their periodic sample can be matched to those in our sample with single periods but that are neither double dip nor shape changer. There are 366 of those, or ∼45% of the periodic sources, in our member sample. (b) Single but unstable sinusoids would be analogous to our shape changers, without the double dips; we find 164 of them, 20% of the periodic sources. (c) They rarely find two discrete peaks in the power spectrum; this would be analogous to our resolved close and distant peaks combined, and we have 15% of our periodic sample falling into this category. We agree that in some cases at least, this could be attributable to latitudinal differential rotation, but we suspect (also see discussion in Rebull et al. 2016a) that at least some (most of the multi-peaked M stars) are binaries. (d) Lastly, their category where power is moved into the first harmonic is analogous to our double-dip category, which is 20% of our periodic sample.

Perhaps a more fair comparison is to work just with the 152 stars that are in common between the two studies. For that sample, we obtain (a) monoperiodic sinusoids: 6%; (b) single unstable sinusoids: 23%; (c) resolved peaks: 16%; and (d) double dips: 47%. The closest match in terms of sample fraction is this last category.

Assuming that we have correctly captured the relationship between our classes and those in Kovács et al. (2014), we have roughly similar sample fractions for the same characteristics.

We calculated the amplitude of the periodic signal in the same fashion as we did in the Pleiades; we assembled the distribution of all points in the LC, took the log of the 90th percentile flux, subtracted from that the log of the 10th percentile flux, and multiplied by 2.5 to convert to magnitudes. Figure 11 plots that amplitude against both P and $(V-{K}_{{\rm{s}}})$ for the periodic LCs. Note that this is not necessarily the amplitude of a sinusoid overlaid or fit to the periodic signal, but the amplitude of the overall LC, which necessarily includes long-term trends.

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In the Pleiades, there was no obvious trend of amplitude with color or period. However, in Praesepe, there is a trend in both of these panels. On the left of Figure 11, longer period stars have lower amplitudes. This is consistent with expectations in that, certainly by this age, more slowly rotating stars should be less active and therefore have smaller spots. Note, however, that nearly all of the turn-down at the longest periods are higher-mass G and K stars; likewise, the shorter P stars are primarily mid- to late-M stars. The bimodal distribution of periods seen in Figure 5 is apparent here, too. There are more multi-period low-amplitude sources than single-period low-amplitude sources (though several of these are likely pulsators, which explains the low amplitude in those bluest and smallest P cases). On the whole, however, there is no clear distinction in the left panel of Figure 11 between the distributions of singly and multi-periodic sources.

On the right of Figure 11, it can be seen that bluer stars (corresponding to $(V-{K}_{{\rm{s}}})$ < 1.3, blue points in Figure 7) are generally multi-periodic and have smaller amplitudes, not all of which are likely pulsators. There are several lower amplitude LCs with colors redder than $(V-{K}_{{\rm{s}}})$ ∼ 1.3. This, and the trend toward larger amplitudes at even redder colors ($(V-{K}_{{\rm{s}}})$ > 4.5), could be an observational bias in that stars need to be "bright enough" for a period to be derivable; fainter stars need larger amplitudes to be seen as periodic, and brighter stars with smaller amplitudes are more easily detected. Note, however, that there is substructure within the right panel of Figure 11—there is a "clumping" of the distribution for 1.3 ≤ $(V-{K}_{{\rm{s}}})$ < 4.5 (corresponding to the yellow points in Figure 7) that moves to lower amplitude as the color gets redder. Then, it turns around and moves to larger amplitude as the color gets redder for $(V-{K}_{{\rm{s}}})$ > 4.5 (the green/red points in Figure 7). For both of these color regimes, the amplitude gets smaller as the period gets longer. This makes sense in the standard rotation-activity sense, if the spot filling factor (at least for the non-axisymmetrically distributed component) decreases for longer periods.

In the Pleiades, we found a strong correlation between multiple periodicities and $(V-{K}_{{\rm{s}}})$—most of the earlier stars were multiply periodic and nearly all the later stars were singly periodic. The distribution in Praesepe is different than in the Pleiades; see Figure 12. Through most of the sample, the fraction that has multiple periods is roughly constant with color. Clearly the bins for $(V-{K}_{{\rm{s}}})$ ≲ 1 and $(V-{K}_{{\rm{s}}})$ ≳ 6.5 are significantly affected by sample completeness.

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There is a transition between where multiple periods dominate to where single periods dominate, and it is at $(V-{K}_{{\rm{s}}})$ ∼ 1.5 (∼G0). The Pleiades sample extends to bluer colors, and the transition between where multiple periods dominate to where single periods dominate is at ${(V-{K}_{{\rm{s}}})}_{0}$ ∼ 2.6 (early K).

Inclusion of the likely pulsators in this analysis affects the results, particularly because of the sample differences for $(V-{K}_{{\rm{s}}})$ ≲ 1 (see Figure 9). Dropping the likely pulsators causes the sample fraction of multi-period sources to plummet blueward of $(V-{K}_{{\rm{s}}})$ ∼ 1. However, the multi-period sources still dominate for 1 ≲ $(V-{K}_{{\rm{s}}})$ ≲ 1.5.

If latitudinal differential rotation dominates blueward of this $(V-{K}_{{\rm{s}}})$ ∼ 1.5 transition point (which was one of our hypotheses in the Pleiades for the transition point there), then this transition has moved to more massive types by the age of Praesepe. This transition is also essentially where the bluest "branch" turns down in the P versus $(V-{K}_{{\rm{s}}})$ diagram in Figure 6 above, again suggesting that something physically different happens blueward of this color. Stronger differential rotation is expected for hotter stars (e.g., Kitchatinov & Olemskoy 2012), so this may be the dominant effect. However, identification of a star as multi-periodic at all may be limited by the number of complete cycles in the K2 campaign, as discussed in Section 4.2.

Figure 13 shows where the multi-period stars fall in the P versus $(V-{K}_{{\rm{s}}})$ and ${K}_{{\rm{s}}}$ versus $(V-{K}_{{\rm{s}}})$ parameter spaces. This figure is in direct analogy to Rebull et al. (2016a), their Figure 12; in the Pleiades, most of the early-type stars were multi-periodic, and the later types were nearly all singly periodic, and the multi-periodic later type stars were nearly all photometric binaries.

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In Praesepe, the single-period stars are distributed more uniformly throughout the diagram, consistent with Figure 12. The earlier-type stars are dominantly multiply periodic for $(V-{K}_{{\rm{s}}})$ ≲ 1.5–2, which is a smaller range than for the Pleiades, again consistent with Figure 12. It is still true that most of the multi-period stars with $(V-{K}_{{\rm{s}}})$ ≳ 3.5–4 seem to be dominated by photometric binaries, given the position in the ${K}_{{\rm{s}}}$ versus $(V-{K}_{{\rm{s}}})$ diagram. This is consistent with the M stars still rotating as solid bodies at Praesepe's age.

The panels in Figure 14 break down where the individual LC classes appear in the ${K}_{{\rm{s}}}$ and P versus $(V-{K}_{{\rm{s}}})$ diagrams; the analogous Pleiades figures in Rebull et al. (2016a) are Figures 13 and 14.

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Beaters and complex peaks were both, in the Pleiades, found to dominate the stars with $(V-{K}_{{\rm{s}}})$ < 3.7. In Praesepe, the region where they dominate has shifted blueward, to $(V-{K}_{{\rm{s}}})$ ≲ 2.5–3, consistent with the discussion associated with Figure 12 above. In both cases, beaters that are also M stars are more likely to be binaries than single stars, based on position in the CMD. Shape changers dominated in the Pleiades for 1.1 < $(V-{K}_{{\rm{s}}})$ < 3.7, with a significant fraction of the stars in the "disorganized region" (with 3.7 < $(V-{K}_{{\rm{s}}})$ < 5) being in this category. In Praesepe, there are a higher fraction of shape changers, and they dominate the "middle branch" (yellow points in Figure 7) with 1.3 ≤ $(V-{K}_{{\rm{s}}})$ < 4.5. If the shape changers reflect rapid spot evolution and/or differential rotation, this happens more frequently in Praesepe than it does in the Pleiades. Note that the stars that exhibit shape-changing behavior in Praesepe are on average rotating much more slowly than in the Pleiades. The longer periods on the slowly rotating branch for 1.3 < $(V-{K}_{{\rm{s}}})$ < 4.5 for Praesepe relative to Pleiades combined with the fixed campaign length for K2 makes it likely that stars will be moved from the resolved close peak category into the shape-changer or moving double-dip category, explaining at least some of the differences in LC class distributions we see.

The LCs with two distinct periods can be found throughout the P versus $(V-{K}_{{\rm{s}}})$ diagram for both clusters, and for M stars, tend to be on the brighter side of the cluster distribution on the CMD. For the M stars in particular, these are more likely to be binaries. Resolved distant peaks are distributed throughout the P versus $(V-{K}_{{\rm{s}}})$ diagram, but resolved close peaks tend to cluster in the bluest portion of the P versus $(V-{K}_{{\rm{s}}})$ diagram (and to some extent in the M stars' fast-rotating clump). Resolved close peaks and bluer colors may be differential rotation and/or spot evolution. (See Section 6 below.)

Finally, moving double-dip stars dominate the "middle branch" (yellow points in Figure 7) of the Praesepe P versus $(V-{K}_{{\rm{s}}})$ diagram to a much larger extent than in the Pleiades. There are fractionally many more moving double-dip stars in Praesepe. There are fractionally far fewer M stars exhibiting any double-dip behavior in Praesepe than in the Pleiades.

There are 12 objects that we identified in the process of LC inspection as candidate EBs (or planet candidates) in this data set. When we report a period here for these objects, we believe that the P we report and use in our analysis is a rotation period. All of the EBs are listed in Table 4.

We note explicitly here that one object is a known W Uma-type EB according to Whelan et al. (1973): EPIC 211918335/KW244 = TX Cnc. Kovács et al. (2014) also report this as an EB. For this LC, we do not include our 0.191446 day period in our analysis of rotation periods.

Mermilliod et al. (1999, 2009) includes stars in Praesepe that were monitored for spectral binarity (RV variations) over ∼20 years. Kovács et al. (2014) also report photometric periods they attribute to binary components. Barrado et al. (1998) identified spectral binaries from single-epoch observations. Table 5 includes a comparison of the periods derived here to those derived for the binaries in these papers. Many of the periods given in the literature are ≫35 days, so there is no way that we could have obtained that period from these data. For cases where the reported literature periods are both close (and ≲35 days), more often than not, we do not recover both periods. It could be that one of the two stars does not have spots, or organized enough spots, to create a detectable signal.

Table 5.  Binaries in the Literature

EPIC	Other Name	Coord	Period(s)	Period(s)a	Notes
		(J2000)	(here, day)	(literature, day)
Kovács et al. (2014)
211935509c	KW539 = S4	083648.96+191526.4	7.475	3.72481, 5551	HAT-269-0001402; our ${P}_{\mathrm{rot}}$ is twice theirs, and we believe ours to be correct
211977390b	KW55 = S27	083749.95+195328.8	6.789	7.13827, 0.838434	HAT-269-0001352; our peak is broad and the waveform has changes over the campaign; both 6.8 and 7.1 may be consistent; no peak at 0.8 days; also fast outlier in P versus $(V-{K}_{{\rm{s}}})$ diagram
211949097c	BD+19d2061	083924.97+192733.6	3.940, 4.261	4.14405, 1.41563	HAT-269-0000761; our peak is broad and both of our periods are probably consistent with theirs; no peak at 1.4 days
211950081c	KW184	083928.58+192825.1	10.175	10.4866, 4.52284	HAT-269-0001490; well-matched ${P}_{\mathrm{rot}};$ no peak at 4.5 days
211918335d	KW244 = TXCnc	084001.70+185959.4	(0.1914)	0.1915	HAT-269-000058; WUma EB (Whelan et al. 1973); also listed in EBs table
211933215c	S137	084041.90+191325.4	6.112	3.11867, 897	HAT-269-0000850; our peak is correct for our data
211972627c	KW368 = S155	084110.30+194907.0	9.054	9.00657, 8.50340	HAT-269-0001570; our waveform undergoes many changes over the campaign, but we do not have two distinct peaks at 9.0 and 8.5 days
211935518c	KW434 = S184	084154.36+191526.7	4.184	4.13291, 0.951565	HAT-269-0001333; well-matched to ${P}_{\mathrm{rot}};$ we do not find a 0.95 day peak; also fast outlier in P versus $(V-{K}_{{\rm{s}}})$ diagram
211947631	BD+19d2087	084305.93+192615.2	4.736	4.65095,7.71605	HAT-269-0000465; well-matched ${P}_{\mathrm{rot}};$ no peak at 7.7 days
211969494c	S213	084320.19+194608.5	6.382	6.18544, 648	HAT-269-0000913; well-matched ${P}_{\mathrm{rot}}$
211896596	JS655 = FVCnc	084801.74+184037.6	2.976	2.92535, 1.01926	HAT-318-0000612; well-matched ${P}_{\mathrm{rot}};$ we do not find a 1.02 day peak; also appears Mermilliod et al. (1990) with P = 2.98 days; also fast outlier in P versus $(V-{K}_{{\rm{s}}})$ diagram
Mermilliod et al. (2009)
211958646	HD73081	083702.02+193617.2	4.413, 2.093	45.97	Longer than we can detect
211971690	S12	083711.48+194813.2	8.242	5.86628	Our waveform undergoes many changes over the campaign, but no peak is apparent at ∼6 days
211929178	KW34 = BD+19d2050	083728.19+190944.3	0.840, 0.819	7383	Longer than we can detect
211959522	S16	083727.54+193703.1	8.873	144	Longer than we can detect
211955820	BD+20d2130	083727.94+193345.1	3.618, 6.605, 2.572, 2.694	47.44	Longer than we can detect
211947686	S25	083746.61+192618.0	5.933	458	Longer than we can detect
211936163	HD73210	083746.76+191601.9	0.082	35.90	We should have seen a ∼36 day period, but none are apparent
211977390b	KW55 = S27	083749.95+195328.8	6.789	1268	Longer than we can detect
211983461	KW58 = S29	083752.09+195913.9	7.933	5567	Longer than we can detect
211942703	S38	083814.26+192155.4	7.225, 9.554, 6.369	117	Longer than we can detect
211990866	KW100 = S42	083824.29+200621.8	4.255, 4.604, 3.971	4.92946	Comparable to measured periods, but it is unclear whether these are ${P}_{\mathrm{rot}}$ or ${P}_{\mathrm{orb}}$.
Mermilliod et al. (1999)
212034371	JS102 6	083556.94+204934.7	1.21051	...	Listed as a binary; also fast outlier in P versus $(V-{K}_{{\rm{s}}})$ diagram
212001830	S11	083711.66+201704.9	...	1149.5	${P}_{\mathrm{orb}}$ very much longer than we can detect. Noisy and not detected as periodic
211927313	KW47 = BD+19d2052	083742.36+190801.5	3.077, 3.303, 3.626	34.619	${P}_{\mathrm{orb}}$ comparable to our max P but this is not detected
211988454	KW127 = S51	083850.00+200403.4	5.909, 8.514	13.2803	The waveform is very complicated, with more than one period and dense power spectra; 13.3 days is not recovered, but could be embedded in the signal
211898181	KW547	084037.88+184200.4	2.524	...	Listed as SB1; also fast outlier in P versus $(V-{K}_{{\rm{s}}})$ diagram
211975006d	KW367 = BD+19d2077	084109.60+195118.6	3.0675	1659.	Sinusoidal signal at 3.0675 day very strong; also fast outlier in P versus $(V-{K}_{{\rm{s}}})$ diagram
211912407	KW439 = HD73994	084157.81+185442.2	2.0297, 2.102	457.8	Longer than we can detect; noisy
211989010	BD+20d2198	084407.33+200436.9	3.682, 7.03816	40.6649	Longer than we can detect but no hint of structure on ∼40 days
Barrado et al. (1998)
211958646	HD73081	083702.02+193617.2	4.4134, 2.093	...	Listed as SB
211991571	KW146 = HD73429	083905.23+200701.8	0.7856	...	Listed as SB
211956096	KW236	083959.82+193400.2	11.509	...	Listed as SB
212003469	KW297	084028.63+201844.8	8.1555	...	Listed as SB
211909748	KW401	084130.70+185218.7	2.422	...	Listed as SB; also fast outlier in P versus $(V-{K}_{{\rm{s}}})$ diagram
211956984	BD+20d2193	084244.41+193447.8	0.4794, 0.9488, 0.9937	...	Listed as SB
211947631	BD+19d2087	084305.93+192615.2	4.7357	...	Listed as SB

Notes.

^aThe period given from Mermilliod et al. (2009) is P_orb. The two periods given from Kovács et al. (2014) are in the order P_rot, P_orb. ^bThe same star appears explicitly in Kovács et al. (2014) and Mermilliod et al. (2009), as well as Mermilliod et al. (1999) and Barrado et al. (1998). ^cAlso appears in Mermilliod et al. (1999). ^dAlso appears in Barrado et al. (1998).

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Table 7.  Likely Pulsators

EPIC	Other Name	Coord. (J2000)	$(V-{K}_{{\rm{s}}})$ (mag)	Periods (days)	Notes
211953002	BR Cnc = HD 73175	083740.70+193106.3	0.588	0.0872, 0.1219, 0.0965	F0; δ Scuti (e.g., Breger 1970)
211983602	CY Cnc = HD 73345	083837.86+195923.1	0.530	0.051, 0.058, 0.053, 0.067	F0V; δ Scuti (e.g., Hauck 1971)
211951863	HD 73397	083846.95+193003.3	0.779	0.5125, 0.5950, 0.6150, 0.7568	F4
211957791	BS Cnc = HD 73450	083909.09+193532.7	0.625	0.0587, 0.0644	A9; δ Scuti (e.g., Breger 1970; Breger et al. 2012)
211994121	HD 73616	083958.37+200929.6	0.884	0.8876, 0.2713, 0.5718, 0.6137	F2
211984704	39 Cnc = HD 73665	084006.41+200028.0	2.219	1.2411	G8III (dropped as too bright)
211941583	HD 73712	084020.13+192056.4	0.735	0.1413, 0.4815, 0.1179, 0.1395	Spectroscopic binary; A9V
211931309	BV Cnc = HD 73746	084032.96+191139.5	0.686	0.0608, 0.0637, 0.0494, 0.0624	F0; δ Scuti (e.g., Hauck 1971)
211973314	HD 73854	084110.67+194946.5	0.850	0.7522, 0.6970, 0.7937, 0.7409	F5
211979345	HD 73872	084113.76+195519.1	0.556	0.0644, 0.0437, 0.0771, 0.0649	A5; δ Scuti (e.g., Breger et al. 2012)
211935741	HI Cnc = HD 73890	084118.40+191539.4	0.627	0.070, 0.078, 0.089, 0.059	A7V; δ Scuti (e.g., Hauck 1971)
211945791	BX Cnc = HD 74028	084206.49+192440.4	0.529	0.058, 0.126, 0.136, 0.055	A7V; δ Scuti (e.g., Breger 1970)
211914004	BY Cnc = HD 74050	084210.80+185603.7	0.557	0.055, 0.053, 0.047, 0.052	A7V; δ Scuti (e.g., Breger 1970)
211954593	BD+20d2192	084240.71+193235.4	1.081	0.4142, 0.6202, 1.2397, 1.2719	F2III
211956984	BD+20d2193	084244.41+193447.8	1.089	0.4794, 0.9488, 0.9937	F6
212033939	HD 74135	084253.07+204909.1	0.777	0.4757, 0.3128, 0.4135, 0.2734	A9III
211909987	HD 74589	084520.53+185231.3	0.724	0.0434, 0.0486, 0.0552, 0.0535	F0
212008515	HD 74587	084528.25+202343.5	0.767	0.0587, 0.0513, 0.0519, 0.0801	A5; δ Scuti in SIMBAD and Kovács et al. (2014) but unclear if identified as δ Scuti prior to Kovács et al. (2014)

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