K2 rotation periods for low-mass Hyads and a quantitative comparison of the distribution of slow rotators in the Hyades and Praesepe

We analyze K2 light curves for 132 low-mass ($1\ \gtrsim\ M_*\ \gtrsim\ 0.1$~${M_{\odot}}$) members of the 600--800~Myr-old Hyades cluster and measure rotation periods ($P_{rot}$) for 116 of these stars. These include 93 stars with no prior $P_{rot}$ measurement; the total number of Hyads with known $P_{rot}$ is now 232. We then combine literature binary data with Gaia DR2 photometry and astrometry to select single star sequences in the Hyades and its roughly coeval Praesepe open cluster, and derive a new reddening value of $A_V = 0.035$$\pm$$0.011$ for Praesepe. Comparing the effective temperature--$P_{rot}$ distributions for the Hyades and Praesepe, we find that solar-type Hyads rotate, on average, 0.4~d slower than their Praesepe counterparts. This $P_{rot}$ difference indicates that the Hyades is slightly older than Praesepe: we apply a new gyrochronology model tuned with Praesepe and the Sun, and find an age difference between the two clusters of 57~Myr. However, this $P_{rot}$ difference decreases and eventually disappears for lower-mass stars. This provides further evidence for stalling in the rotational evolution of these stars, and highlights the need for more detailed analysis of angular-momentum evolution for stars of different masses and ages.


INTRODUCTION
The Hyades and Praesepe open clusters are benchmarks for determining the dependence of stellar rotation on age. The Hyades was one of the first open clusters for which photometric rotation periods (P rot ) were measured for low-mass stars ( < ∼ 1 M ⊙ ; Radick et al. 1987Radick et al. , 1995. The two clusters are sufficiently nearby such that many photometric P rot across the full FGKM mass range have now been measured for both from ground-and space-based photometric monitoring (e.g., Agüeros et al. 2011;Delorme et al. 2011;Hartman et al. 2011;Douglas et al. 2014Douglas et al. , 2016Douglas et al. , 2017Rebull et al. 2017).
Empirical efforts to establish the functional form of the rotation-age relation, sometimes referred to as gyrochronology (Barnes 2003), have commonly assumed that the mass dependence can be separated from the age dependence, such that P rot (M ⋆ , t) = f (M ⋆ ) × g(t). This was famously proposed by Skumanich (1972), who found that solar-type stars spin down as P rot ∝ t n , where the braking index n ≈ 0.5. Barnes (2003Barnes ( , 2007 accounted for the dependence on mass by adopting photometric color as its observational proxy, then fit coefficients for a simple analytic function from observations of rotators with a range of masses in young nearby clusters. The resulting model implied that lower-mass stars spin down more rapidly than their solar-type counterparts. Later authors (e.g., Mamajek & Hillenbrand 2008;Meibom et al. 2009;Angus et al. 2015) have adjusted the coefficients and braking index, but otherwise have assumed the same functional form as Barnes. However, an examination of the figures in Barnes (2003) shows that this fixed relation between P rot , t, and color is insufficient to describe stellar spin-down for stars with a range of masses.
More recent P rot measurements for G and K dwarfs in open clusters have shown that P r ot evolution cannot be described by separating the mass and age dependence. Using the Skumanich relation, Meibom et al. (2011b) tested whether Hyades rotators could be spun up to match the observed distribution of P rot in M34, which is 220 Myr old. These authors determined that while the distribution of spun-up solar-type Hyads did match that of their younger cousins, spinning up Hyades K dwarfs by the same factor resulted in these stars having faster P rot than those observed in M34. Comparing P rot measured for GKM stars in various open clusters from 100 Myr to 1 Gyr leads to a similar conclusion: Skumanich-like spin-down works well for solar-type stars, but K dwarfs spin down more slowly (Meibom et al. 2011b;Cargile et al. 2014;Agüeros et al. 2018).
Furthermore, while re-tuning the coefficients for the Barnes (2007) gyrochronology equation, Angus et al. (2015) could not simultaneously fit Praesepe and the Hyades. When including Praesepe, these authors' fit resulted in a multi-modal distribution for their color singularity term, which controls the downturn toward rapid rotation for bluer/hotter/more massive stars (which have thinner convective envelopes, resulting in relatively weaker magnetic dynamos and braking efficiency). This is additional evidence that the shape of the slow-rotator sequence can vary from cluster to cluster.
A complication in using the Hyades and Praesepe for calibrating gyrochronology is that their absolute and relative ages, usually determined from isochrones, are still debated (see Table 1 for examples of ages derived for the two clusters). Most authors agree that the clusters are either coeval, or that the Hyades is slightly older, and that their ages range from ≈600 to ≈800 Myr. But the disagreements among these ages do not provide much hope that we can successfully calibrate gyrochronology using isochronal cluster ages. And it creates confusion for gyrochronology studies: some authors separate the two clusters when comparing data to theoretical models (Brown 2014;Matt et al. 2015;Garraffo et al. 2018), while others combine them (Reiners & Mohanty 2012;Angus et al. 2015).
Our goal is to compare the shapes of the slow-rotator sequences in the Hyades and Praesepe and to determine whether they can be combined into a single benchmark sample for gyrochronology. Delorme et al. (2011) carried out a similar anaysis. These authors compared P rot distributions in the Hyades, Praesepe, and Coma Berenices (thought to be of similar age). Using a simple linear fit to the color-period relation, they found the Hyades to be ≈50 Myr older than the other two clusters. However, Delorme et al. (2011) did not have access to Gaia data for membership, nor did they have the wealth of new P rot measurements enabled by K2 's observations of the Hyades and Praesepe. We use up-  David & Hillenbrand (2015) fit two different isochrone models; we give both results from their summed PDF analysis in log age space. b Cummings et al. (2018) fit two different isochrone models; we list both results. c Gossage et al. (2018) fit models with different rotation parameterizations to both (B,V ) and (J,Ks) photometry. We give the results from fitting the model with a free rotation parameter and the model with a fixed rotation parameter but a spread in rotation to both color-magnitude diagrams.
dated catalogs of rotators in both clusters to carry out our analysis. We describe our membership and archival P rot catalogs for the two clusters in Section 2 before deriving masses (M * ) and effective temperatures (T eff ) for these stars in Section 3. In Section 4, we identify binaries among our K2 targets. Binary companions can impact the rotational evolution of a star, and therefore confuse interpretation of the mass-period distribution of a cluster. We then present new P rot measurements for 116 Hyads from K2 Campaign 13 in Section 5. Finally, we derive single-star sequences in both clusters using the second Gaia data release (DR2; Gaia Collaboration et al. 2018b), obtain a new reddening value for Praesepe, and derive a differential gyrochronological age for the Hyades in Section 6. We discuss our results and their potential implications for calibrating angular momentum evolution in Section 7, and conclude in Section 8.

Hyades Membership and Rotation Catalog
As in Douglas et al. (2014Douglas et al. ( , 2016, we use the Röser et al. (2011) and Goldman et al. (2013) catalogs as the basis for our work. To these we add 13 stars identified using reduced proper motions and parallaxes from Hipparcos, bringing us to 786 total Hyads. Since archival data for the Hyades are generally of high quality, and since our pre-Gaia catalog was used to select our K2 Campaign 4 and 13 targets (Guest Observer proposals 4095 and 13064), we do not attempt to update the full cluster membership list using Gaia DR2.
Furthermore, since our sample consists of variable stars and includes probable binaries, these stars will have increased photometric variability and possibly also high astrometric excess noise. This variability and excess noise will impact the availability of the Gaia data, as well as the determination of appropriate quality cuts. Indeed, 188 stars in our original catalog do not pass the quality cuts recommended by the Gaia Collaboration (Gaia Collaboration et al. 2018a), and >80 of these are confirmed or candidate binaries.
In Douglas et al. (2014Douglas et al. ( , 2016, we assembled P rot measurements for Hyads from Radick et al. (1987Radick et al. ( , 1995; Prosser et al. (1995); Delorme et al. (2011);Hartman et al. (2011); and from an analysis of All Sky Automated Survey (ASAS; Pojmański 2002) data (A. Kundert & P. Cargile, private communication, 2014) 1 into a catalog of 102 rotators. We then added 37 new P rot from our analysis of K2 Campaign 4 data in Douglas et al. (2016), bringing the total number of known Hyades rotators to 139. With a few exceptions, these surveys generally measure consistent P rot ; for details, see Douglas et al. (2014Douglas et al. ( , 2016. The mass-period relationship for these 139 Hyads is shown in Figure 1. In the second half of this paper, we consider only single, slowly rotating Hyads, and we use Gaia data to select these stars. We match our Douglas et al. (2016) Hyades catalog to Gaia DR2, and select the nearest neighbor. We then check this match by computing synthetic Gaia G magnitudes from UCAC r, i magnitudes (Zacharias et al. 2010), SDSS r, i (Alam et al. 2015), Douglas et al. (2014) we cited these Prot as Kundert et al. in prep, and in Douglas et al. (2016) as Cargile et al. in prep. These periods were measured by A. Kundert as an undergraduate while being supervised by co-author P. Cargile. The paper was never completed, however, and additional ASAS data have become available for Hyades members in the last few years. We therefore give the existing ASAS Prot measurements in Table 3, but further details will be provided in a later paper, where we will re-analyze the expanded ASAS data set. Since we find that ground-based Prot generally, and ASAS Prot specifically, are consistent with K2 periods, we feel justified in continuing to include the current ASAS periods in our analysis. We also include the uncertainties on M * , which are dominated by distance uncertainties even in the Gaia DR2 era. The error bars only represent systematic uncertainties from our mass calculation, and do not reflect, e.g., systematics in the model or excess K-band flux due to an unresolved companion.
2MASS J, K (Skrutskie et al. 2006), and/or Tycho2 B, V (as given in 2MASS). We require that at least one of these synthetic magnitudes match the measured Gaia G value to within 1 standard deviation (σ) for optical photometry or to within 2σ for 2MASS photometry. Of the 786 stars in our catalog, only 10 fail this test: three stars lack photometry to compute synthetic G magnitudes, two lack Gaia counterparts, and five fail the G magnitude test. However, none of those 10 stars has a measured P rot or is a K2 target, so they do not impact our analysis and are excluded from all tables.

Praesepe Membership and Rotation Catalog
We continue to use the Douglas et al. (2017) Praesepe membership catalog, which is based primarily on Kraus & Hillenbrand (2007). Our catalog includes 1130 cluster members with P mem ≥ 50% from Kraus & Hillenbrand (2007), supplemented by 39 previously cataloged members too bright to be identified by those authors. We assign these bright stars P mem = 100%.
We match this list of Praesepe rotators to Gaia DR2 and again select the nearest neighbor. Only three rotators in our catalog lack a DR2 match within 0. ′ 1: EPIC 211970974 and EPIC 211907026 are both rapidly rotating M dwarfs, and EPIC 211954582 is overluminous by −1.18 mag, which suggests that it might be a triple system. Since our analysis focuses on single, slowly rotating stars, the lack of a DR2 match in these three cases does not affect this work.

Stellar Masses
As in previous work, we estimate stellar masses by linearly interpolating between the M K and M ⋆ points given by Kraus & Hillenbrand (2007), who list M ⋆ and spectral energy distributions (SEDs) for B8-L0 stars.
We calculate distances (D) to individual stars using Gaia DR2 or Hipparcos (Perryman et al. 1998) parallaxes, or the secular parallaxes from Röser et al. (2011) or Goldman et al. (2013). For stars passing the Gaia quality cuts, we use Gaia parallaxes. For the remaining stars, we use Hipparcos parallaxes or secular parallaxes. We then use these distances to compute M K .
We also propagate the m K and D uncertainties for each star to determine the M * uncertainties, σ M * . The uncertainties are typically small, on the order of a few percent. In our previous work, a few stars had large uncertainties in D, which led to large mass uncertainties. The improved parallaxes from Gaia have remedied this. Our stated σ M * are only the systematic uncertainties resulting from our calculation and the chosen model; they do not take into account other sources of uncertainty, such as our choice of model or K-band excesses due to a binary companion.

Effective Temperatures
In Section 6, we also compare the two clusters' P rot -T eff relations. For solar-type stars with 4700 < T eff < 6700 K, we derive an empirical color-T eff relation using a Gaia DR2 match to the Califor-nia Planet Survey catalog (Brewer et al. 2016). For warmer stars, we supplement this with Hyades members from Gaia Collaboration et al. (2018a) with T eff from DR2/Apsis (Andrae et al. 2018). For cooler stars, we combine the benchmark K and M dwarfs from Mann et al. (2015) and Boyajian et al. (2012). That sample only reaches T eff > 3056 K, so we also adopt the Rabus et al. (2019) M G -T eff relation for stars with 2600 < T eff < 4000 K. At T eff = 4000 K, our color-T eff relation predicts a value only 9 K different from the Rabus et al. (2019) formula when using our fit to the Hyades main-sequence to convert between color and absolute magnitude.

BINARY IDENTIFICATION
We search binaries among known rotators in the Hyades because they can bias our analysis of the P rot distribution. Binary companions may exert tidal or other physical effects on the primary star (e.g., Meibom & Mathieu 2005;Meibom et al. 2007;Zahn 2008;Douglas et al. 2016Douglas et al. , 2017. In addition, when two (or more) stars are blended in a given image, the second star may dilute the rotational signal and/or add flux that will cause us to overestimate L bol and M * . These effects can cause stars to be misplaced in the mass-period plane, leading us to misidentify trends or transitions in the period distribution. Finally, shortperiod binaries are susceptible to tidal interactions, which can cause atypical angular momentum evolution. Binaries with orbital periods under ∼10 days might be circularized and locked, but others with orbital periods up to 30 days could still be affected. We therefore wish to identify as many binary systems as possible among our Hyades K2 targets. We denote all confirmed and candidate binaries in our analysis, and provide a brief overview of our binary identification methods below. For more details, see Douglas et al. (2016Douglas et al. ( , 2017.

visual identification:
We examine a co-added K2 image, a Digital Sky Survey (DSS) red image, and a 2MASS (Cutri et al. 2003) K-band image of each target to look for neighboring stars (see Figure 4). We use a flag of "Y" for yes, "M" for maybe, and "N" for no to indicate whether the target and a neighbor have blended point spread functions (PSFs) on the K2 chip. Stars flagged as "Y" are labeled candidate binaries; we find 38 such targets, or 29% of stars with K2 P rot .
By searching 12 regions of the nearby sky in Gaia DR2, we find the rate of chance alignments with G ≤ 20 mag stars within 10 ′′ to be ≈6-58%. We find a range in potential contamination rates because the Hyades is so large on the sky: part of Note-This table is available in its entirety in machine-readable form.
a Index in the Röser et al. (2011) catalog the cluster sits close to the Galactic Plane, but it also extends well away from the Plane. At typical Hyades distances, 10 ′′ corresponds to ≈400-550 AU; it is possible that all of the blends we identify are chance alignments, or that up to 23% of Hyads have a companion within ≈400-550 AU (for comparison, we determined that ≈10% of Praesepe members likely have a bound companion within 10 ′′ , or 10 3 -10 4 AU; Douglas et al. 2017).
For consistency with our previous work, we continue to label probable blends as candidate binaries.
2. photometric identification: As in previous work, we identify candidate unresolved binaries that are overluminous for their color. In Douglas et al. (2014Douglas et al. ( , 2016Douglas et al. ( , 2017, we selected binaries that were overluminous in a r′ vs (r′ − K S ) colormagnitude diagram (CMD), using Hipparcos Perryman et al. (1998) parallaxes or secular parallaxes from Röser et al. (2011) and Goldman et al. (2013). As in Section 3.1, we now update the r′ vs (r′ − K S ) selection using Gaia DR2 parallaxes when the data passes the quality cuts defined in Gaia Collaboration et al. (2018a). We also select new photometric candidate binaries using Gaia DR2 photometry, discussed further in Section 6.1.2. This method is biased towards binaries with equal masses, and we are certainly missing candidate binaries with lower mass ratios. Our binary selections are shown in Figure 2; in Section 5 we flag all photometric candidate binaries, but in Section 6 we reject only candidates selected from Gaia photometry.
3. multiperiodic K2 stars: In binaries where the components have roughly equal brightness, variability from both stars can appear in the K2 light curve. However, we may also detect two P rot and/or an obvious beat pattern when a single star exhibits differential rotation. As discussed in Section 5, we assume that the two periods come from different components of a binary if the periods are different by >20%. This cutoff is based on the maximum period separation for differentially rotating spot groups on the Sun. We find multiple P rot , indicating probable unresolved binaries, in 11 K2 targets.

literature identifications:
We searched the literature for Hyades binaries among known rotators and K2 Campaign 4 targets in Douglas et al. (2016). We update this list with binaries among our Campaign 13 targets. We also add binaries identified or confirmed through observations with the Tillinghast Reflector Echelle Spectrograph (TRES) on the 1.5-m Tillinghast telescope at the Smithsonian Astrophysical Observatory's Fred L. Whipple Observatory on Mt. Hopkins, AZ (R. Stefanik, private communication, 2018).
We consider all visual and photometric pairs, as well as multiperiodic K2 stars, to be candidate binaries in our analysis. For other literature binaries, we follow the confirmed versus candidate nomenclature used in the source paper. The resulting list of confirmed and candidate binaries is given in Table 2.
a Index in the Röser et al. (2011) catalog b Quality of the Prot detection. 0 is a high-confidence measurement, 1 is questionable, 2 is not trusted, and 3 indicates that there were no significant periodogram peaks.
c Presence of multiple periods in the light curve. Y, M, and N represent "yes", "maybe", and "no", respectively d Presence of a blended neighbor. Y, M, and N represent "yes", "maybe", and "no", respectively e Flag for the Prot source selected. "R": Radick et al. (1987Radick et al. ( , 1995, "P": Prosser et al. (1995), The distribution of Hyades targets in K2 Campaigns 4 and 13 is shown in Figure 3.
We use detrended light curves generated using the K2 Systematics Correction method (K2SC; Aigrain et al. 2016) for our analysis. Aigrain et al. (2016) developed a semi-parametric Gaussian process model to simultaneously correct for the spacecraft motion and model the stellar variability. As discussed in Douglas et al. (2017), we find that this approach is best at removing instrumental signals and trends while leaving stellar periodic signals intact. 2 We ran the K2SC code on the K2 PDC pipeline light curves ourselves since the processed K2SC light curves for Campaign 13 are not yet on MAST. We downloaded the pipeline light curves in March 2018.
We follow the same period measurement method used in Douglas et al. (2017), and only summarize it here. We use the Press & Rybicki (1989) FFT-based Lomb-Scargle algorithm 3 to measure P rot . We compute the Lomb-Scargle periodogram power for 3×10 4 periods ranging from 0.1 to 70 d (approximately the length of the Campaign). We also compute minimum significance thresholds for the periodogram peaks using bootstrap 2 For more information, see Aigrain et al. (2016) and the MAST high level science product page, https://archive.stsci.edu/missions/hlsp/k2sc/hlsp_k2sc_k2_llc_all_kepler_v1_readme.txt.
3 Implemented as lomb_scargle_fast in the gatspy package; see https://github.com/astroML/gatspy. re-sampling, and only consider a peak to be significant if its power is greater than the minimum significance threshold for that light curve. We take the highest significant peak as our default P rot value; only three of our targets show no significant periodogram peaks.

Period Validation
We employ several automated and by-eye quality checks to validate the P rot identified above. We inspect each phase-folded light curve to confirm that the detected P rot appears astrophysical and not instrumental. Clearly spurious detections are flagged as Q = 2, and questionable detections as Q = 1. A Q = 3 flag indicates that there were no significant periodogram peaks. We also plot the full light curve with vertical dashed lines at intervals corresponding to the detected P rot , to ensure that light curve features repeat over several intervals. Finally, we check for cases where there is a doubledip in the light curve, and the highest periodogram peak likely corresponds to half of the true P rot . This is caused by two similar spot groups on opposite sides of the star. We then select the correct peak as the final P rot . Figure 4 shows an example of the plots we use to inspect the data; we include a figure set showing these plots for every target in our sample online. The left column shows the (r ′ − Ks) color we used in previous work, but using Gaia parallaxes where available to determine M r ′ . The right column shows one of the Gaia colormagnitude diagrams (CMDs) we used to update our candidate binary list for this work. Top-CMD with our selected main sequence (solid line) and binary cuts overlaid: the dotted line shows the nominal binary main sequence, and the dot-dashed line gives the minimum magnitude above which we consider a star to be a candidate binary. In previous work (left) we use the model SEDs assembled by Kraus & Hillenbrand (2007), and in this work (right) we use a polynomial fit to the Hyades main sequence. Middle-residuals between observed and expected absolute magnitudes; the horizontal lines are the same thresholds given above. Candidates identified in Douglas et al. (2016) are shown in in green, and new candidates identified from Gaia DR2 photometry are given in blue. It is clear that the improved Gaia parallaxes have removed some of our previously identified candidate binaries. Bottom-the same as the middle panel, but now confirmed multiples (black stars) and literature candidates (orange diamonds) are also shown. While our photometric selection is useful for identifying additional candidates, there are still many confirmed binaries that show no photometric offset from the main sequence. We find 13 stars with significant periodogram peaks but no believable P rot . In six cases, the light curve is just noise or displays only a long trend, without any detected periodic variability. In the remaining seven cases, there is some probable spot-induced variability, but the phasefolded light curves do not actually match up and there is no clear period. In these cases, we are likely observing rapid spot evolution, perhaps on two stars in a binary.
For 18 other stars, the highest periodogram peak does not appear to correspond to the true P rot . In some cases, as above, the highest periodogram peak comes from a campaign-long trend, and the true period is detected at a weaker power. In other cases, we find a doubledip light curve with almost no difference between the central (half-period) dip and the primary (full-period) dip. In these cases, the phase-folded light curve for the longer period shows the double-dip pattern clearly, even though it is detected at a lower periodogram power.
EPIC 210741091 and EPIC 247337843 are two very interesting cases: it is hard to define a period because the spot modulation only appears in half the campaign. For EPIC 210741091, there is initially some variability but no clear periodic signal; a V-shaped dip suggesting a single large spot (Bopp & Evans 1973;Eker 1994) appears about halfway through the campaign. Nonethe-less, we measure P rot = 11.78 d for this star, very close to the P rot = 11.98 d value we measured in Campaign 4. EPIC 247337843 develops rapidly from cycle to cycle, from a slight double-dip at the beginning of the campaign to variability with no clear period by the second half. Given this variability and parial lack of signal for both stars, we assign Q = 1 for their Campaign 13 P rot .
Finally, in 11 light curves we detect two signals with periods differing by at least 20%. We consider these stars to be candidate binaries. Several other stars exhibit two close but distinct periodogram peaks, and the light curves have obvious beat patterns. This suggests that in these cases we are observing differential rotation of two spot groups at different latitudes.

Summary: New K2 Periods for the Hyades
We obtain robust P rot measurements for 116 Hyades members, including 93 members with no prior P rot measurement. The vast majority of these periods are for rapidly rotating M dwarfs, and bring the total number of Hyads with P rot to 232. Our P rot values, flags, and analysis outputs are found in Table 3. Our new rotation periods, along with literature values, are shown as a function of stellar mass in Figure 5.
Only 23 stars have P rot measured here and in previous studies, including five with a P rot measurement from K2 Campaign 4 (Douglas et al. 2016). Figure 6 shows a comparison of the existing data with our new measurements. In two cases (EPIC 210554781 and EPIC 246806983), the literature period is also detected as a secondary period in the K2 light curve. In two other cases (EPIC 210558541 and EPIC 246714118), we detect a short P rot in K2 and do not detect the longer literature P rot at all. In general, however, we find that ground-and space-based P rot measurements agree to within 10%, similar to our results in Praesepe (Douglas et al. 2017).

COMPARING THE HYADES AND PRAESEPE
Based on the similarity of their color-magnitude diagrams (CMDs) and their activity, rotation, and lithium abundance data, the Hyades and Praesepe are often assumed to be coeval clusters (e.g., Douglas et al. 2014;Cummings et al. 2017). Here, we test this assumption using our expanded rotator samples paired with the high-precision data from Gaia DR2 for each cluster. First, we identify likely single-star members of each cluster. Then we apply a new gyrochronology model tuned with the Praesepe slow-rotating sequence and the Sun to infer a precise, relative, gyrochronological age for the Hyades.   A comparison of Figure 5 and figure 7 in Douglas et al. (2017) shows that the color-P rot distributions for the Hyades and Praesepe appear qualitatively similar to each other. Most stars follow a common slow-rotator sequence from the late-F stars down to early M, followed by a sharp transition from slow to rapid near the fully convective boundary at ≈M4. However, many stars are outliers and appear to be rotating more rapidly or slowly than the slow-rotating sequence.

Defining Single-Star Sequences
Where possible, it is important to reject outliers following membership and multiplicity criteria, instead of removing them based on their position in color-period space. The primary reason is that we wish to show that ≈700-Myr-old stars follow a single-valued color-P rot relation from mid-F down to early M, and that any rapid stars in this mass range are rapid for a reason unrelated to single-star angular-momentum evolution (e.g., because they are binaries, blends, or interlopers, or have poor data). Since the Hyades and Praesepe samples of rotators are large, we can apply strict physical (e.g., based on positions, kinematics, or luminosity excesses) and data-quality criteria (e.g., poor astrometric solutions, blended light curves resulting in multiple period detections) to select stars with Kepler and Gaia data consistent with single-star membership without overdepleting the color-period plane at any color. We describe our selection criteria below; each criterion is applied independently and the outputs combined to create our final list of single members. The results are summarized in Figure 7, and Tables 3 and 4 include flags indicating which tests were passed by each star

Kinematics
For the Hyades, we select candidate single stars first by rejecting confirmed binaries identified in the literature, and then by considering the Galactic U V W space velocities for stars with six-parameter positions and kinematics from Gaia DR2. We calculate the cluster median U V W velocities from the Hyades membership list in Gaia Collaboration et al. (2018a); 4 next, we compute the absolute velocity deviation, ∆v, for the 101 rotators in our sample with six-parameter positions and kinematics by subtracting off the cluster median values for each U V W component and then adding the residuals in quadrature. The Hyades's internal velocity dispersion is estimated to be only 0.3 km s −1 (Gunn et al. 1988;Perryman et al. 1998), which is comparable to the DR2 Hyades members are plotted as gray points, stars with measured Prot are gray points outlined with black circles, and the subset we identify as single-star members are shaded orange. Top right-Prot for the Hyades are plotted against T eff , which we calculated from Gaia DR2 photometry using empirical color-temperature relations described in Section 3.2. Filtering out rotators that are known spectroscopic binaries, or have multiple periods detected in K2 light curves, or are astrometric or photometric non-single members from Gaia DR2 removes all rapid outliers from the diagram, revealing a cleanly converged slow sequence down to T eff ≈ 3500 K. Bottom left-Gaia Collaboration et al. (2018a) Praesepe members are plotted as gray points, stars with Prot are gray points outlined with black circles, and the subset we consider to be single-star members are shaded blue. Bottom right-Applying similar filtering as described for the Hyades removes all slow outliers in Praesepe's Prot distribution, and all rapid outliers down to T eff ≈ 4000 K. The slow sequence appears converged down to T eff ≈ 3600-3800 K, depending on the still uncertain multiplicity of a few rapid stars in that range.
radial velocity (RV) error. We adopt a more conservative threshold for identifying non-single members of ∆v > 2 km s −1 , which eliminates 26 stars. We also consider stars with DR2 RV errors σ RV > 2 km s −1 to be non-single members, which cuts an additional four stars, so that in the end we have 71 single-star rotators in our sample. For Praesepe, we first remove the 43 binaries confirmed in the literature. Then, we filter non-singlemember stars using proper motions separately from RVs. This is possible because Praesepe is more distant than the Hyades and useful because 719 of 743 rotators have DR2 proper motions, whereas only 185 have DR2 RVs. The distribution of proper motions for our rotator sample can be approximately described by a Gaussian with σ = 1.25 mas yr −1 (the median proper motion error is 0.2 mas yr −1 ). We set our threshold at twice this value and reject stars with absolute proper motion deviations larger than this 2.5 mas yr −1 . 5 This eliminates 146 of 719 stars with DR2 proper motions. Separately, we reject 48 stars with ∆RV > 2 km s −1 from the cluster median value quoted by Gaia Collaboration et al. (2018a), 6 and 46 stars with σ RV > 2 km s −1 . In total, we reject 196 unique non-single members, and retain 523 singlestar rotators.

Photometry
We use the Gaia Collaboration et al. (2018a) Hyades catalog to generate a fiducial cluster CMD, and then iteratively fit the resulting main-sequence with a cubic basis-spline. We then generate a new CMD using our full rotator list, and determine each star's deviation from the fiducial main-sequence.
We fit two CMDs: absolute G magnitude, M G , versus both (G BP − G RP ) and (G − G RP ). We analyze (G − G RP ) to account for the larger uncertainty in G BP for redder/fainter stars. We then calculate the photometric deviation from these empirical main-sequences for our rotator sample, d cmd = |M G,observed − M G,predicted |, and label all stars that are consistent with at least one of the empirical isochrones as photometric single-member stars. We set a threshold of d cmd < 0.375 mag for all stars, which is half of the offset for an equal-mass binary (e.g., Hodgkin et al. 1999). We find that 176 of 222 Hyads with DR2 photometry are consistent with being single-star members.
For Praesepe, we adjust the fiducial Hyades CMD fit according to its interstellar reddening/extinction that we derive in Appendix A (A V = 0.035) and the distance modulus implied by inverting the cluster parallax (̟ = 5.371 mas; Gaia Collaboration et al. 2018a). We find 525 of the 741 stars with DR2 photometry are consistent with being single-star members.

Astrometric data quality
The Gaia DR2 astrometric solution for each star assumes it is a single point source. Objects that are inconsistent with this assumption can have excess astrometric noise (ǫ i ), and we remove those with ǫ i > 1 and G < 19 mag from our samples. This includes 40 stars in the Hyades and 48 stars in Praesepe. Most were already filtered by our kinematic and photometric selection criteria.

Prot quality and corrections
For rotators with K2 light curves, we remove those for which we detect multiple periods, which, again, we interpret as either physically unassociated blends or cluster binaries (see Sections 4 and 5.2).
For Praesepe, an additional step is required: several periods in the literature need to be corrected. In Douglas et al. (2017), we assembled literature periods and our own K2 periods, and then recommended which source to use for each star. We recommended Delorme et al. (2011) for EPIC 211995288, andEislöffel (2007) Douglas et al. (2017) measurements for these two stars are half-period harmonics, caused by presumably by nearly symmetric spot patterns on opposite-facing hemispheres. We therefore double the old P rot for these stars.
Finally, EPIC 211950227 was originally given a period of 13.15 d (Delorme et al. 2011). However, the Campaign 16 light curve shows that the dominant modulation signal has a period of P rot = 1.76 d. We see no ≈13 d signature in its Campaign 16 light curve and conclude that the K2-derived P rot is the correct one.

Resulting CMDs and T eff -P rot Distributions for the Hyades and Praesepe
The resulting CMDs for the two clusters are shown in the left column of Figure 7, with their T eff -P rot distributions in the right column. Applying the cuts described above yields a nearly clean P rot distribution for both  (2007) b Flags for selecting single stars in Section 6.1. Entries correspond to astrometry, photometry, K2 multiperiodic, RV, and confirmed binary selection: "Y" indicates the star passes a given test, "N" indicates failure, and "-" indicates that we lack the data to perform a particular test. We only retain stars flagged "YYYYY" or "YYY-Y".
clusters. Overall, 118 Hyades rotators out of 232 satisfy our single-star-membership criteria. When examining the cluster's P rot distribution, we find no rapid outliers relative to the cleaned, slow-rotating sequence for M ⋆ 0.57 M ⊙ (T eff 3789 K), and only three moderately faster rotators for M ⋆ 0.5 M ⊙ (T eff 3620 K). The transition to completely rapid rotation in the Hyades occurs at M ⋆ ≈ 0.35 M ⊙ (T eff ≈ 3420 K, M3), which is slightly warmer than the T eff -radius discontinuity at T eff = 3200-3340 K identified by Rabus et al. (2019).
For Praesepe, we find that 496 of the 743 rotators are consistent with being single-star members. None of these stars appears significantly more rapid than the slow converged sequence for T eff 3845 K (M ⋆ 0.6 M ⊙ , M0). Of the 43 single members on our list with 3600 < T eff < 3850 K, 10, or 23%, are rapidly rotating outliers that have P rot faster than the slow sequence by at least 3 d. The transition to all rapid rotators happens around M ⋆ ≈ 0.4 M ⊙ (T eff ≈ 3500 K), but is not as well defined as in the Hyades.

A Precise Differential Gyrochronology Age for the Hyades
We now turn to the question of whether Praesepe and the Hyades are truly coeval. We search the literature and tabulate recent isochrone ages for the two clusters derived using a variety of photometry, constraints, models, and methods (see Table 1). From these, we calculate an age for the Hyades of 728±71 Myr (median and 1σ of thirteen values), and for Praesepe of 670±67 Myr (median and 1σ of eleven values). Since this suggests that Praesepe is the younger of the two clusters, we then calibrate an empirical gyrochronology model by fitting the Praesepe T eff -P rot sequence, and then tune the age-dependence with the Sun. Finally, we compare the T eff -P rot sequences of the Hyades and Praesepe, and derive a precise, differential age according to our empirical model.
We summarize our assumed values for the Sun here. We take the Sun's P rot = 26.09 d, measured from periodic modulations in the Mount Wilson Ca II H & K S-index by Donahue et al. (1996).
We take its age to be 4567±1±5 Myr (Chaussidon 2007). Based on observations of solar twins derived from the updated Spectroscopic Properties of Cool Stars (SPOCS; Brewer et al. 2016) catalog, we derive a solar color of (G BP − G RP ) ⊙ = 0.817 mag, consistent with the value of (G BP − G RP ) ⊙ = 0.82 estimated by Casagrande & VandenBerg (2018). A more detailed discussion of our derivation of this color can be found in Appendix B.
Our analysis also makes the following assumptions: 1. The Sun has slowed down continuously since it was 670 Myr old (our adopted age of Praesepe). According to van Saders et al. (2016), magnetic braking efficiency plummets at a critical Rossby number (the ratio of P rot to convective turnover time) of R crit = 2, approximately the current solar value. We assume that the Sun has not yet reached this threshold and that it has therefore spun down continuously with a single-valued time dependence.
2. The difference in metallicity between the Sun and Praesepe does not appreciably affect spin-down and that comparing equal-color stars is valid, even though a solar-mass star in Praesepe is cooler than the Sun's current temperature. 7 We fit a sixth order polynomial to Praesepe's cleaned and dereddened DR2 color-period sequence for stars with ((G BP − G RP )) < 2.4 (T eff ≈ 3500 K, M ⋆ ≈ 0.42 M ⊙ , M2V). This color limit stops our model before the sharp drop to rapid rotation around the fully convective boundary. The sixth order polynomial is necessary as lower-order polynomials fail to accurately track the rapid change in P rot from the F to G dwarfs.
The Praesepe fit predicts a period at the solar color of P rot = 8.09±0.25 d. We calculate this value using a T eff -P rot diagram de-reddened by our A V = 0.035 value, while the uncertainty comes from assuming either A V = 0 (no reddening) or A V = 0.084 (Taylor 2006). We use the age for Praesepe derived from the literature of 670 Myr, and calculate that the braking index n = 0.619.
We now apply our new gyrochronology formula to the cleaned stars in the Hyades with 0.7 <(G BP − G RP ) 0 < 1.1, where gyrochronology should be viable at this age (Agüeros et al. 2018, Curtis et al. in prep.). If Praesepe is 670 Myr old and its A V = 0.035, and if it is chemically identical to the Hyades, then the Hyades is 57 Myr older. We find the Hyades age to be 727±75 Myr (median and 1σ), based on 25 cluster members. (For 49 analogous Praesepe stars, 1σ = 69 Myr.) Recall that we calculate an isochrone age difference of 58 Myr by computing the difference between the median of various isochronal ages for each cluster; this is essentially identical to our differential gyrochronology result. Figure 8 shows the T eff -P rot diagram for the cleaned Praesepe and Hyades samples, and their corresponding gyrochronology ages using our recalibrated formula. Derived ages for individual stars are given in Table 6. 7. DISCUSSION 7 Stars do not spin down through T eff -Prot space along perfectly vertical lines, since they warm as they age. Differences in metallicity will also modify moments of inertia, convective turnover times, and other physical ingredients that are critical to understanding angular-momentum evolution. Theoretical models are the appropriate way of accounting for metallicity and stellar-evolution effects (e.g., van Saders & Pinsonneault 2013), but we presently lack sufficient coeval benchmarks with different metallicities to validate their predictions. Also, all available models fail to represent the cluster sample, aside from the most Sun-like G dwarfs (e.g., Agüeros et al. 2018, Curtis et al. submitted, and this work). Since our primary goals are to test if the Hyades and Praesepe are truly coeval and to measure a differential age, any systematic inaccuracies in the model will propagate to both cluster ages equally.
New P rot measurements from K2 and precise Gaia data have enabled us to compare the rotation distributions in Praesepe and the Hyades in detail. Whereas in previous work we assumed that the clusters have overlapping P rot sequences, we now find that is not the case for solar-type stars. Overall, we find that Hyades FG stars rotate more slowly than their Praesepe counterparts, corresponding to a differential gyrochronological age of 57 Myr. This difference is consistent with the 47 ± 17 Myr difference between the clusters found by Delorme et al. (2011), who used a linear fit to the P rot vs. (J − K s ) relation in the Hyades and Praesepe. The 57 Myr age difference suggests that the two clusters should be separated when considering the evolution or effects of stellar rotation in solar-type stars and when accuracy below the 10% level is required.
Interestingly, the age discrepancy between the two clusters is largest for T eff > 5200 K and decreases as we move to cooler stars. We fit the gyrochronology ages of Hyades stars with locally weighted scatterplot smoothing (LOWESS) as a function of T eff , and compare it to the fiducial Praesepe model (Figure 9). Between 5250 > T eff > 4900 K, the differential gyro ages decrease, so that cooler Hyads converge with the Praesepe sequence. The late K and early M dwarfs do not brake appreciably from the age of Praesepe to that of the Hyades. This contradicts the common assumption that braking timescales increase as mass decreases. Our work therefore adds to prior evidence that low-mass stars follow a different, more complex braking timeline than their solar-type counterparts.
Several other authors have reached similar conclusions. Meibom et al. compared M35 (≈150 Myr;Meibom et al. 2009), M34 (≈220 Myr; Meibom et al. 2011a), and NGC 6811 (≈1 Gyr; Meibom et al. 2011b) to the Hyades, and find that K dwarfs must spin down less efficiently than FG stars. Cargile et al. (2014)    We therefore provide concrete evidence that K stars spin down at a variable rate, as opposed to existing empirical models which show them spinning down continuously from the time they reach the main sequence. This stalling is apparent even over ∼50 Myr timescales. Previous empirical work has assumed a fixed functional form for the dependence of P rot on mass or (B − V ) at all ages. For example, Delorme et al. (2011) fit a line to the P rot vs. (J − K s ) distributions in clusters, and Barnes (2003Barnes ( , 2007 fit other analytic functions. These efforts assumed that it was possible to decouple the mass and age dependencies, but our results demonstrate that rotation evolves at different rates for stars of different masses. Barnes (2010) presented the only empirical gyrochronology relation that allowed more complicated mass-dependent evolution by including a dependence on the convective turnover time τ , instead of color. That model accurately described the M ⋆ > 0.85 M ⊙ stars in the 2.5 Gyr NGC 6819 cluster (Meibom et al. 2015). However, it actually predicted that K dwarfs spin down more rapidly than G dwarfs, instead of more gradually as indicated by the open cluster data. Mamajek & Hillenbrand (2008), Meibom et al. (2009), and, more recently, Angus et al. (2015) simply recalibrated the model presented by Barnes (2003Barnes ( , 2007, without considering more complex mass-dependent rotational evolution.
One probable reason that empirical models have not included a mass dependence is the paucity of > ∼ 1-Gyr-old benchmarks for K and M dwarf rotators. P rot have been published for solar-type members of NGC 6819 and M67, but not their lower-mass members. We show that this dependence is present even over short timescales, but the field of gyrochronology requires additional benchmarks at older ages to properly calibrate braking timescales for stars of different masses. Future work on NGC 6819 and Ruprecht 147, also ≈2.5 Gyr old, will provide further constraints on mass-dependent evolution at older ages.
For the time being, when examining effects at a single age, we can consider the low-mass rotators in the Hyades and Praesepe as a single ensemble. The low-mass rotators deserve additional consideration in future work, but this will first require comprehensive binary surveys of late K and early M dwarfs to disentangle evolutionary effects from multiplicity effects in these clusters. Several authors have found tentative evidence that binaries rotate faster than single stars (e.g., Meibom et al. 2007;Douglas et al. 2016Douglas et al. , 2017, which is one reason why we remove known binaries from our sample above. The Hyades and Praesepe, however, have not been uniformly surveyed for binaries, particularly at the lowmass end. In our K2 analysis, we identify candidate binaries via blends and multiple periods detected in a single light curve. However, these candidates could be chance alignments or (when the two periods are very similar) a signal of latitudinal differential rotation.
NASA's ongoing Transiting Exoplanet Survey Satellite mission (TESS ; Ricker et al. 2015) will also provide an excellent opportunity for expanding the P rot catalog for Hyades M dwarfs. Many Hyades M dwarfs lie on the outskirts of the cluster (with many more potentially found in unbound tidal tails; Röser et al. 2019), far enough from the ecliptic to be observed by TESS. Although there will certainly be issues with systematics given the standard 27.4 d observing cadence, we expect to measure P rot for ≈200 Hyads in the Southern Hemisphere alone (TESS Program G011197). Many more Hyads, as well as members of another approximately coeval Coma Ber cluster (Collier Cameron et al. 2009), will be observed by TESS in the Northern Hemisphere. Since one current challenge in comparing the Hyades and Praesepe is the much smaller Hyades P rot catalog, future TESS measurements will be invaluable for differentiating the behavior of M dwarfs in these similarly aged clusters.

CONCLUSIONS
We analyze K2 Campaign 13 light curves for 132 members of the Hyades open cluster. We measure P rot for 116 (88%) of these stars, including 93 members with no prior P rot measurements, bringing the total number of Hyads with known P rot to 232. As in our last two papers (Douglas et al. 2016(Douglas et al. , 2017, we find that groundbased P rot measurements are generally consistent with space-based measurements. The primary difference is that space-based observatories can observe a wide field of view nearly continuously while simultaneously reaching even faint members of nearby open clusters.
We then use Gaia DR2 data and literature binary information to define a clean sequence of single-star Hyads in color-magnitude space. We then apply this procedure to data for the Praesepe open cluster, which is generally thought to be coeval with the Hyades. As a result, we obtain two clean sequences of slowly rotating FGK stars in T eff -P rot space for both clusters.
There are far fewer known binaries among the M dwarfs in these two clusters. But our cuts also produce a nearly clean slow-rotator sequence for early M dwarfs, with only a few rapidly rotating members in this mass range in both clusters. These remaining rapid rotators highlight the need for additional binary surveys of M dwarfs in these clusters, especially Praesepe.
We use these single-star sequences to derive a reddening value of A V = 0.035±0.011 mag for Praesepe, assuming that the Hyades experiences no reddening. This value is intermediate between the oft-assumed A V = 0.0 and the A V = 0.084 mag derived by Taylor (2006) for Praesepe. We then derive a polynomial fit to the slow rotator sequence in Praesepe, as a function of dereddened Gaia DR2 (G BP −G RP ) 0 color. We use this fit as the basis for a new empirical model for gyrochronology, where we assume that stars begin on the Praesepe sequence at 670 Myr and their periods evolve as P rot ∝ t n . By comparing the Praesepe sequence to the Sun, we derive a value of n = 0.619.
Finally, we compare the slow-rotator sequence in the Hyades to this model we have generated based on Praesepe. We find that, if we only consider the F and G stars, the Hyades is 57 Myr older than Praesepe. We also find, however, that the difference between the Hyades and Praesepe sequences decreases towards lower-mass stars, so that the K and early M dwarfs in the two clusters are indistinguishable. This provides further evidence for stalling in the rotational evolution of these stars, and highlights the need for more detailed analysis of spindown over time for stars of different masses.
S.T.D. acknowledges support provided by the NSF through grant AST-1701468.
J.L.C. acknowledges support provided by the NSF through grant AST-1602662. M.A.A. acknowledges support provided by NASA through K2GO4 grant NNX17AF73G and by the NSF through grant AST-1255419. We also thank the anonymous referee for comments which improved the manuscript.
We thank R. Stefanik for sharing his Hyades binary detections with us. We also thank D. Latham for contributing to those observations and for providing valuable advice.
This research has made use of NASA's Astrophysics Data System Bibliographic Services, the SIMBAD database (Wenger et al. 2000), operated at CDS, Strasbourg, France, and the VizieR database of astronomical catalogs .
This paper includes data collected by the K2 mission. Funding for the K2 mission was provided by the NASA Science Mission directorate.
This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gai processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.cosmos.esa.int/web/gaia/dpac/co Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. The Two Micron All Sky Survey was a joint project of the University of Massachusetts and IPAC. The Digitized Sky Survey was produced at the Space Telescope Science Institute under U.S. Government grant NAG W-2166. The images of these surveys are based on photographic data obtained using the Oschin Schmidt Telescope on Palomar Mountain and the UK Schmidt Telescope. The plates were processed into the present compressed digital form with the permission of these institutions.  Figure 10. The difference between photometric (T eff,phot ) and spectroscopic effective temperatures (T eff,spec ) for 20 FGK members of the Hyades (orange diamonds) and nine of Praesepe (blue circles) are plotted against T eff,spec . T eff,phot values are estimated based on the relationship between the Gaia DR2 color (G BP − G RP ) and T eff,spec for the Hyads, which we assume appear to us un-reddened. We interpret stars with T eff,phot < T eff,spec as reddened and extinguished by interstellar dust. Based on Praesepe's median negative offset, we estimate AV = 0.035 ± 0.01 for that cluster.
Interstellar reddening is often constrained with color-color diagram or CMD analyses. We take an alternative approach using spectroscopy. Co-author J. Brewer has observed members of the Hyades and Praesepe with Keck/HIRES for a separate project, and analyzed the spectra with Spectroscopy Made Easy (Valenti & Fischer 2005) following the Brewer et al. (2015) procedure (see also Brewer et al. 2016;Brewer & Fischer 2018). We match their target list with Gaia DR2 and filter out non-single star members according to their proximity to the empirical cluster main-sequence defined by the Gaia Collaboration et al. (2018a) membership list and their astrometry. We also only focus on those stars with 5000 < T eff < 6200 K, giving us 20 FGK stars in our Hyades sample and nine in our Praesepe sample.
We fit an empirical color-temperature relation to the Hyades sample, and define its reddening to be zero. Figure  10 compares the Praesepe stars with their Hyades analogs, and shows that the Praesepe stars have photometric temperatures that are systematically cooler than their spectroscopic temperatures. Spectroscopic and photometric temperatures for individual stars are given in Table 5. We then calculate the necessary reddening values for each star in the Hyades and Praesepe needed to align their photometric and spectroscopic temperatures. We find A V = 0.035±0.011 (median and 1σ) for Praesepe. Our result splits the difference between the Taylor (2006) value and the oft-assumed zero reddening.

B. THE SUN'S GAIA DR2 COLOR
Since Gaia cannot observe the Sun's disk-integrated light, we must instead estimate its Gaia color with analogous field stars. We select stars in the updated SPOCS catalog (Brewer et al. 2016) with spectroscopic properties most similar to the Sun's, identifying 11 stars with T eff within 100 K of 5777 K (the solar T eff adopted by SPOCS), log g > 4.3 dex, [Fe/H] within 0.05 dex of solar, and log R ′ HK < −4.8 dex. We then fit a cubic polynomial relating T eff to color for these stars, finding that T eff = 5777 K predicts a solar color ((G BP − G RP )) ⊙ = 0.817 mag. This empirical value is in excellent agreement with that of Casagrande & VandenBerg (2018), who estimated the solar color from a variety of spectral templates to be ((G BP − G RP )) ⊙ = 0.82 mag.
The SPOCS star that we decided was most similar to the Sun is HD 103828 (Gaia DR2 845471463339146496). It has the following spectroscopic properties in Brewer et al. (2016): T eff = 5771 K, log g = 4.39 dex, metallicity [M/H] = Table 5. Praesepe and Hyades members used to derive the differential reddening between the two clusters