Stellar Rotation in the Gaia Era: Revised Open Clusters' Sequences

The Monitor Project ⁴ performed an optical photometric survey of several young open clusters using 2 and 4 m class telescopes with wide field cameras (Irwin et al. 2007b). The derived lightcurves allowed for searches of transiting planets (Miller et al. 2008) and eclipsing binaries (Irwin et al. 2007a), as well as measurements of rotation periods.

Of the clusters observed by the Monitor Project, we include NGC 2547, M50, and NGC 2516 in our sample. All three of these clusters offer interesting constraints on stellar rotation evolution. With an age of ∼35 Myr, NGC 2547 is a well-populated system where solar analogs have just arrived on the main sequence, and data in this age range (younger than the Pleiades and older than star-forming regions) is a sensitive test of theory. With ages of ∼130 and 150 Myr, M50 and NGC 2516 offer natural comparison points for the benchmark Pleiades cluster. Additionally, the contamination rates quoted for these clusters are in the ∼40%–60% range (Irwin et al. 2007b, 2008, 2009), which hints that a revision of their rotational sequences could be significant.

NGC 2547, M50, and NGC 2516 were studied in a similar fashion by Irwin et al. (2008), Irwin et al. (2009), and Irwin et al. (2007b), respectively. These authors surveyed regions out to ∼50', ∼30', and ∼60' from the clusters' center, identified candidate members on the basis of V versus V − I color–magnitude diagram (CMD) selections, and reported rotation periods for 176, 812, and 362 stars, respectively.

M37 (age of ∼550 Myr) is one of the very few open clusters with a large rotational sample that are older than 500 Myr. This, in addition to its age being close to the classical ∼580 Myr age for Praesepe, along with a quoted ∼20% contamination rate (Hartman et al. 2009b), make it an interesting system to include in our sample.

Hartman et al. (2008b) observed the M37 cluster with the 6.5 m MMT telescope in order to constrain its parameters, study variable stars (Hartman et al. 2008a), measure rotation periods (Hartman et al. 2009b), and study the occurrence rate of transiting planets (Hartman et al. 2009a). Given the exquisite optical observations, thorough lightcurve analysis, and the aforementioned expected contamination rate, we include M37 in our sample.

Hartman et al. (2009b) surveyed regions out to ∼20' from the cluster center, identified candidate members on the basis of two CMD selections (r versus g − r and g − i), and reported rotation periods for 575 stars. Of these, however, only 367 are considered by Hartman et al. (2009b) to be the clean periodic sample, where the different algorithms for determining periods showed good agreement with each other, and their results did not differ by more than 10%. We take this clean sample as our nominal M37 data set.

The Kepler spacecraft, in both its original 4 yr mission and subsequent K2 mission, observed several open clusters of a range of ages. These observations produced lightcurves with exquisite photometric precision, and have been used to construct rotation period catalogs for a number of clusters.

Of the clusters observed by Kepler and K2, we include the Pleiades, Praesepe, and NGC 6811 in our sample. Given their proximity, the Pleiades (distance of ≈136 pc; age of ∼125 Myr) and Praesepe (distance of ≈186 pc; age of ∼580 Myr) clusters have been used for rotation studies for several decades (e.g., Anderson et al. 1966; Kraft 1967b; Dickens et al. 1968; Stauffer et al. 1984, 2016; Stauffer & Hartmann 1987; Queloz et al. 1998; Terndrup et al. 1999, 2000; Scholz & Eislöffel 2007; Agüeros et al. 2011; Delorme et al. 2011; Douglas et al. 2014; Rebull et al. 2016a, 2016b, 2017), and the periods derived from the K2 observations comprise their state-of-the-art rotational samples. While the number of NGC 6811 stars with Kepler periods is lower compared to the aforementioned clusters, its old age (∼1 Gyr) makes it a remarkably interesting system for stellar rotation studies (e.g., Meibom et al. 2011a; Curtis et al. 2019a; Rodríguez et al. 2020; Velloso et al. 2020).

• 1.  
  Pleiades: Rebull et al. (2016a) performed a comprehensive lightcurve analysis of the stars observed by K2 in regions out to ∼350' from the cluster center. Rebull et al. (2016b) expanded on this by meticulously classifying and identifying the different structures present in the periodograms and lightcurves, and Stauffer et al. (2016) then used this to study the Pleiades in the context of angular momentum evolution. The sample of candidate members used by these studies comprises 759 stars with measured periods that were classified as Best+OK members by Rebull et al. (2016a) on the basis of proper motions and position in the CMD selections. We take this sample as our nominal Pleiades data set. We note that, although the K2 observations missed part of the northern region of the cluster, this is unlikely to bias the period distribution (Rebull et al. 2016a).
• 2.  
  Praesepe: a study similar to that of the Pleiades was carried out for Praesepe by Rebull et al. (2017). In this case, the candidate members extend out to ∼400' from the cluster center, and they were selected on the basis of proper motions and CMD position. The final Rebull et al. (2017) sample comprises 809 candidate members with measured periods.
• 3.  
  NGC 6811: Meibom et al. (2011a) studied candidates that were previously vetted using radial velocity (RV) data in a region out to 30' from the cluster center, and reported periods for 71 stars. Curtis et al. (2019a) surveyed a region of 60' radius, selected candidate members on the basis of Gaia CMD position and astrometry, and reported periods for 171 stars. We complement these with the Santos et al. (2019, 2021) Kepler field catalogs, which reported periods for 194 NGC 6811 candidate members. To maximize the number of stars with measured periods, we combine all of these catalogs and end up with 235 stars after accounting for repetitions. For stars in common among the references, we prioritize the Santos et al. (2019) and Santos et al. (2021) periods first, Curtis et al. (2019a) second, and Meibom et al. (2011a) third (although a comparison of these values showed excellent agreement). While the periods of our joint NGC 6811 catalog are derived from slightly different techniques, the studies carried out by these four references are all based on the same Kepler data.

In Section 3.1.1, we describe the Gaia data; in Section 3.1.2, we use them to characterize the cluster and the field in parallax and proper motion space by fitting a model that allows for intrinsic dispersions; and in Section 3.1.3, we classify the stars in different categories and calculate memberships probabilities, identifying highly likely cluster members.

3.1.1. Gaia DR2 Data

3.1.2. Cluster Fitting

Our method is similar to membership probability studies found in the literature, and its goal is to separate the cluster population from the field population in phase space. For instance, Vasilevskis et al. (1958) and Sanders (1971) used a 2D version of this method using proper motions to compute memberships for a number of clusters. Other studies that have used similar versions of this method include Francic (1989), Jones & Walker (1988), Jones & Stauffer (1991), and Jones & Prosser (1996). A 3D version including parallaxes as well as individual star measurement errors and correlations, in addition to allowing for intrinsic dispersions, has been recently used by Franciosini et al. (2018) and Roccatagliata et al. (2018, 2020).

In our method, a given star i has a probability of belonging to either population P (field F or cluster C), with the total likelihood being:

$$\begin{eqnarray}&&{{ \mathcal L }}_{i}={f}_{F}{{ \mathcal L }}_{F,i}+{f}_{C}{{ \mathcal L }}_{C,i},\end{eqnarray} \tag{ 1 }$$

where f_F and f_C represent the fraction of stars that belong to the field and cluster (normalized such that f_C = 1 − f_F ), and ${{ \mathcal L }}_{F,i}$ and ${{ \mathcal L }}_{C,i}$ are the likelihoods of each population, respectively. We assume that the likelihood of each population P can be described by a 3D multivariate Gaussian:

$$\begin{eqnarray}&&{{ \mathcal L }}_{P,i}=\displaystyle \frac{1}{{\left(2\pi \right)}^{3/2}| {{\rm{\Sigma }}}_{i}{| }^{1/2}}\exp \left[-\displaystyle \frac{1}{2}{\left({{\boldsymbol{x}}}_{i}-{{\boldsymbol{x}}}_{P}\right)}^{T}{{\rm{\Sigma }}}_{i}^{-1}({{\boldsymbol{x}}}_{i}-{{\boldsymbol{x}}}_{P})\right],\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&{{\boldsymbol{x}}}_{i}-{{\boldsymbol{x}}}_{P}=\left(\begin{array}{c}{\varpi }_{i}-{\varpi }_{P}\\ {\mu }_{{\alpha }_{i}}-{\mu }_{{\alpha }_{P}}\\ {\mu }_{{\delta }_{i}}-{\mu }_{{\delta }_{P}}\end{array}\right)\end{eqnarray} \tag{ 3 }$$

and Σ_i = C_i + Σ_P , where C_i is the individual covariance matrix, and Σ_P is the matrix of intrinsic dispersions:

$$\begin{eqnarray}{C}_{i}=\left(\begin{array}{ccc}{\sigma }_{{\varpi }_{i}}^{2} & {\rho }_{\varpi {\mu }_{\alpha },i}{\sigma }_{{\varpi }_{i}}{\sigma }_{{\mu }_{{\alpha }_{i}}} & {\rho }_{\varpi {\mu }_{\delta },i}{\sigma }_{{\varpi }_{i}}{\sigma }_{{\mu }_{{\delta }_{i}}}\\ {\rho }_{\varpi {\mu }_{\alpha },i}{\sigma }_{{\varpi }_{i}}{\sigma }_{{\mu }_{{\alpha }_{i}}} & {\sigma }_{{\mu }_{{\alpha }_{i}}}^{2} & {\rho }_{{\mu }_{\alpha }{\mu }_{\delta },i}{\sigma }_{{\mu }_{{\alpha }_{i}}}{\sigma }_{{\mu }_{{\delta }_{i}}}\\ {\rho }_{\varpi {\mu }_{\delta },i}{\sigma }_{{\varpi }_{i}}{\sigma }_{{\mu }_{{\delta }_{i}}} & {\rho }_{{\mu }_{\alpha }{\mu }_{\delta },i}{\sigma }_{{\mu }_{{\alpha }_{i}}}{\sigma }_{{\mu }_{{\delta }_{i}}} & {\sigma }_{{\mu }_{{\delta }_{i}}}^{2}\end{array}\right),\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}{{\rm{\Sigma }}}_{P}=\left(\begin{array}{ccc}{\sigma }_{{\varpi }_{P}}^{2} & 0 & 0\\ 0 & {\sigma }_{{\mu }_{{\alpha }_{P}}}^{2} & 0\\ 0 & 0 & {\sigma }_{{\mu }_{{\delta }_{P}}}^{2}\end{array}\right).\end{eqnarray} \tag{ 5 }$$

By using these equations on the Gaia data, we can derive the cluster and field parameters using a maximum likelihood fit.

To accurately derive these parameters, we apply a set of quality and geometric cuts. We highlight that these cuts are applied only when defining the subset of stars we use to derive the cluster and field astrometric parameters, but in Section 3.1.3 we apply our membership classification to the full Gaia sample (i.e., the stars excluded during this exercise are still classified later on). We run our maximum likelihood calculation using the subset of stars with astrometric_excess_noise = 0 and apparent G < 18 mag. In this way, we only fit the stars with well-defined astrometric solutions, and derive representative cluster parameters that are not affected by uncertain measurements of individual faint stars. We additionally note that the photometric monitoring survey that provides rotation period information searched for candidate members far in the outskirts of the cluster. In our cluster parameter calculation, however, we perform our fit using a subset of stars located closer to the cluster center (but not so close that the intrinsic dispersions we derive are affected by this decision).

We use Python's minimize function (method = SLSQP) to run our maximum likelihood calculation and derive the cluster and field parameters (i.e., ϖ_C , ${\mu }_{{\alpha }_{C}}$, ${\mu }_{{\delta }_{C}}$, ${\sigma }_{{\varpi }_{C}}$, ${\sigma }_{{\mu }_{{\alpha }_{C}}}$, ${\sigma }_{{\mu }_{{\delta }_{C}}}$ for the cluster, and the analogous set of parameters for the field population). We report the calculated cluster parameters in Appendix B. All in all, although our algorithm to derive cluster parameters could have a higher degree of complexity (e.g., by calculating the parameters in a magnitude-dependent way), the membership classification presented in what follows is robust.

3.1.3. Membership Probability Calculation

Now that we have calculated the cluster and field parameters, we seek to evaluate Equation (1) for individual stars and calculate membership probabilities. First, however, the availability and quality of the Gaia astrometry need to be considered, as the data are not of equal precision for all stars, and they strongly depend on the target brightness. For instance, typical parallax (proper motion) uncertainties are of order ∼0.05 mas (0.05 mas yr⁻¹) for a G = 16 mag star, and of order ∼0.5 mas (1.5 mas yr⁻¹) for a G = 20 mag star.

Additionally, the presence of binary stars needs to be considered. Binary stars with measured periods are interesting targets for rotation studies (e.g., Stauffer et al. 2018; Tokovinin & Briceño 2018), and can provide important clues when investigating rotation in stellar populations (e.g., Simonian et al. 2019, 2020). Because of this, and to not bias our rotational sequences, we purposely avoid discarding binary stars with our astrometric quality cuts.

We proceed as follows. From the full Gaia sample, we select the stars with available positions, proper motions, and parallax values. Additionally, following Gaia Collaboration et al. (2018a), we select stars with visibility_periods_used > 8 and with $\sqrt{{\chi }^{2}/(\nu ^{\prime} -5)}\lt 1.2\max (1,\exp (-0.2(G\,-19.5)))$, where χ²≡astrometric_chi2_al and ν' ≡ astrometric_n_good_obs_al. The latter of these quality cuts removes most of the artifacts while retaining genuine binaries (where naturally the single-star solution does not provide a perfect fit; e.g., see Belokurov et al. 2020).

The stars excluded by these quality cuts do not have enough information for a reliable membership probability to be calculated. Nonetheless, we do not remove them from our sample, as some of them could correspond to stars with measured rotation periods. We assign them the classification flag of no info stars and include them in the NGC 2547 rotational sequence shown in Section 5.

The stars that survive these astrometric quality cuts have enough information such that we can perform a thorough membership classification. At this point, we separate stars that could be cluster members from those that are most likely not members. For a given star i, we calculate the quantity:

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{i}=\displaystyle \frac{1}{\sqrt{3}}\sqrt{\displaystyle \frac{{\left({\varpi }_{C}-{\varpi }_{i}\right)}^{2}}{({\sigma }_{{\varpi }_{C}}^{2}+{\sigma }_{{\varpi }_{i}}^{2})}+\displaystyle \frac{{\left({\mu }_{{\alpha }_{C}}-{\mu }_{{\alpha }_{i}}\right)}^{2}}{({\sigma }_{{\mu }_{{\alpha }_{C}}}^{2}+{\sigma }_{{\mu }_{{\alpha }_{i}}}^{2})}+\displaystyle \frac{{\left({\mu }_{{\delta }_{C}}-{\mu }_{{\delta }_{i}}\right)}^{2}}{({\sigma }_{{\mu }_{{\delta }_{C}}}^{2}+{\sigma }_{{\mu }_{{\delta }_{i}}}^{2})}},\end{eqnarray} \tag{ 6 }$$

where values with the C index represent the cluster parameters derived in Section 3.1.2. We classify the stars with values of Δ_i > 3 as non members, as they are located outside the cluster's 3σ ellipsoid in phase space.

All of the remaining unclassified stars have Δ_i ≤ 3 and are technically consistent with the cluster phase-space parameters (at the 3σ level). This population, however, has a contribution from objects that have large astrometric uncertainties (particularly in parallax), for which a reliable classification is hard to determine with the existing data. Therefore, another selection criterion is required in order to separate the likely cluster members from this ambiguous population. For this purpose, we again use the method described in Section 3.1.2, and calculate membership probabilities for all objects with Δ_i ≤ 3. Following Equation (1), for a given star i, the cluster membership probability is calculated as

$$\begin{eqnarray}&&{P}_{C,i}=\displaystyle \frac{{f}_{C}{{ \mathcal L }}_{C,i}}{{{ \mathcal L }}_{i}}.\end{eqnarray} \tag{ 7 }$$

Conversely, this corresponds to 1 − P_F,i , where P_F,i is the field membership probability for the star i. The distribution of cluster membership probabilities for all stars with Δ_i ≤ 3 in the NGC 2547 field is shown in Figure 1.

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For NGC 2547, we find the distribution of cluster membership probabilities to be strongly bimodal, with most of the sample having probabilities close to either 0 or 1. Although not explicitly shown, the membership probability distribution has a strong dependence on the stars' apparent G-band magnitude, with the distribution being mostly bimodal for bright stars and becoming blurrier for fainter stars (with increasingly more stars having intermediate probability values; e.g., see Figure 5 of Jones & Prosser 1996). This magnitude-dependent outcome is typical, as bright stars tend to have more precise astrometric measurements, and the method provides a yes or no answer regarding their memberships. On the other hand, faint stars typically have larger astrometric uncertainties, which only allow the method to provide an intermediate answer.

With this in mind, we apply a final selection criterion and classify the stars with P_C ≥ 0.90 as probable members and those with P_C < 0.90 as possible members. We show the resulting phase-space projections and apparent CMD of these populations, in addition to the non member population, in Figures 2 and 3. (The no info population is absent from these figures, as no astrometric information is available for those stars.)

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We choose the probability threshold of P_min = 0.90 to separate between probable and possible members as a compromise between completeness in our sample of probable cluster members on the one hand, and the presence of contamination on the other. Several tests with different P_min values showed that the phase space and CMD of the probable members seemed to lose bona fide cluster stars when using higher P_min values, while lower values included stars with incoherent kinematics and photometry. We nonetheless note that the overall number of probable and possible members do not considerably change unless very high (P_min ≳ 0.95) or very low (P_min ≲ 0.05) threshold values are used (see Figure 1). Furthermore, Appendix C reports the Gaia data for the probable and possible members and their cluster membership probabilities, and the reader is free to experiment with customized threshold values.

Figure 2 shows that the probable members (blue points) tend to be clustered in phase space with a small number of stars located in the outskirts. On the other hand, the non members (i.e., the field; red points) occupy a much larger region in the phase-space projections (with stars populating virtually all corners of the diagrams), but tend to be concentrated at smaller parallax and proper motion values. The possible members (cyan points) appear as an intermediate population of stars, located in between the cluster and the field.

Figure 3 complements what is seen in Figure 2. The probable members form a tight isochrone-like sequence in the apparent CMD (with the exception of the color turnover for apparent G ≳ 18.5 mag; see discussion below), with even a parallel sequence being visible indicating the presence of photometric binaries (e.g., Hurley & Tout 1998). The non members behave as expected from the field, with the two main branches (main-sequence and giant branch) being clearly visible in the data. The possible members mainly populate the bottom part of the plot, demonstrating that this population is dominated by faint stars that, consequently, have large astrometric uncertainties, which in turn causes them to have Δ_i ≤ 3 values (differentiating them from the non members). While some of these possible members could very well be real cluster members, the current precision of their astrometric information does not allow for a more stringent classification.

We highlight that, even though our classification method is completely agnostic to the photometry of the stars, and it is solely based on kinematics, it seems to appropriately select stars that form a main sequence in the CMD. We take this as a confirmation that our method is properly identifying likely cluster members and distinguishing them from the field.

A peculiar feature can be seen in Figure 3 for apparent G ≳ 18.5 mag. At this apparent magnitude, instead of continuing to the bottom right part of the diagram, the probable members start to turn to bluer colors for fainter magnitudes. This feature has already been found by other works when using the Gaia DR2 photometry (e.g., Arenou et al. 2018; Lodieu et al. 2019a, 2019b; Smart et al. 2019), and it arises from an overestimation of the flux in the G_BP passband for faint, red sources (Riello et al. 2018, 2021).

Now that we have kinematically classified every Gaia DR2 star in the field of NGC 2547 as either a probable, possible, or non member, or else a no info star, we proceed to connect this with the rotation sample of Irwin et al. (2008).

We crossmatch both catalogs using the CDS X-Match Service on VizieR. ⁶ First, we download all the Gaia DR2 matches to a given NGC 2547 star within 3''. This produces matches for all stars, with some of them (∼10%) being matched to two Gaia sources. The distribution of angular separations is strongly peaked at small values, with ∼80% of the stars having a match within 02, a tail extending out to 05, and some matches extending past this value out to 3''.

To select the best possible matches for stars matched to two Gaia sources, we compare the V magnitudes from Irwin et al. (2008) with the G magnitudes from Gaia. This exercise helps us break the ties and decide which star is the correct match. In all cases, the Gaia star with the more similar brightness corresponds to the closer match in angular separation. The second matched star is typically several magnitudes fainter, with angular separations greater than 1'' (the outliers of the separation distribution). We only keep one Gaia match per star, and are left with an angular separation distribution where ∼95% of the stars have a match within 03 and all of them have a match within 09.

In order to test the reliability of our crossmatching approach, we use the Pleiades cluster as a benchmark for which a completely independent crossmatching is available. We first replicate the same crossmatching procedure used in NGC 2547 with our Pleiades data. We then take the 2MASS IDs for the Pleiades stars from Rebull et al. (2016a) and look them up in the precomputed 2MASS–Gaia DR2 crossmatch (gaiadr2.tmass_best_neighbor) that can be found in the ESA's webpage of the Gaia Archive. ⁷ We find an excellent agreement (>99.7% of coincidence) between ESA's crossmatch and ours for the stars found by both methods (and moreover, VizieR produces matches for 10 Pleiades stars not found by ESA). We take this as a confirmation that our crossmatching is properly combining the periodic samples with the Gaia sources.

We follow the method described in Section 3.1 for NGC 2547, and apply it to the other six clusters listed in Table 1. We report the astrometric parameters we calculate for the clusters in Appendix B, where we also compare them with the values reported in the literature. We find our astrometric parameters to be in good agreement with those reported by other studies, with fractional differences being typically at the ∼1%–2% level or less.

For each cluster, we classify every Gaia star contained within the region where the corresponding rotation period catalog reports candidate members into one of the four previously described categories (no info, probable, possible, or non member). We report tables with the corresponding list of probable and possible members for every cluster in Appendix C. Analogously to Figures 1 and 3 for NGC 2547, Figures 4 and 5 show the membership probability distribution and apparent CMDs for all the clusters we study. The phase-space projections for all clusters, analogous to Figure 2 for NGC 2547, can be found in Appendix D.

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Figure 4 shows that, for every cluster, in the sample of Gaia stars that satisfy Δ_i ≤ 3, there is a population of stars with membership probabilities close to 1. This population represents a set of likely cluster members, and we classify those with P_C ≥ 0.90 as probable members (and the rest as possible members). On the low-probability end, however, clear cluster-by-cluster differences can be observed. While clusters such as the Pleiades and Praesepe show only a small number of possible members, M50 and NGC 6811 are in the opposite situation. These differences are explained by considering how similar or different the astrometric parameters of a given cluster are, compared with the field (which is typically a diffuse background population with small parallax and low proper motion). In other words, a cluster with a large parallax and proper motion (e.g., the Pleiades or Praesepe) will be more distinct and separated from the field in phase space, and therefore it is highly unlikely that unassociated field stars will have similar astrometry (i.e., Δ_i ≤ 3), and hence a small number of possible members is expected. On the other hand, field stars are more likely to mimic the astrometry of a cluster that has a small parallax and a low proper motion (e.g., M50 or NGC 6811), and therefore the number of stars that are kinematically consistent with the cluster increases.

The apparent CMDs of all our clusters, shown in Figure 5, display an isochrone-like sequence of probable members, with clear binary sequences being identified in all of them. The older clusters M37, Praesepe, and NGC 6811 show not only a main-sequence population but also a few giant stars (with Praesepe even showing a clear white dwarf sequence). We highlight that, in order to obtain unbiased cluster memberships, our method is completely agnostic to the Gaia photometry, and these results are a consequence of our careful astrometric analysis. Additionally, as previously found for NGC 2547, we observe the turnover to bluer G_BP − G_RP colors at apparent magnitudes G ≳ 18.5 mag in the probable member sequence of all the clusters. We take this as further evidence that this corresponds to an artifact of the Gaia photometry (see Section 3.1.3 for details).

As discussed in Section 3.1.3, our ability to cleanly separate between possible and probable members is a strong function of the stars' apparent G-band magnitude, as the quality of their astrometry decreases with decreasing brightness. This can be clearly seen, for instance, in the CMD of NGC 6811, where all the stars fainter than apparent G ≃19 mag are classified as either possible or non members, and none of them are classified as probable members. Real NGC 6811 members with G > 19 mag are simply too faint for Gaia DR2 to provide precise astrometry, preventing us from reliably differentiating them from the field stars.

The crossmatching between the periodic samples and the Gaia DR2 data is done in identical fashion for all of the clusters, following the approach of Section 3.2. Unlike the case of NGC 2547, however, we do not find a Gaia DR2 counterpart for every periodic star in most of the other clusters. Our recovery rates (and number of stars not found in the crossmatching over the number of stars with measured periods) are: 100% (0/176) for NGC 2547, 99.9% (1/759) for Pleiades, 94.3% (46/812) for M50, 99.4% (2/364) for NGC 2516, 96.7% (12/367) for M37, 99.8% (2/809) for Praesepe, and 100% (0/232) for NGC 6811. The only clusters with recovery rates below 99% are M50 and M37, which correspond to distant systems where the periodic samples extend to magnitudes fainter than the apparent G ≈ 21 mag limit of Gaia DR2. In the following, we classify the stars not found in the crossmatching as no info stars, as their membership is uncertain and we do not have evidence to discard them.

To perform meaningful intercluster comparisons of stellar rotation, reliable cluster properties are needed. In the pre-Gaia era, classic works such as Denissenkov et al. (2010), Gallet & Bouvier (2013), and Gallet & Bouvier (2015) thoroughly compiled periodic samples as well as properties for a large number of clusters. Nevertheless, the heterogeneous nature of their parameter compilation (adopting values from varied techniques and data sets) can result in inhomogeneities in the adopted masses, ages, and metallicity scales. Nowadays, the systematic observations of star clusters by spectroscopic surveys (Netopil et al. 2016; Gaia-ESO, Magrini et al. 2017; Spina et al. 2017; APOGEE, Majewski et al. 2017; Jönsson et al. 2020), as well as the uniform all-sky Gaia data, provide an opportunity to study these systems in a more consistent and homogeneous fashion. In the following, we attempt, to the extent that it is possible, to compile and derive cluster properties that are in a consistent and uniform scale.

The mass and temperature values we calculate in Section 4.2, as well as the results and discussion presented in Sections 5 and 6, require four cluster properties to be known: distance modulus, reddening, age, and metallicity. The former two are needed to translate the observed photometry to absolute and dereddened CMD space, while the latter two are required to compare the data with the appropriate stellar evolutionary model. For distance modulus, we adopt the values derived from our astrometric analysis (considering the parallax zero points described in Section 3.1.1), and for metallicity, we compile values reported from recent high-resolution spectroscopic results. In particular, we adopt the state-of-the-art values reported by Netopil et al. (2016) for Pleiades, NGC 2516, M37, Praesepe, and NGC 6811, and the value reported by the Gaia-ESO survey (Spina et al. 2017) for NGC 2547. The only cluster without a spectroscopic metallicity measurement is M50, for which we adopt a solar value. We now describe our procedure to obtain cluster reddenings and ages.

Instead of adopting reddening values from the literature, we calculate our own using a dedicated fitting routine. For a given cluster, we start by correcting the observed G-band magnitudes by distance modulus. Once the spectroscopic metallicity is known, we download a set of PARSEC models ⁸ (Bressan et al. 2012; Chen et al. 2014) of that composition with a range of plausible ages (e.g., 100, 200, and 300 Myr for NGC 2516). Then, we define a region in the absolute CMD where the different models agree with each other, which excludes the phases near and more evolved than the turnoff. We then iterate over a range of possible reddenings and find the value that produces the best agreement between the observed data with the series of models.

For a given E(B − V) value, we assume A_V = 3.1 × E(B − V) (Cardelli et al. 1989) and deredden the photometry as follows. For stars with measured photometry in the three Gaia bands, we deredden the observed magnitudes following the approach of Gaia Collaboration et al. (2018a), where the extinction coefficients depend on G_BP − G_RP color and the extinction itself. For stars that lack a measured G_BP − G_RP color, we adopt the PARSEC coefficients ⁹ for a G2V star (A_G /A_V = 0.85926, ${A}_{{G}_{\mathrm{BP}}}/{A}_{V}=1.06794$, and ${A}_{{G}_{\mathrm{RP}}}/{A}_{V}=0.65199$).

For every E(B − V) value we are iterating over, we deredden the probable cluster members' photometry, define a set of bins in the ${({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}})}_{0}$ color coordinate, and calculate the mean colors and 75th magnitude percentiles (which accounts for the photometric binaries). We then find the E(B − V) value that minimizes the sum of the squared difference of the data minus the models given the bins, and adopt this as our global cluster reddening. Finally, we compare the CMD location of the full sample of probable members with models of varying ages, and estimate the age following the envelope of hot stars below and near the turnoff (typically stars hotter than ∼7000 K). We estimate that this procedure leads to typical errors of ∼20% for reddenings and ∼10%–20% for ages.

We summarize the revised cluster properties we adopt in Table 2. Note that, for the nearby and young clusters NGC 2547 and the Pleiades, we adopt the lithium depletion ages from Jeffries & Oliveira (2005) (see also Naylor & Jeffries 2006) and Stauffer et al. (1998), respectively. This technique yields reliable and almost completely model-independent ages in the 10–200 Myr range. These adopted ages are in good agreement with the values we derive from our CMD analysis.

4.1.1. Comparing Our Cluster Properties with the Literature

In order to perform comprehensive comparisons of the rotational sequences of the open clusters, we need to place them on a common scale. For this, we derive mass and temperature values for the cluster members by comparing the photometric data with state-of-the-art evolutionary models in absolute magnitude space. Although the papers that reported periods also reported photometry in at least two bands for most of our clusters (e.g., V and I for NGC 2547; see Section 2), the specific filters vary on a cluster-by-cluster basis. Instead of using these, we take advantage of the uniform G, G_BP, and G_RP photometry provided by Gaia DR2. This also allows us to derive masses and temperatures for all the Gaia cluster members, not just the subset with measured periods.

We again illustrate our procedure using NGC 2547 as a working example. We use its E(B − V) and distance modulus values (see Section 4.1) and calculate absolute and dereddened photometry. We show the extinction-corrected absolute CMD of the probable and possible NGC 2547 members in Figure 6. The data can now be directly compared with stellar models to infer mass and temperature values. We use a PARSEC isochrone with the corresponding NGC 2547 age and metallicity, and show it as the orange line in Figure 6.

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The bright probable cluster members show an excellent agreement with the model in Figure 6. As discussed in Figure 3, the G_BP − G_RP color misbehaves for apparent G ≳ 18.5 mag, and the probable member sequence becomes bluer for fainter magnitudes. This limit is translated to the absolute and dereddened CMD and shown as the horizontal dashed line in Figure 6, with the dotted line showing a similar apparent G = 17.5 mag limit. We use these two apparent magnitude limits, here translated to absolute magnitude space, to define different regimes in our mass and temperature calculation.

Additionally, whenever possible, the effects of binarity on the CMD need to be considered. Photometric binaries appear as brighter/redder stars compared to the main-sequence locus, and to account for them we follow the procedures of Stauffer et al. (2016) and Somers et al. (2017). In this approach, when the Gaia ${({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}})}_{0}$ color is reliable, stars are projected down (or up) onto the main-sequence locus at fixed color, effectively removing the contribution of close companions (i.e., our mass and temperature estimates for these stars correspond to those of the primary star). We designate the projected absolute magnitudes as ${M}_{{G}_{0,\mathrm{proj}}}$, and describe its calculation in detail below.

To acknowledge the misbehavior of the G_BP − G_RP color and account for the contribution of photometric binaries when possible, we define three regimes to calculate projected magnitudes ${M}_{{G}_{0,\mathrm{proj}}}$ in the absolute CMD of Figure 6:

• 1.  
  Bright regime: For stars brighter than the dotted line (apparent G = 17.5 mag translated to absolute magnitude), we calculate ${M}_{{G}_{0,\mathrm{proj}}}$ by projecting the stars ${M}_{{G}_{0}}$ value down (or up) onto the model at the measured ${({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}})}_{0}$ color.
• 2.  
  Faint regime: For stars fainter than the dashed line (apparent G = 18.5 mag translated to absolute magnitude), or stars that lack a G_BP or G_RP magnitude, we cannot rely on the ${({G}_{\mathrm{BP}}-{G}_{\mathrm{RP}})}_{0}$ color to do a deprojection. Instead, we simply keep the calculated ${M}_{{G}_{0}}$ value as it is, and define ${M}_{{G}_{0,\mathrm{proj}}}$ to be equal to it. This is therefore a regime where we do not account for unresolved companions.
• 3.  
  Intermediate regime: For stars in between the dotted and dashed lines (17.5 ≤ G ≤ 18.5 mag range translated to absolute magnitude), we do an intermediate, ramp-like calculation. We calculate the values we would have obtained following the bright and faint regimes separately, and combine them in a progressive way with linear weighting. In this way, the extremes match the corresponding methods, effectively avoiding a sharp transition between the different regimes.

For every star, this procedure collapses the observed photometry to the single quantity ${M}_{{G}_{0,\mathrm{proj}}}$. We use this value to interpolate the corresponding NGC 2547 PARSEC model in the magnitude–mass and magnitude–temperature space. We only interpolate in the regime of stars fainter than the main-sequence turnoff (i.e., we do not report masses for evolved stars) and down to the faintest ${M}_{{G}_{0}}$ magnitude allowed by the model. We report our derived masses and temperatures for the NGC 2547 probable and possible members in Appendix C.

For the subset of NGC 2547 stars with period measurements, Irwin et al. (2008) calculated masses using their I-band magnitudes and the NextGen models (Baraffe et al. 1998). These values offer a completely independent reference point to validate our own masses, and we compare our estimates with theirs in Figure 7. It is important to highlight one key difference between both methods: we account for the presence of photometric binaries (when possible), while Irwin et al. (2008) do not. Accordingly, we find the mass difference to be a strong function of mass itself. For stars in the faint regime, where we do not account for binaries, we see a good agreement but with an approximately constant offset of ΔMass ≈−0.05M_⊙ that is likely due to the different underlying models used in the interpolation. For higher masses, the scatter increases and many stars have positive mass differences, indicating cases where Irwin et al. (2008) did not account for the contribution of unresolved companions. Overall, we find a good agreement, with mean and standard deviation values of ΔMass being ≈0.01 and 0.08 M_⊙, respectively.

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The above comparison validates our method and suggests that masses derived in this way are subject to a systematic uncertainty of order ∼0.05 M_⊙. Statistical uncertainties are of order ∼0.01–0.02 M_⊙, considerably smaller than the systematic ones. More importantly, by using the Gaia photometry, we have defined a procedure to calculate the masses and temperatures that can be uniformly applied to all our clusters.

Analogously to Figure 6 for NGC 2547, Figure 8 shows the extinction-corrected absolute CMD for the probable and possible members of all the clusters. Some discrepancies arise for low-mass stars, where the probable members appear brighter/redder than the model (e.g., see the ∼ 0.4M_⊙ stars in Pleiades and Praesepe). As other studies suggest, these differences could arise due to the known radius inflation problem in cool dwarfs (e.g., Torres & Ribas 2002; Clausen et al. 2009; Torres et al. 2010; Kraus et al. 2011; Somers & Stassun 2017; Jackson et al. 2018, 2019). Accounting for the underlying physical process that causes this in the models is beyond the scope of this paper, but we refer interested readers to the works by Somers & Pinsonneault (2015) and Somers et al. (2020). Regardless, while evidently not perfect, Figure 8 shows a good overall agreement of the data with models across the HR diagram for all the clusters.

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We follow the procedure described for NGC 2547, and use the absolute CMDs to calculate masses and temperatures for the probable and possible cluster members. We report these values in Appendix C. For the subset of stars with independent mass and temperature estimates from other studies, we compare these values with ours in Appendix E. Globally, we find our masses and temperatures to be in good agreement with those reported by the period references. The comparisons suggest possible systematic uncertainties of ∼0.05–0.1M_⊙ in mass and ∼150 K in temperature, which are modest considering the different methods, photometric data, and underlying stellar models employed.

At this point, for every cluster in our sample, we have classified the periodic stars using our astrometric analysis (Section 3), presented a set of revised properties that are needed to perform meaningful inter-cluster comparisons (Section 4), and discussed the effects that removing the field contamination has on the individual rotational sequences (Section 5). Now, we combine all of these to construct an updated portrait of the evolution of stellar rotation.

We illustrate this in Figure 10, where we show the revised period versus mass diagrams as a function of age. The color-coding is the same as the revised sequences of Figure 9, and the non member contaminants have been removed. This represents the state of the art for rotation studies in terms of clean samples with stellar masses, temperatures, and ages derived in a consistent scale. We make these data publicly available in Appendix F.

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One of the major results of our study can be seen in the rotational sequence of M50 in Figure 10. When the non member contamination is removed, the revised M50 sequence clearly resembles that of the Pleiades, as we would expect given their similar ages (150 and 125 Myr, respectively). Incidentally, the low-mass end of the M50 sequence coincides with the high-mass end of the co-eval NGC 2516 sequence (≈0.65 M_⊙). This is illustrated in their respective panels, where we show the M50 sequence in blue and green while the NGC 2516 sequence is shown in gray, and vice versa. Given their indistinguishable ages, the merged M50 and NGC 2516 data can provide a valuable comparison point for the benchmark Pleiades cluster. Likewise, a similar comparison can be made for the older M37 and Praesepe clusters. Ultimately, the above demonstrates a decisive finding: when careful membership analysis are performed, the rotational sequences that have been constructed from ground-based observations (e.g., M50 by Irwin et al. 2009) can be as informative for stellar rotation studies as those constructed from space-based observations (e.g., the Pleiades by Rebull et al. 2016a). In this context, future dedicated ground-based monitoring combined with careful astrometric selections using Gaia could provide unprecedented constraints for angular momentum evolution (e.g., Curtis et al. 2020). This conclusion is particularly relevant in the post Kepler and K2 era, and considering that only a fraction of the TESS targets are being observed with long baselines.

Regarding the field contaminants, our membership analysis yields varying degrees of contamination rates. As mentioned in Section 2, we expected the clusters observed from the ground to show higher contamination rates compared to those observed from space. This turns out to be the case for NGC 2547 and M50, where we find the contamination to be ≈25% and 36%, respectively. While high, these values are lower than the ∼40%–60% rates anticipated by Irwin et al. (2008, 2009), which were based on predictions from Galactic models. For the rest of the sample, we find the contamination rates to be lower, on the order of ≲5% (see Section 5).

Interestingly, although our membership study has predominantly removed rotational outliers at both long and short periods for several clusters, many outliers are nonetheless classified as probable members and remain in the revised sequences. An interesting example of this is provided by the slowly rotating stars (periods ≈ 7–10 days) in the young NGC 2547 (age of ∼35 Myr), hinting at strongly mass-dependent initial conditions for stellar rotation (see also Somers et al. 2017; Rebull et al. 2018, 2020). Similarly, all three ≳0.5 Gyr old clusters show confirmed members that are rotating faster (∼1 day) and slower (∼20 days) than their slowly rotating branches. Now that the membership statuses of these outliers have been confirmed, their presence can no longer be ignored or attributed to field contaminants.

The rapid rotators in systems with a converged sequence cannot be explained by single star evolution in current models. The most plausible explanation is that they have experienced tidal synchronization or are merger products. Regarding the slowly rotating outliers, a quick inspection reveals that most appear as typical cluster members in astrometric and photometric regards, but a fraction of them are photometric binaries in the CMDs. ¹⁰ We suspect that they could correspond to the long-period end of the distribution of tidally synchronized binaries (e.g., see Lurie et al. 2017). Another explanation could be that they were born with unusually low angular momentum, but given that the young Pleiades sequence does not show many of these stars, we find this hypothesis less likely. Ultimately, unless explained by modern theories of angular momentum evolution, these outliers could potentially weaken the applicability of gyrochronology in field stars.

We now quantify the trends that can be obtained from the revised sequences of Figure 10. We display this in three different approaches, and these are shown in Figures 11–13. Additionally, we calculate the percentiles of the rotational distributions of our clusters in different mass bins, and we make them publicly available in Appendix G.

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Figure 11 shows, for each cluster in our sample, the period versus mass diagram of the median rotators in 0.1M_⊙ bins. This allows us to simultaneously analyze the mass and age dependence of stellar rotation. For stars with masses below 0.5M_⊙, the median rotation periods for the young NGC 2547 (35 Myr) are longer than those seen in the older Pleiades and NGC 2516 clusters (125 and 150 Myr, respectively). This is consistent with expectations, considering that stars of these masses take a few hundred million years to reach the ZAMS, and are therefore still contracting by the age of NGC 2547. By the age of Praesepe (700 Myr), these stars have spun down in a strongly mass-dependent fashion, with lowest masses still showing periods below 1 day, and ∼0.4M_⊙ stars rotating at ≳10 days.

The stars more massive than 0.5M_⊙, on the other hand, show a monotonic behavior with time in most of the age range probed by our clusters. They consistently spin down as they age from 35 to 700 Myr, and their median rotator trends in Figure 11 do not intersect each other, in agreement with expectations from a Skumanich-type spin-down (i.e., period ∝ age^1/2). The notable exception to this, however, is the comparison of Praesepe and NGC 6811. For these two clusters, their rotational sequences do not seem to be simple translations of each other to longer or shorter periods. Although their age difference is non-negligible (∼250 Myr), their median rotators are virtually overlapping at ∼1.1 M_⊙ and in the 0.8–0.6M_⊙ range. The latter of these has already been noted by Curtis et al. (2019a) (see also Meibom et al. 2011a), and a similar overlapping was reported by Agüeros et al. (2018) for a smaller sample in the 1.4 Gyr old cluster NGC 752. In terms of the interpretation of this feature, Agüeros et al. (2018) and Curtis et al. (2019a) have formulated it as a temporary epoch of stalling in the spin-down of K dwarfs, and Spada & Lanzafame (2020) have proposed that it arises from the competing effect between magnetic braking and a strongly mass-dependent internal redistribution of angular momentum. We leave a detailed examination of this as future work, but given the considerable difference in metallicity between both clusters ([Fe/H] = +0.16 dex for Praesepe and +0.03 dex for NGC 6811; Netopil et al. 2016), we highlight the importance of incorporating chemical composition in comprehensive models of angular momentum evolution (e.g., Amard & Matt 2020; Amard et al. 2020; Claytor et al. 2020).

Figure 11 can also be used to remark upon global properties of stellar rotation. First, regardless of the age, no sharp transition is seen in the rotation periods near the boundary between partially convective and fully convective stars (∼0.35M_⊙). Second, as illustrated by NGC 2547 in our sample, and by the young Upper Scorpius and Taurus associations in Somers et al. (2017) and Rebull et al. (2018, 2020), the initial conditions of stellar rotation are strongly mass-dependent and need to be accounted for in models of angular momentum evolution. Third, when considering the young ages probed by our sample (∼35–200 Myr), we note that the stars in the 1.0–0.6M_⊙ range populate a global maximum in terms of rotation periods. In other words, the median rotators in this mass interval never rotate more rapidly than ∼3–4 days. Given the intimate connection between stellar rotation and activity (e.g., Wright et al. 2011, 2018; Lehtinen et al. 2020), where more rapidly rotating stars tend to expose their planets to higher levels of potentially harmful radiation, this feature could provide an optimal window in the search for habitable worlds.

In Figure 12, we show the rotational sequences for the clusters we study, but in this case separating the data into different percentiles of rotation. For each cluster, across the mass range where they have period information, we show a band of rapid rotators (star within the 10th to 25th percentiles of the period distribution; light gray region), intermediate rotators (25th to 75th percentiles; black region), and slow rotators (75th to 90th percentiles; dark gray region). At young ages, we observe a well-defined upper limit in the rotation periods, with virtually no stars showing periods longer than ∼10 days. At late ages, we clearly see the convergence of rotation periods to a tight sequence in a heavily mass-dependent fashion, in agreement with expectations. The convergence in our revised sequences is so strong that the upper and lower edges of the distribution in M37, Praesepe, and NGC 6811 in the 1.0–0.8M_⊙ range are barely visible. Additionally, in both Figures 11 and 12, there is another important clue about the torques from magnetized winds. For all clusters, there is: (1) a characteristic mass where the distribution has collapsed down into a narrow range (the unsaturated domain, where the torque is dJ/dt ∝ ω³); and (2) a mass range just below it where stars still retain a range of surface rotation rates (with the rapid rotators still being in the saturated domain, where the torque is dJ/dt ∝ ω). For this latter mass range, the width of the interquartile range is broader in the older systems (≳500 Myr) than it is in the young ones (≲150 Myr). This indicates a relative divergence in the surface rotation rates, in contrast to the convergence that is predicted by canonical models of angular momentum loss (e.g., van Saders & Pinsonneault 2013; Matt et al. 2015).

In other words, the data in Figure 12 appear to require lower torques for rapid rotators than for intermediate rotators, which acts to create the bimodal distribution that has been noted in the literature (and which is now clearly seen in our sample). One possible explanation for this, as proposed by Garraffo et al. (2018), is that there is a change in field topology, such that rapid rotators have more complex magnetic field configurations than intermediate rotators and are therefore less efficient at losing angular momentum. Although this concept is interesting, the data could also be fit with a wind model that incorporates a supersaturation mechanism, where the overall field strength declines. Alternatively, it could indicate a dependence of the convective overturn timescale on the rotation rate at fixed mass, as loss laws are typically parameterized by the ratio of the rotation period to this timescale (i.e., the Rossby number). Ultimately, a number of additional physical properties that are involved in the loss of angular momentum could be influencing the empirical sequences (e.g., mass-loss rate, dipole field strength, Alfvén radius). It will be informative to compare different proposed mechanisms, but an adjustment to canonical models is clearly required. We note that the late time behavior of the distribution is not sensitive to the treatment of this feature, as older stars have lost memory of their initial conditions, and typical semi-empirical approaches are tuned to reproduce the spin-down of the slow and rapid rotator branches of the data. However, the properties of intermediate rotators in systems that retain rapid rotators will not be correctly modeled by classical methods.

Finally, in Figure 13, we show the temporal evolution of the rotational percentiles for three different masses (0.5, 0.8, and 1M_⊙). We frame this as a comparison with Gallet & Bouvier (2015) ¹¹ in terms of the 25th, 50th, and 90th percentiles of the distribution of angular velocities (ω = 2π/P_rot), normalized to the solar value (ω_⊙ = 2.87 × 10⁻⁶ s⁻¹). Our membership analysis plays a role in shifting our values with respect to the literature ones in both axes, as the revised cluster properties change the ages (Section 4) and the revised rotational sequences change the angular velocity distributions (Section 5). In general, our percentile evolution appear less noisy and shows smoother patterns. In the 0.5 and 0.8M_⊙ stars, our data capture the same broad trends. Importantly, the overall slow rotation in young stars remains, and this provides further evidence for the need for core-envelope decoupling in angular momentum models (e.g., Denissenkov et al. 2010). Before our work, the classification of all of the slow rotators in NGC 2547 as field contaminants was a real possibility, and while our analysis did remove a number of them, some still remain. For the 1M_⊙ stars, our data suggests a faster rotation in all percentiles (as our analysis has predominantly removed slow rotators), and a globally steeper spin-down slope that remains approximately constant (and similar to the Skumanich value) in the entire age range.

Investigating the behavior of binary stars in empirical rotation studies is important to better understand stellar populations and angular momentum evolution. In the regime of wide separations, binaries can be used to place novel gyrochronology constraints in unexplored age and metallicity domains (Chanamé & Ramírez 2012; Janes 2017; Godoy-Rivera & Chanamé 2018). In the regime of close separations, recent studies have found a correlation between rapid rotation and field binaries (Simonian et al. 2019, 2020). Specifically in the M-dwarf domain, Stauffer et al. (2018) found that, in the young Upper Scorpius association (∼8 Myr) and the Pleiades cluster, the stars with multiple period measurements from K2 are predominantly photometric binaries (see also Rebull et al. 2016a, 2017, and Tokovinin & Briceño 2018). Furthermore, Stauffer et al. (2018) showed that the rotation periods of these young, unresolved binaries, are much shorter as well as more similar to each other than otherwise expected for single M dwarfs. This hints at the crucial role that binarity plays in star formation and the rotation rates imprinted at birth (see also Messina 2019).

In this context, in Figure 14, we examine the rotational sequences and CMDs of the no info stars in our most nearby clusters (i.e., the targets that did not pass our astrometric quality cuts; see Section 3.1.3). In the revised rotational sequences of M50 and M37, the stars classified as no info predominantly occupy their very lowest T_eff (and therefore apparent magnitude) values, which indicates that their classification is purely due to low-quality astrometry. In contrast to this, for the Pleiades and Praesepe, the no info stars span the entire T_eff range (see Section 5), and we would have expected them to have high-quality Gaia astrometry. Considering the previous selections made by Rebull et al. (2016a) in the Pleiades and by Rebull et al. (2017) in Praesepe, and given the low contamination rates found for these systems in Section 5, we expect most of these no info stars to be real members of these nearby clusters.

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The CMDs of Figure 14 show that the no info stars appear predominantly as photometric binaries with respect to the sequences of probable members in both the Pleiades and Praesepe. This result hints at biases in the Gaia DR2 data against these unresolved systems, even with conservative quality cuts, where their separation is seemingly too close for the current astrometry to properly distinguish their orbital and systemic motions. Interestingly, many of these no info stars appear as rapid rotators in the Pleiades rotational sequence. By the age of Praesepe, however, they are overlapping with the bulk of the probable members on the slowly rotating branch. This indicates that, while their separations are close enough to be unresolved in Gaia, these binaries are not synchronized and they are actually spinning down in a fashion similar to that of the single stars. This finding is similar to that of Stauffer et al. (2018), who reported that the rapid rotation of stars in binaries with respect to single stars is gone by the age of Praesepe, but in this case we extend it to higher-mass stars, beyond the M-dwarf regime.

Figure 14 also provides an interesting distinction in our membership classification. While presumably some of the stars classified in the no info category could turn out to be field contaminants, the stars classified as non members appear to occupy different locations in the rotational sequences and CMDs. Most of these non members appear below the track of probable members on the CMD, and correspond to rotational outliers in the converged branch of Praesepe. In addition to their astrometric distinction, we take this as further confirmation that the non members and no info stars correspond to markedly different populations.

Finally, although our astrometric selections were designed to avoid biases against binary stars, we conclude that newer Gaia data will be needed to fully resolve these photometric binaries. In the meanwhile, although they are not classified as probable members by our membership analysis, we advise future rotation studies of binary stars to include these no info stars in their samples, for completeness purposes.