STELLAR RADIAL VELOCITIES IN THE OLD OPEN CLUSTER M67 (NGC 2682). I. MEMBERSHIPS, BINARIES, AND KINEMATICS^*

After exploratory observations of a few stars, the initial CfA sample was defined in 1982 as all stars with Sanders (1977) proper-motion membership probabilities greater than 50%, $V\lt 12.8$ and $(B-V)\gt 0.40$. This sample comprised the top of the main sequence, subgiants and the giant branch. (The Sanders 1977 study covers a square of approximately 80 arcmin on a side, centered on the cluster.) A study to monitor the RVs of the 13 classical blue stragglers in M67 was being pursued independently (Latham & Milone 1996), and the relevant results from that survey are included in this paper as a convenience to the reader. Also, a subset of 28 giants and subgiants were observed more intensively from Spring 1982 to Spring 1983.

By Spring 1988 the target sample expanded substantially to include all Sanders (1977) members to V = 14, and all members in the cluster core (radius $\lt 10$ arcmin) to V = 16 (only observable with the MMT), totaling 432 stars. The last surviving CfA Digital Speedometer, on the 1.5 m Tillinghast Reflector, was retired in the summer of 2009. Over the following five observing seasons, TRES was used to continue the RV observations of targets (mostly binaries) from both the CfA and the WIYN samples. Whenever the observations accumulating for a target suggested that the velocity was not constant, additional observations were scheduled at a frequency designed to reveal its orbital parameters. For some systems the CfA observations span almost 35 years.

Importantly, Roger Griffin and James Gunn also had a RV program for M67 from 1971 to 1982 at the Palomar Hale 5 m telescope, which they supplemented with contemporaneous observations obtained with the CORAVEL instrument at Haute Provence for five of the binaries. Their target sample was very similar to our initial sample, and the combination of the two data sets to expand the time baseline was a natural and straightforward step. The integration of the Palomar and CORAVEL data with the CfA data is discussed in detail in Mathieu et al. (1986), where the entirety of the Palomar data as well as the CfA data to that date are presented, and in Mathieu et al. (1990).

Hereinafter, we will refer to the target stars and the measurements taken at all telescope/instrument pairs except WIYN/Hydra as the CfA sample and data. As of 2015 February, there are 447 stars in the CfA sample. The radius–V-magnitude distribution of the stars observed by these telescopes is shown in Figure 1 (top), and the middle and bottom panels of the same figure present the chronology of observations graphically.

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As evident in Figure 1, the CfA sample is not comprehensive in radius for the fainter stars in the sample. Furthermore, the Sanders (1977) proper-motion study was progressively more incomplete and less precise with increasing magnitude. The Hydra Multi-object Spectrograph (MOS) on the WIYN 3.5 m telescope, combined with modern target lists, has been able to provide a complete magnitude and spatially limited sample, as described here.

WIYN observations of M67 began on 2005 January 15 as part of the WIYN Open Cluster Study (WOCS; Mathieu 2000). The current WIYN target list contains all stars with $10\leqslant V\leqslant 16.5$ and within 30 arcmin in radius from the cluster center. We drew our sample (including astrometric positions) from the 2MASS catalog, supplemented with photometry from Montgomery et al. (1993). The Hydra MOS on WIYN has a 1° field of view, which sets the radial extent of the WIYN survey.

We can derive reliable RVs with our WIYN observing setup for stars with ${(B-V)}_{0}\gt 0.4$. (Rapid rotation and a diminished number of absorption features often hinders our ability to derive precise RVs for earlier-type stars.) The only stars in M67 that are bluer than this limit and bright enough to be within our sample are BSS. Because of their scientific interest, we included these stars in the WIYN stellar sample. In total there are 1278 stars within the WIYN stellar sample; 382 of these stars are also in the CfA stellar sample. We show the radius–V-magnitude distribution for stars observed at WIYN in the top panel of Figure 2.

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As seen in Figure 2 (middle), the V magnitude range of the stars observed at WIYN has evolved with time. Initially we chose to make use of WIYN's advantage at faint magnitudes and prioritized stars from $12.5\lt V\lt 16.5$. (The Hydra MOS has a dynamic range limit of 4 mag.) Later, to improve the completeness of our observations within the combined WIYN and CfA sample, we extended our WIYN sample to include all stars within a 30 arcmin radius from the cluster center and with $10\leqslant V\leqslant 16.5$. For a few epochs, we focused exclusively on the brighter stars in our list in order to build this complete sample.

We also note that we removed from our sample the few high-proper-motion stars in the M67 field whose coordinates have changed significantly during the course of our survey, as these stars are certainly not cluster members.

In order to detect binaries (and higher-order systems), we require multiple observations at multiple epochs with known RV precisions. Our sample contains observations of mainly narrow-lined stars within a modest magnitude range, but taken at multiple different telescopes with different instruments. Therefore to facilitate our subsequent analysis of binarity (Section 5.1), we estimate single-measurement precision values for RVs derived from each respective telescope.

The majority of the CfA measurements come from the MMT and the Tillinghast Reflector (both using the Digital Speedometers). We follow the same procedure as Geller et al. (2008) to empirically determine the typical single-measurement precision for observations made at these two telescopes and at the WIYN Observatory with Hydra. Specifically we fit a ${\chi }^{2}$ distribution of two degrees of freedom to the distribution of standard deviations of the first three RV measurements for each single-lined star in these samples, respectively. The results for all stars observed $\geqslant 3$ times at these telescopes are shown in Figure 3. The dashed lines in each panel of Figure 3 show the best fitting ${\chi }^{2}$ distribution functions, respectively, which yield typical single-measurement precision values of 0.8 km s⁻¹ for RVs from the Tillinghast Reflector + DS, 0.7 km s⁻¹ for RVs from the MMT, and 0.5 km s⁻¹ for RV from WIYN. These single-measurement RV precision values for the CfA telescopes are in good agreement with those found by Mathieu et al. (1986), and also agree with the value found by Hole et al. (2009) for observations of NGC 6819 taken at the MMT and the Tillinghast Reflector with the same instrumental setup. Likewise, the single-measurement RV precision for observations at WIYN found here is similar to that found for the narrow-lined stars in NGC 188, NGC 6819 and M35 observed with this same setup (Geller et al. 2008, 2010; Hole et al. 2009).

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For these three telescopes (WIYN, MMT and Tillinghast + DS), we also have sufficient data to characterize our RV precision as a function of V magnitude (which, in general, correlates with the signal-to-noise ratio (S/N) of the measurement for a given integration time). As also found by Geller et al. (2008) and Mathieu et al. (1986) the single-measurement RV precision degrades toward fainter magnitudes. In Figure 4 we show the single-measurement precision values resulting from ${\chi }^{2}$ distribution fits to observations from these three telescopes in bins of the stars' V magnitudes. The lines show linear fits to these data. To avoid unrealistic precision values, we impose a precision floor at the values in the bin containing the brightest stars, specifically at 0.4 km s⁻¹ for WIYN RVs and 0.6 km s⁻¹ for MMT and Tillinghast + DS RVs. We then use these fit lines to determine our single-measurement precision values for stars observed at these telescopes (given their respective V magnitudes) in later analyses.

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We also note that observations of the few rapidly rotating stars in our sample have poorer precision (see Geller et al. 2010 for a detailed discussion of the effects of stellar rotation on the WIYN RV precision). For these few stars, we do not attempt to derive single-measurement RV precisions.

We lack sufficient observations on the remaining telescopes to reliably utilize the ${\chi }^{2}$ distribution fitting technique. For the Palomar Hale 5 m RVs, the typical standard deviation of the measurements of a given star is 0.3 km s⁻¹(Mathieu et al. 1986); we use this value as our single-measurement RV precision for stars observed at Palomar. For observations from the Wyeth 1.5 m, there are only 12 stars with at least three RV measurements, and a third of these stars are in binaries (with orbital solutions). The mean of the standard deviations of the first three RVs for the remaining eight stars is 0.4 km s⁻¹, and we take this as an estimate of the single-measurement RV precision for observations from the Wyeth. CORAVEL and TRES observations were primarily of binaries. For CORAVEL RVs, we estimate the RV precision based on the $(O-C)$ residuals from the orbital fits in Mathieu et al. (1990), and find a typical precision of 0.5 km s⁻¹.

The correction of the TRES velocities for shifts in the instrumental zero-point from month to month and year to year is now reliable at the level of 0.02 km s⁻¹ or better (e.g., see Quinn et al. 2014), much better than the precision achieved for most of our TRES spectra of M67 stars due to S/N limitations set by photon noise. Nevertheless, most of our TRES observations of slowly rotating single-lined stars in M67 do yield a velocity precison on the order of 0.1 (or perhaps 0.2) km s⁻¹. We assume a single-measurement RV precision of 0.1 km s⁻¹ for TRES observations of narrow-lined stars in the following analyses.

TRES observations of rapidly rotating stars have proven to be much more precise than the results for the same stars observed with the CfA Digital Speedometers. We attribute this to the higher S/N and better wavelength coverage provided by TRES. In particular, TRES has allowed us to derive reliable RVs for two of the rapidly rotating BSS in M67 that eluded success with our other facilities.

The long tails in the distributions extending beyond the best fitting curves shown in Figure 3 are populated by binary (and higher-order) systems. Moreover, because the standard deviation of the RVs from a binary should be significantly larger than the single-measurement RV precision, we use the results of this analysis to identify binaries in our sample. We discuss our criteria for identifying binaries in Section 5.1.

We define a "primary stellar sample" which extends from the brightest stars in the cluster to $V\leqslant 15.5$ and out to a 30 arcmin radius from the cluster center. In total, there are 903 stars in our primary sample. We have at least one RV measurement for 902 of these stars, and $\geqslant 3$ RV measurements for 899 of these stars. Moreover, there is only one star without any RV measurements; 2038 (S2312)¹¹ is the second brightest star in our primary sample, is brighter than our WIYN sample, and is a Sanders (1977) proper-motion non-member. The other three stars with $\lt 3$ RV measurements are 1016 (S1306), 7033 (S1594), 7054 (F1295). 1016 is the brightest star in our sample and is a proper-motion non-member from three sources. We have two observations of 1016 that are outside of the cluster RV distribution. We have observed 7033 and 7054 multiple times, and they both appear to be rapid rotators. We have been unable to derive reliable RVs from the majority of these measurements. 7033 has one proper-motion membership probability (from Sanders 1977) of 0%. 7054 has no proper-motion measurements. Our one tentative RV measurement of 7054 is outside of the cluster RV distribution, but is uncertain due to the rotation. In short, we have RVs for all likely cluster members within our complete sample.

Figure 6 shows the percentage of stars with RV measurements in our stellar sample as functions of V magnitude, $(B-V)$ color and distance from the cluster center. There is no significant trend in our completeness within the primary sample with magnitude, color or radius. We show our completeness for stars with $15.5\leqslant V\leqslant 16.5$ in the left panel of Figure 6 within the gray region. There are 418 stars within this magnitude range that are within 30 arcmin from the cluster center. We have at least one RV measurement for 295 (71%) of these stars, and $\geqslant 3$ RV measurements for 182 (44%) of these stars. As shown in Figure 1, the CfA sample also contains observations of stars located at >30 arcmin from the cluster center. We do not include these stars in our primary sample used in the subsequent analyses, but we include all observed stars in our RV summary table (Table 2, presented in Section 5).

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We identify binaries in our M67 data using the e/i statistic, which is the ratio of the standard deviation (e) to the expected precision (i) of the RVs for a given star. Members of binaries with orbital periods short enough to show significant RV variation in our data will have higher standard deviations than expected for single stars. Therefore high e/i values indicate binaries or higher-order systems. As a large number of stars in our sample have RV measurements from multiple telescopes, and therefore multiple precision values, we use the formalism from Bevington & Robinson (1992) to derive the e and i values for data with multiple precision values, as in Hole et al. (2009).

We derive an e/i value for all single-lined stars with $\geqslant 3$ RV measurements. Previous work by Geller et al. (2008) shows that stars with e/i > 4 can be securely identified as binaries (or higher-order systems), and we initially followed this same cutoff for identifying binaries here. In practice, however, we have derived RV orbital solutions for all proper-motion members in our primary sample with $3\;\leqslant$ e/i $\lt \;4$. (There are only three proper-motion non-members in our primary sample with $3\;\leqslant$ e/i $\lt \;4$; each of these have mean RVs outside of the cluster distribution.) Therefore here, we identify binaries in our data as having e/i $\geqslant$ 3.

The uncertainties for RVs in double-lined (SB2) binaries are not well established, and therefore we do not derive e/i values for these stars. Instead we identify SB2s as binaries directly by inspection of the spectra and the morphology of the peaks of the cross-correlation functions. (We derive the uncertainties on the mean RVs, ${\mathrm{RV}}_{e}$, for SB2s without orbital solutions using the measurement precisions that we derive above for single-lined stars; we suspect that the RV uncertainties for such SB2s quoted here are in fact lower limits.)

In addition to the e/i statistic, we also provide the $P({\chi }^{2})$ value. This statistic tests the hypothesis that a given star's distribution of RVs is consistent with a constant value at the mean RV. Members of binaries with short enough orbital periods to show significant RV variations in our data will be inconsistent with a constant RV and therefore will have small $P({\chi }^{2})$ values. To derive the $P({\chi }^{2})$ value, we first calculate the standard ${\chi }^{2}$ statistic, using the weighted mean RV and the respective precision of each RV measurement (discussed in Section 3.1). For each observed star, we use the number of RV measurements minus one for the degrees of freedom. Then $P({\chi }^{2})$ is the corresponding probability for obtaining a value of ${\chi }^{2}$ greater than or equal to the observed value with the given degrees of freedom. Again, we only derive $P({\chi }^{2})$ values for single-lined stars with $\geqslant 3$ measurements.

In the following analysis we will identify binaries by having e/i $\geqslant \;3$, or having a binary orbital solution. (The additional $P({\chi }^{2})$ statistic is provided for interested readers who prefer to use these values to perform their own selection of binaries.) All stars with e/i $\lt \;3$ for which we have not derived binary orbital solutions, are labeled as "single." However, some of these stars are undoubtedly in long-period binaries, currently beyond our detection limit.

Finally, we also identify five likely triple stars in our sample, and label them as such in Table 2. In most cases the systems are double lined, where the RVs for one star vary on a much shorter time scale than those of the other star. For a few, we see hints of tertiary velocities at low signal-to-noise, but we have yet to analyze these spectra in detail to derive all three velocities simultaneously. Additional triple stars may be detectable in our sample, for instance within the RV residuals from binary orbital solutions, but we save this analysis for a future paper.

In Figure 7 we show the distribution of weighted mean RVs for all stars in our sample with $\geqslant 3$ RV measurements and e/i $\lt 3$, and γ-RVs for binaries with orbital solutions. For stars with RVs from multiple telescopes, we use the respective RV precision values to calculate the weighted means, and use those results here. The cluster RV distribution is clearly distinguished from that of the field as the narrow distribution peaked at a mean RV of $+33.6$ km s⁻¹.

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In order to derive membership probabilities, we first fit simultaneous one-dimensional Gaussian functions to the cluster and field RV distributions, ${F}_{c}(v)$ and ${F}_{f}(v)$, respectively. (We exclude from this fit binaries without orbital solutions, as their γ-RVs are unknown.) The resulting combined fit to the cluster and field distributions are shown in the blue line in the left panel of Figure 7, and the fit parameters are given in Table 3 (fit to a histogram with a bin size of 0.5 km s⁻¹). We then use the following equation,

$$\begin{eqnarray}&&{P}_{\mathrm{RV}}(v)=\displaystyle \frac{{F}_{c}(v)}{{F}_{f}(v)+{F}_{c}(v)}\end{eqnarray} \tag{ 1 }$$

(Vasilevskis et al. 1958) to calculate the RV membership probability ${P}_{\mathrm{RV}}$(v) for a given star in our sample.

Table 3.  Gaussian Fit Parameters For Cluster and Field Radial-velocity Distributions

Parameter	Cluster	Field
Ampl. (Number)	147.8 ± 0.7	2.38 ± 0.10
$\bar{\mathrm{RV}}$ (km s−1)	33.615 ± 0.005	23.4 ± 2.2
σ (km s−1)	0.854 ± 0.005	46.7 ± 2.2

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We only compute membership probabilities for stars with $\geqslant 3$ observations. For non-RV-variable stars, we use the weighted mean RV in the calculations. For binaries with orbital solutions, we use the γ-RV. For RV-variable stars without orbital solutions, we cannot calculate a reliable RV membership probability, as the γ-RV is unknown. For these stars we instead provide a preliminary membership classification, described in Section 5.3.

We plot the resulting distribution of RV membership probabilities in the right panel of Figure 7. Cluster and field stars are cleanly separated, and we choose a cutoff of ${P}_{\mathrm{RV}}$ $\;\geqslant \;50\%$ to define our M67 cluster member sample. Using our sample of single cluster members with e/i $\lt \;3$ and binary cluster members with orbital solutions, we find a mean cluster velocity of $+33.64$ km s⁻¹ (with an internal precision of ±0.03 km s⁻¹), in good agreement with previous RV surveys (e.g., see Yadav et al. 2008, and references therein).

As stated in Sections 3 and 4, the RVs in this paper are all on the native CfA system, which is found to be shifted by $-0.14$ km s⁻¹ compared to the IAU system. Therefore, the mean cluster velocity quoted here may also have this offset compared to the IAU system (i.e., $+0.14$ km s⁻¹ may need to be added to our velocities to get onto the IAU system).

The ratio of the areas under the Gaussian fits to the cluster and field distributions provides an estimate of the field star contamination. At a membership probability of 50%, we expect a 3.5% contamination from field stars in our RV cluster member sample (i.e., ∼20 stars).

5.2.1. Comparison of RV and Proper-motion Membership Probabilities

For stars that show no significant RV variability and binaries with orbital solutions, we can calculate precise RV membership probabilities that allow us to separate cluster members from field stars, as described above. However, for binary stars that do not have RV orbital solutions, we cannot calculate reliable RV membership probabilities because we do not know their γ-RVs. Thus we follow a similar method to Geller et al. (2008, 2010) and Hole et al. (2009) in order to provide a qualitative classification of the membership and variability for each observed star.

Our classification scheme is defined in Table 4, where we identify our eight qualitative membership classes, the selection criteria for each class, and also the number of stars that reside in each class. The selection criteria depend on the number of RVs, the e/i value, the RV membership probability (${P}_{\mathrm{RV}}$, determined either using the mean RV, $\bar{\mathrm{RV}}$, or the γ-RV, for binaries with orbital solutions), and the proper-motion membership "vote" described in Section 5.2.1. For the proper-motion membership vote, we use "${P}_{\mathrm{PM}}$ = M" to indicate proper-motion members, and "${P}_{\mathrm{PM}}$ = NM" to indicate proper-motion non-members. If a star does not have a proper-motion measurement, we use only the RV criteria to classify the star.

Table 4.  Description of Stars or Star Systems Within Each Membership Class

Class	Description	Criteria	Number
SM	Single Member	$\geqslant 3$ RVs, e/i $\lt \;3$, ${P}_{\mathrm{RV}}$ $(\bar{\mathrm{RV}})\geqslant 50$% AND ${P}_{\mathrm{PM}}$ = M	420
SN	Single Non-member	$\geqslant \;3$ RVs , e/i $\lt 3$, ${P}_{\mathrm{RV}}$ $(\bar{\mathrm{RV}})\lt 50$% OR ${P}_{\mathrm{PM}}$ = NM	498
BM	Binary Member	binary orbit, ${P}_{\mathrm{RV}}$ $(\gamma$-RV $)\geqslant 50$% AND ${P}_{\mathrm{PM}}$ = M	108
BN	Binary Non-member	binary orbit, ${P}_{\mathrm{RV}}$ $(\gamma$-RV $)\lt 50$% OR ${P}_{\mathrm{PM}}$ = NM	17
BLM	Binary Likely Member	$\geqslant \;3$ RVs , e/i $\geqslant 3$, ${P}_{\mathrm{RV}}$ $(\bar{\mathrm{RV}})\geqslant 50$% AND ${P}_{\mathrm{PM}}$ = M	23
BU	Binary with Unknown RV Membership	$\geqslant 3$ RVs, e/i $\geqslant \;3$, ${P}_{\mathrm{RV}}$ $(\bar{\mathrm{RV}})\lt 50$% AND the range in RV measurements includes the cluster mean RV AND ${P}_{\mathrm{PM}}$= M	11
BLN	Binary Likely Non-member	$\geqslant 3$ RVs, e/i $\geqslant \;3$, ${P}_{\mathrm{RV}}$ $(\bar{\mathrm{RV}})\lt 50$% AND all RV measurements are either at higher or lower RV than the cluster distribution, OR ${P}_{\mathrm{PM}}$ = NM	58
U	Unknown RV Membership	$\lt 3$ RVs	143

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In short the single members/non-members (SM/SN) or binary members/non-members (BM/BN) are stars with secure membership status; these are the only stars for which we can provide reliable RV membership probabilities. For RV variable stars with $\geqslant 3$ RV measurements that do not have a binary orbital solution, we divide our classification into three groups, including binary likely members (BLM), binaries with unknown RV membership (BU) and binary likely non-members (BLN). We anticipate that eventually orbital solutions derived for BLM binaries will place them in the BM category, while those for the BLN binaries will place them in the BN category (since many of these sources are proper-motion non-members, and for those that are not, it is unlikely that an orbital solutions will place their γ-RVs within the cluster distribution). Binaries with unknown RV membership (BU's) are proper-motion members. Therefore here we assume that these are indeed cluster members (unlike in other WOCS papers, where proper-motion memberships were unavailable). Stars with <3 RV measurements have unknown RV membership (U), as these stars do not meet our minimum criterion for deriving RV memberships or e/i values. In the following analyses we restrict our cluster member sample to only include the 562 stars classified as either SM, BM, BLM or BU.

Finally, as mentioned above, stars that show broadened spectral features (e.g., due to rapid rotation) do not have secure single-measurement RV precision values, and therefore in most cases we cannot confidently classify such stars as binaries or singles. We provide our best assessment of the binarity of these sources in Table 2 and indicate our uncertainty in their class with parentheses, e.g., (BL)M, (S)N, etc.

In Figure 8 we plot the CMD of all stars in our stellar sample with $V\lt 15.5$ (and available $(B-V)$ colors; left) and only the confirmed cluster members within the same magnitude range (right). Without removing non-members, the main-sequence of the cluster is visible, but the BSS and giant populations cannot be distinguished from the field. Applying both our proper-motion and RV membership criteria reveals a rich cluster containing well populated main-sequence, subgiant and giant branches as well as a large population of BSS (blue points), four yellow giants (red points), and two sub-subgiants (green points). Specifically, we plot here stars that we classify as SM, BM, BLM and BU, and we also include here stars in the U category as well as rapid rotators that are proper-motion members.

For comparison, we also plot a 4 Gyr isochrone and a zero-age main-sequence isochrone (solid lines), as well as an equal-mass binary line (dashed line). The isochrones are from Marigo et al. (2008), and use solar metallicity, ${(m-M)}_{V}=9.6$ and $E(B-V)=0.01$, consistent with recent results derived in the literature (see Section 1). We include the isochrones simply to help guide the eye; they are not meant as a fit to the observed data.

Binaries with orbital solutions are circled, and binaries without orbital solutions are marked with diamonds. As is seen clearly here, and also noted by Latham & Milone (1996) and Latham (2007), the "exotic" stars (i.e., the BSS, yellow giants, and sub-subgiants) have a remarkably high binary frequency. In total we identify 14 BSS, four yellow giants (one of which is outside of a 30 arcmin radius from the cluster center) and two sub-subgiants in our stellar sample. At least 16/20 (80% ± 20%) of these exotic stars are RV variables (and others may also be binaries, e.g., with long-period orbits that are currently outside of our detection limit). In comparison, 122/538 (22.7% ± 2.1%) of the "normal" stars, located in more typical positions in the CMD, show RV variability indicative of binarity. Thus the exotic stars have a significantly higher frequency of binaries than the normal stars. This result is similar to that found in the old (7 Gyr) open cluster NGC 188, where Mathieu & Geller (2009, 2015) find that 80% of the NGC 188 BSS have binary companions (roughly three times the binary frequency of the main-sequence stars in NGC 188).

We also note the very tight red giant sequence, first discussed in detail by Janes & Smith (1984). Such a tight red giant sequence is expected for a single coeval population, but, interestingly, the giants of NGC 188 (e.g., Geller et al. 2008) and the intermediate-age (2.5 Gyr) open cluster NGC 6819 (e.g., Hole et al. 2009) show a much larger scatter than the giants in M67. The origin of the scatter in these other open clusters is unknown.

Within our cluster member sample, there are a number of intriguing stellar populations, from those that lie far from the predicted locus of single stars from stellar evolution theory (including the well known BSS, Section 6.2.1, and anomalous giants, Section 6.2.2), to those that have physical characteristics very similar to those of our Sun (Section 6.2.3), and even three exoplanet host stars (Section 6.2.4). In the following we briefly identify and discuss these stars of note.

6.2.1. Blue Stragglers

6.2.2. Anomalous Giant Stars

6.2.3. Solar Twins

6.2.4. Exoplanet Host Stars

The spatial distribution of stars in M67 has been studied in detail (e.g., Mathieu & Latham 1986; Zhao et al. 1996; Sarajedini et al. 1999; Bonatto & Bica 2003; Davenport & Sandquist 2010). M67 is mass segregated, as is expected for a 4 Gyr cluster with a half-mass relaxation time of 100 Myr (Mathieu & Latham 1986). In light of our new RV memberships and identification of binaries, we briefly re-investigate the spatial distribution of cluster member populations using our primary sample. In Figure 9 we show the projected radial distribution for the single main-sequence stars, binary main-sequence stars, single giants and BSS. Singles are stars classified as SM, and binaries are classified as either BM, BLM or BU.

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A Kolmogorov–Smirnov (K–S) test shows that the main-sequence binaries are centrally concentrated with respect to the main-sequence single stars at the 98.7% confidence level. Comparing all single stars to all binaries shows that the binaries are more centrally concentrated at the 99.8% confidence level. Because the total mass of a given binary is generally more massive than that of a single star in our primary sample, this results confirms that M67 is mass segregated.

Mathieu & Latham (1986) find the BSS to be centrally concentrated with respect to the single stars near the cluster turnoff. Our BSS sample is slightly different than that of Mathieu & Latham (1986). Nonetheless, the result is the same. A K–S test comparing the BSS to the single main-sequence stars shows that the BSS are centrally concentrated at the 99.7% confidence level. This result suggests that the BSS are more massive than the main-sequence stars in the cluster, as was noted by Mathieu & Latham (1986).

In many globular clusters and also the old open cluster NGC 188, BSS show a bimodal projected radial distribution (e.g., Ferraro et al. 1997, 2012; Mapelli et al. 2006; Geller et al. 2008). We do not observe evidence for a bimodal radial distribution in our M67 BSS sample. However, our sample only extends to 6 or 7 core radii. In NGC 188, which is of a similar dynamical age to M67, the halo BSS population begins at roughly 6 core radii and extends to about 13 core radii (at least). Thus if M67 and NGC 188 have similar BSS radial distributions, we would only expect to see the inner population of M67 in our current sample, and would require a survey extending to roughly twice the current radial extent to search for a bimodal structure.

Interestingly, the giants appear to follow a similar spatial distribution as the binaries, despite having very similar masses to the upper main-sequence stars included in our primary sample. However, given the relatively small sample size of the single giants, a K–S test comparing the spatial distributions of the single giants and single main-sequence stars returns a distinction at only the 95.9% confidence level. For comparison, the NGC 188 giants do not follow the distribution of the binaries, and instead follow closely to the single cluster members (see Figure 9 in Geller et al. 2008), both when considering the entire spatial extent of the RV survey and when limiting to a similar number or core radii as our M67 sample. This difference is intriguing, but the small sample sizes of the giants in both clusters make such comparisons uncertain.

We follow the method of Geller et al. (2010) to calculate the RV dispersion of the M67 single main-sequence, subgiant and giant members in our primary sample. (We exclude the binaries and BSS because these have significantly different spatial distributions, likely due to their higher masses.) This method assumes that the observed RV dispersion is composed of two components, one which we will call the "combined RV dispersion" and one for the observational error, and also assumes that the distributions of RVs and errors are Gaussian. The observed dispersion is directly measured, and by subtracting off the component from observational error, we recover the combined RV dispersion. The combined RV dispersion itself contains contributions from both the true RV dispersion of the cluster and undetected binaries, which also artificially inflates the observed RV dispersion. We aim to recover the true RV dispersion from our data.

First we calculate the observed RV dispersion, which is simply the standard deviation of the observed mean RVs for each star about the cluster mean RV (calculated in Section 5.2). In our calculation, we use the weighted mean RVs for each star in the given sample (see Table 2), which utilize the single-measurement precision values from the given telescope for each individual RV value (see Section 3.1). For the single main-sequence, subgiant and giant stars in our primary sample, the resulting observed velocity dispersion is 0.86 ± 0.11 km s⁻¹. After correcting for the observational error (following the method of McNamara & Sanders 1977 and McNamara & Sekiguchi 1986), we find a combined RV dispersion of 0.80 ± 0.04 km s⁻¹.

We then follow a similar method to that of Geller et al. (2010) to correct this combined RV dispersion for the contribution from undetected binaries. Briefly, we run a Monte Carlo analysis to create many realizations of our M67 observations. For each realization, we generate a population of synthetic single and binary stars (given an input binary frequency), and produce synthetic RVs for these stars on the true observing dates from our M67 survey, which we analyze in the same manner as our real observations. We run this analysis for a range of true RV dispersion values, and for each value, we derive the difference between the synthetic combined RV dispersion and the input true RV dispersion. This difference is the contribution from undetected binaries (β in Equation (5) of Geller et al. 2010).

For this analysis, we improve upon the technique of Geller et al. (2010) in our treatment of the binaries, and therefore, for clarity, we explain the method in some further detail here. In order to recover the true velocity dispersion, we must also estimate the binary frequency of the cluster. We use a similar Monte Carlo method to Geller & Mathieu (2012) to account for our incompleteness in binary detections out to the hard-soft boundary. As in Geller & Mathieu (2012), we assume that the orbital parameters of the M67 binaries follow the same distributions as observed for the solar-type binaries in the Galactic field from Raghavan et al. (2010), except here we use a circularization period of 12.1 days (Meibom & Mathieu 2005). We also make a few updates to the method of Geller & Mathieu (2012). First, rather than choosing one specific primary mass, we draw from an inferred primary-mass distribution within our M67 primary sample (derived by comparisons to a Marigo et al. 2008 isochrone). Second, we attempt to recreate the correlation between binary mass ratio and orbital period observed for field binaries by Raghavan et al. (2010, see Figure 17 and related discussion). Specifically, for binaries with periods between 100 days and 1000 years, we choose mass ratios from a uniform distribution with 10% of systems having a mass ratio of unity (e.g., twins). For binaries with periods greater than 1000 years we draw mass ratios from a uniform distribution limited to be less than 0.95. For binaries with periods less than 100 days, we enforce ∼19% to have mass ratios between 0.2 and 0.45, ∼44% to have mass ratios between 0.45 and 0.9, and the remainder to have mass ratios between 0.9 and 1, all drawn from uniform distributions between the respective mass ratio limits. Finally, we limit the field log-normal orbital period distribution so that the binaries are detached (using radii estimates from a Marigo et al. 2008 isochrone), and the periods are less than the hard-soft boundary.

The assumed location of the hard-soft boundary, which is at roughly 10⁵–10⁶ days in M67, is especially important for the undetected binary correction. We do not expect to detect binaries at these long periods. Furthermore, the hard-soft boundary is near the peak of the log-normal distribution, and therefore the assumed binary frequency is particularly sensitive to this cutoff. We estimate the hard-soft boundary as the location where a synthetic binary's binding energy is equal to the kinetic energy of a "typical" star moving at an assumed velocity dispersion. For the mass of this typical star, we take the mean mass of an object (single or total mass of a binary) from the Hurley et al. (2005) N-body simulation of M67, which at 4 Gyr is 0.95 ${M}_{\odot }$. We then assume a velocity dispersion, run our analysis to derive the true velocity dispersion in our M67 sample (given the resulting hard-soft boundary and total binary frequency), and iterate until this assumed velocity dispersion is $\lt 0.05$ km s⁻¹ different from the derived true velocity dispersion in M67 (a somewhat arbitrary limit meant to be roughly equivalent to the precision with which we can measure the velocity dispersion). As a starting guess, we use the combined dispersion value of 0.80 km s⁻¹, found above.

Our analysis of the single main-sequence, giant and sub-subgiant stars in this sample requires only two iterations. The resulting orbital period distribution matches closely to those predicted by the N-body open cluster models of Hurley et al. (2005) and Geller et al. (2013), and we estimate the total binary frequency in our sample to be 57% ± 4%. We derive a true velocity dispersion, after correcting for measurement error and undetected binaries, for the M67 single main-sequence stars, subgiants and giants with $V\leqslant 15.5$ of ${0.59}_{-0.06}^{+0.07}$ km s⁻¹.

Using a subsample of 20 of the brightest stars in our sample, Mathieu (1983) measured a cluster RV dispersion, after correcting for measurement errors, of 0.48 ± 0.09 km s⁻¹. After also correcting for undetected binaries (assuming a 50% binary frequency), Mathieu (1983) find an RV dispersion of 0.25 ± 0.18 km s⁻¹. Girard et al. (1989) show that this RV dispersion increases to 0.48 ± 0.15 km s⁻¹ (corrected for both measurement errors and undetected binaries) when including the larger sample from Mathieu et al. (1986), in good agreement with the result we find here.

More recently, Pasquini et al. (2012) derive an RV dispersion for main-sequence stars in M67 with $V\lt 15$ of 0.680 ± 0.063 km s⁻¹, and for M67 giants of 0.540 ± 0.090 km s⁻¹, using HARPS spectra with a typical precision of ∼10 m s⁻¹ (not corrected for undetected binaries). If we divide our sample into the SM main-sequence stars with $V\lt 15$ and SM giant stars, after correcting for measurement errors, we find a combined velocity dispersion for the main-sequence stars of 0.83 ± 0.04 km s⁻¹ and for giants of 0.60 ± 0.08 km s⁻¹. However, the main-sequence stars have a much higher binary frequency than the giants, and therefore a much larger correction for undetected binaries. After correcting for both measurement errors and undetected binaries, we find a true velocity dispersion for main-sequence stars ($V\lt 15$) of ${0.60}_{-0.07}^{+0.08}$ km s⁻¹ and for giants of 0.60 ± 0.10 km s⁻¹. Thus we find essentially identical velocity dispersions for the main-sequence and giant stars. We note that Padova isochrones indicate a mean mass for the main-sequence stars in this sample of 1.11 ${M}_{\odot }$ (with a standard deviation of 0.10 ${M}_{\odot }$), and for the giants of 1.33 ${M}_{\odot }$ (with a standard deviation of 0.01 ${M}_{\odot }$); our sample covers only a narrow mass range.

One-dimensional proper-motion dispersion measurements from the literature are somewhat higher than these RV dispersions (as was also noted by Girard et al. 1989). McNamara & Sanders (1978) find a velocity dispersion of 0.95 km s⁻¹ with a $1\sigma$ upper limit of <1.48 km s⁻¹. Zhao et al. (1996) find a dispersion of 0.96 ± 0.09 km s⁻¹, and Girard et al. (1989) find a dispersion of 0.81 ± 0.10 km s⁻¹. As discussed above, the Girard et al. (1989) proper motions have the highest precision, and therefore here we will compare directly to their result. After correcting for undetected binaries, our true RV dispersion measurement differs from the Girard et al. (1989) proper-motion dispersion measurement at about the $2\sigma$ level (without accounting for an additional uncertainty on the proper-motion dispersion from the range in cluster distances quoted in the literature). Given the uncertainties on these measurements, the different distances assumed in converting from angular proper motions to km s⁻¹, and the different methods used to account for measurement error in the proper-motion dispersions, we conclude that the RV and proper-motion dispersions are in agreement at our level of precision.

We have also examined the true RV dispersion of M67 as a function of radius from the cluster center, shown in Figure 10. We divide this sample of main-sequence, subgiant and giant single members into five equal bins in radius. Crosses, open circles and filled circles show the observed, combined and true RV dispersion values, respectively, at each bin in radius. The correction for undetected binaries is highest in the inner-most bin, as this bin has the highest binary frequency, consistent with our finding that the binaries are mass segregated with respect to the single stars (see Figure 9).

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The radial distribution of the true velocity dispersion within the parameter space covered by our survey is consistent with an isothermal distribution. Zhao et al. (1996) find an increase in the proper-motion dispersion as a function of radius, although they note that this effect may be in part due to increased field-star contamination with radius in their sample. We do not see a similar trend in our data. We have also examined different binnings with no detectable effect on the results presented here.

Given the velocity dispersion, we can estimate the mass of the cluster through the virial theorem. Under the assumption that the cluster is in dynamical equilibrium, the kinetic and potential energies are related by ${V}^{2}=\eta {GM}/R$. From (Spitzer 1987), if this equation is rewritten in terms of the half-mass radius (${r}_{{\rm{h}}}$), $\eta \sim 0.4$ for most systems. Also, for an isotropic velocity dispersion, the observed half-mass radius in projection is given by ${r}_{\mathrm{hp}}\sim \frac{3}{4}{r}_{{\rm{h}}}$, and the one-dimensional RV dispersion is related to the rms velocity V by ${\sigma }_{r}^{2}={V}^{2}/3$. Thus the virial mass of the cluster can be calculated by the equation:

$$\begin{eqnarray}&&{M}_{V}=10\displaystyle \frac{{r}_{\mathrm{hp}}{\sigma }_{r}^{2}}{G}.\end{eqnarray} \tag{ 2 }$$

Fan et al. (1996) find a half-mass radius of M67 for stars with $13.8\lt V\lt 14.5$ of 2.54 ± 0.41 pc and for stars with $15.6\lt V\lt 14.5$ of 2.74 ± 0.29 pc, assuming a distance of 850 pc. The M67 sample used here extends from $V\sim 8$ to $V=15.5$, though the vast majority of the stars have $V\gtrsim 12.5$. We will assume a projected half-mass radius for the stars in our sample of ${r}_{\mathrm{hp}}=2.58\pm 0.45$ pc. This value lies in the middle and extends to the limits of the range in ${r}_{\mathrm{hp}}$ found by Fan et al. (1996) for these two magnitude regimes. Given our true RV dispersion above of ${\sigma }_{r}={0.59}_{-0.06}^{+0.07}$ km s⁻¹, we find a virial mass for M67 of ${2100}_{-550}^{+610}$ ${M}_{\odot }$.

This value is in good agreement with previous dynamical mass estimates. McNamara & Sanders (1978) derive a virial mass of 1600 ${M}_{\odot }$ (with a $1\sigma$ upper limit of 4000 ${M}_{\odot }$), and a direct cluster mass of 1100 ${M}_{\odot }$ based on star counts. Zhao et al. (1996) find a virial mass of 1500 ± 250 ${M}_{\odot }$ using their proper motions. From examination of the cluster luminosity function, Fan et al. (1996) calculate a mass of 1270 ${M}_{\odot }$ within a radius of 2000 arcsec and including stars with masses $\gt 0.5$ ${M}_{\odot }$.

These mass estimates (including ours) are somewhat higher than the mass estimates of Montgomery et al. (1993), Francic (1989) and Mathieu (1983), who find cluster masses of 724 ${M}_{\odot }$, 553 ${M}_{\odot }$ and 903 ${M}_{\odot }$, respectively, all through analyses of the cluster luminosity function. We interpret these measurements as lower limits on the true cluster mass, as each of these surveys were limited either in magnitude or radial extent.