WIYN OPEN CLUSTER STUDY. XXXII. STELLAR RADIAL VELOCITIES IN THE OLD OPEN CLUSTER NGC 188

2.1.1. Chronology

2.1.2. Sample

The Domain Astrophysical Observatory (DAO) data set contains multiple RV measurements of 77 stars within the magnitudes of 10.8 ⩽ V ⩽ 16 covering dates from 1973 December through 1966 November. Observations prior to 1980 were taken by Roger Griffin and James Gunn at the Palomar 5 m telescope (e.g., Griffin & Gunn 1974); later observations were made by Robert McClure, Hugh Harris and Jim Hesser at the DAO (e.g., Fletcher et al. 1982; McClure et al. 1985). All RV measurements were then converted into the DAO Radial-Velocity Spectrometer (RVS) system.

3.1.1. The Telescope and Instrument

3.1.2. Observing Procedure

The DAO RVs were measured using the DAO RVS on the 1.2 m telescope. This instrument observes 43 nm of the spectrum centered at 455 nm with a resolution of 0.2 Å, scanning a mask rapidly across the stellar absorption lines. The properties of the instrument and the observing procedures can be found in Fletcher et al. (1982) and McClure et al. (1985).

Generally the highest signal-to-noise spectra result in the highest CCF peak height values. We use the peak height of the CCF fit as the measure of the quality of a given measurement in the WIYN data set. Only measurements whose fits lie above a certain cutoff value in the CCF peak height are deemed reliable. In order to determine this cutoff value, we first determine the offset of each RV measurement of a star from the mean RV of that star, using only single stars. We then plot these offsets, for all RV measurements of all single stars, against the CCF peak height (Figure 1).

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It is clear from Figure 1 that above a CCF peak height of ∼0.4 the RV offsets are quite small while those measurements whose CCF peak heights are lower than 0.4 show a very large scatter. We therefore adopt a CCF peak height cutoff value of 0.4, and present here only measurements of this quality.

In addition to our CCF peak height quality criterion, we examine the distribution of RVs for each individual star and visually inspect any measurements that are outliers in the distribution. Occasionally we remove measurements whose CCF, though having a peak height above 0.4, clearly provides a spurious measurement (e.g., inadequate sky subtraction).

In order to determine the precision of our WIYN RV measurements, we fit a χ² distribution to our empirical distribution of RV standard deviations (following Kirillova & Pavlovskaya 1963). The theoretical χ² curve models the distribution of errors on measurements of single stars. This function depends on both the precision of all measurements and the number of degrees of freedom (i.e., one less than the number of measurements). For consistency across all stars, many of which have only three measurements, here we have used only the first three measurements of each star to calculate the standard deviation of its measurements.⁹ In order to derive this precision value, we fit a χ² function with two degrees of freedom to the distribution of the RV standard deviations:

$$\begin{equation} \chi ^2(\sigma _{\rm obs}) = \left(\frac{2}{\sigma _i^{2}}\right) \left(\sigma _{\rm obs}\right) \exp \left({-\frac{\sigma _{\rm obs}^{2}}{\sigma _i^{2}}}\right). \end{equation} \tag{ 1 }$$

Here, σ_obs denotes the standard deviation of the RV measurements for a star, and σ_i represents the overall precision of our measurements.

The empirical distribution (Figure 2) represents the combination of two populations: the single stars (with lower σ_obs) and the velocity-variable stars (with higher σ_obs). The χ² function is designed to fit only the single stars. For this reason, we chose a maximum σ_obs for our fit, above which the population is dominated by velocity-variable stars. We used a standard least-squares determination, varying the value σ_i as well as a cutoff value in σ_obs, to fit the χ² function to our data. A maximum σ_obs cutoff of 0.7 km s⁻¹ (shown by the dashed line in Figure 2) and σ_i = 0.4 km s⁻¹ produced the smallest residuals and hence the best fit to the peak at lower σ_obs. Figure 2 shows the RV standard deviation histogram of all stars with ⩾3 WIYN measurements along with the best-fit χ² distribution function. This analysis yields a nominal precision on our WIYN data of σ_i = 0.4 km s⁻¹.

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Note that toward higher standard deviations, the data are populated almost entirely by binary stars. This is reflected in the departure of the data from the theoretical χ² curve. We can therefore use this plot to facilitate our binary identification routine by assuming that if a given star has a σ_obs that is higher than a certain value, the star is a velocity variable. By inspection of Figure 2 we note that the theoretical curve drops to very near zero at >3 times σ_i. We therefore define our velocity-variable stars as those stars whose RV standard deviations (by convention, we use e in place of σ_obs) are greater than 4 times our quoted precision (by convention, we use i in place of σ_i), or e/i > 4. We plot this value of e/i = 4 as the dotted line in Figure 2, and we discuss our velocity variables in further detail in Section 7.2.

Although we have calculated a single value for our WIYN precision, the dispersion of our measurements is affected by photon error, and is thus a function of V magnitude. Figure 3 shows χ² calculated values of σ_i as functions of CCF peak height and V magnitude. For fit quality, our precision ranges from 0.22 km s⁻¹ for the highest quality RVs and degrades to 0.80 km s⁻¹ toward our CCF peak height cutoff value of 0.4. We see a similar range when looking at the precision as a function of V magnitude; the brightest stars are measured to a precision of 0.25 km s⁻¹ while the measurement precision on our faintest stars is reduced to 0.55 km s⁻¹.

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The range in χ²-fit values for our precision agrees very well with that derived from our error budget analysis in Section 5.3. We are therefore confident in quoting our overall precision to be σ_i = 0.4 km s⁻¹, which can then be used to identify our binary stars and to evaluate the velocity dispersion of the cluster.

In this section, we present a thorough empirical analysis of the error sources contributing to our measurement precision; a preliminary report can be found in Meibom et al. (2001). Our internal measurement errors have both systematic and random components that include fiber-to-fiber variations (σ_f–f), photon error (σ_ph(V)), and pointing-to-pointing variations (σ_p–p), each of which we discuss and quantify here.

5.3.1. Systematic Fiber-to-Fiber Variations (σ_f–f)

Sky flats demonstrate the existence of fiber-to-fiber variations of unknown origin. During our 11 years of monitoring the stars in NGC 188, different configurations of the Hydra MOS result in the same star being measured through different fibers. Thus fiber-to-fiber variations represent a source of internal error on the set of measurements for a star. Fortunately this error is systematic and can be corrected.

During each observing run, we take at least one sky flat as a calibration measure, with all fibers arranged in a circle in the focal plane. When corrected for the Earth's rotational and orbital motions, all fibers should provide RVs of 0 km s⁻¹. In reality, it has become clear that each fiber has a unique and consistent offset of <1 km s⁻¹. Specifically, for each sky flat we derive a mean velocity across all of the fibers, and then measure the velocity offset from this mean for each fiber. We take the means of these velocity offsets over ∼10 years of observing NGC 188 for each of the two observed wavelength ranges, and the results are plotted in Figure 4. The offsets for the 513 nm region range from 0.00 km s⁻¹ to 0.70 km s⁻¹. The mean of the standard deviations on each mean offset measurement is 0.13 km s⁻¹. Analysis of the 640 nm range results in similar values. We assume that the derived offset for a given fiber is independent of the position of the fiber on the focal plane. Our RV measurements are corrected for these offsets on a fiber-by-fiber basis.

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5.3.2. Random Photon Errors (σ_ph(V))

As described in Section 3, each RV measurement derives from three combined sequential integrations obtained in the same fiber configuration. Since these three integrations are almost contemporaneous and are made through the same fibers for any given star, the standard deviations of the three velocities measured from each of the three integrations must arise from photon error alone. Figure 5(a) shows the distribution of such standard deviations as a function of V magnitude, and provides a direct measurement of our photon error. It is clear that the photon error is a function of V magnitude, as expected. (Here we have only included 2400 s integrations.) Indeed, if we bin the data shown in Figure 5(a) by intensity (I), the trend is well fitted by I^−1/2.

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5.3.3. Pointing-to-Pointing Variations (σ_{p − p})

In Figure 5(b), we plot the standard deviations of the RV measurements for each single star as a function of V magnitude. For V ≳ 14, the morphologies of Figures 5(a) and 5(b) are nearly identical. For these fainter stars, the dispersion on our measurements is dominated by photon error. However, at V ≲ 14, the values in Figure 5(a) drop below those in Figure 5(b). Here, the photon error is lower than our empirical precision. Indeed, at V < 14, we observe a constant dispersion with an average value of ∼0.3 km s⁻¹ while our average photon error in Figure 5(a) is only ∼0.2 km s⁻¹. We must therefore identify a second component to our random error. We conjecture that this component is due to systematic offsets in each pointing that are randomly distributed in time. We label this component the pointing-to-pointing variation: σ_p–p.

Thus, we define our internal sources of random error as

$$\begin{equation} \sigma _i(V)^2 = \sigma _{\rm ph}(V)^2 + \sigma _{p\hbox{--}p}^2. \end{equation} \tag{ 2 }$$

Because we do not have multiple observations of identical pointing configurations, we need to determine σ_p–p via a statistical approach. We begin by defining the quantity σ_mp, the dispersion of the mean RVs of all pointings. Specifically, for each pointing we take the mean RV of the single-cluster members, which we call the pointing velocity. Then σ_mp is the standard deviation of all of these pointing velocities.

Note that σ_mp is not simply σ_{p − p}, because each pointing velocity derives from a different sample of single-cluster members. The dispersion among pointing velocities is thus a consequence of the pointing-to-pointing variations, the photon errors, and the true motions of the stars. We digress here briefly to discuss the last.

Stars in a gravitationally bound cluster will orbit the center of mass of the system. The orbital motions around the cluster center-of-mass is typically characterized by a velocity dispersion (σ_c). In addition, orbital motions introduced by a binary companion will offset the individual RV measurements of a given star from the binary system's center-of-mass motion. When a binary orbital solution is available the true center-of-mass motion is known. However, there are certainly binaries that we cannot detect spectroscopically (e.g., long-period binaries), and are thus included as single (constant-velocity) stars. These undetected binaries add to the dispersion on our measurements. We characterize the distribution of undetected binary motions as a Gaussian with dispersion σ_b. Therefore, we call the combination of the cluster motions and undetected binary motions the combined velocity dispersion:

$$\begin{equation} \sigma _{\rm cb}^2 = \sigma _c^2 + \sigma _b^2. \end{equation} \tag{ 3 }$$

Thus, we define the measured pointing variation as

$$\begin{equation} \sigma _{\rm mp}^2 = \frac{\big[\sigma _{\rm cb}^2 + \sigma _{\rm ph}(V)^2 \big]}{N} + \sigma _{p-p}^2, \end{equation} \tag{ 4 }$$

where N corresponds to the average number of single-cluster-member stars in a given pointing (within the desired magnitude range).

Using Equation (2), the apparent dispersion of the RV measurements for a set of stars can be written as

$$\begin{equation} \sigma _{\rm obs}^2 = \sigma _{\rm cb}^2 + \sigma _i^2 = \big[\sigma _{\rm cb}^2 + \sigma _{\rm ph}(V)^2 \big] + \sigma _{p-p}^2, \end{equation} \tag{ 5 }$$

where σ_obs is the standard deviation of all single stars' mean velocities (in the appropriate magnitude domain).

Combining Equations (4) and (5) allows us to eliminate the bracketed values and recover the pointing-to-pointing dispersion:

$$\begin{equation} \sigma _{p-p}^2=\frac{N}{N-1} \bigg(\sigma _{\rm mp}^2 - \frac{\sigma _{\rm obs}^2}{N}\bigg). \end{equation} \tag{ 6 }$$

5.3.4. Calculating the Empirical Internal Error (σ_i(V))

In order to check for a potential zero-point offset between the two data sets, we compared 24 single stars with ⩾3 observations in each data set. A star is defined as single if the standard deviation of its measurements is less than four times our measurement precision of 0.4 km s⁻¹ (see Sections 5.2 and 7.2). We find a mean offset of 0.03 ± 0.1 km s⁻¹. We therefore conclude that there is no measurable systematic RV offset between the WIYN and DAO data at this level.

We then investigated the precision of each data set, employing a χ²-fitting algorithm as described by Kirillova & Pavlovskaya (1963) and discussed in Section 5.2. Using this method, we find the precision of the WIYN data set to be 0.4 km s⁻¹ and of the DAO data set to be 1.0 km s⁻¹. The DAO RV errors found here are larger than the typical errors of 0.4 km s⁻¹ normally seen with this instrument (Fletcher et al. 1982). We believe these increased errors are due to photon statistics, as the stars in NGC 188 are faint enough to require long exposure times for this relatively small telescope. We have therefore noted the number of WIYN and DAO observations for each star in our master data table. We have also accounted for this difference in precision when calculating our average RVs and e/i measurements for stars observed at both observatories (as explained in Sections 7.1 and Section 7.2).

The WIYN and DAO RV measurements were thus integrated without modification into a single data set. We note that the combination of the data sets extends our maximum time baseline up to 35 years for some stars.

The basis for our stellar sample is the P03 PM study of NGC 188, which we take to be complete within our magnitude limits. We note that 28 of these 1498 stars lack PM measurements.

As mentioned in Section 2, our original cluster sample was composed of the PM members from the Dinescu et al. (1996) study, and, as such, our observations are most complete within this sample. We have ⩾3 observations of 99% of the Dinescu et al. (1996) sample with PM membership probabilities >70%. The Dinescu et al. PM members (P_PM > 70%) comprise 27% of their entire sample. We have also observed 47% of the remainder of the Dinescu et al. sample ⩾3 times, and 65% at least once.

Our current stellar sample consists of 1498 stars, including 624 with non-zero PM membership probabilities, taken from the P03 study. We have ⩾3 observations of 96% of these non-zero PM members, and at least one observation of >99% of these members. We have also observed 27% of the remainder of the Platais et al. sample ⩾3 times, and 46% at least once. We have at least one observation of 69% of our entire stellar sample.

In total, we have calculated RV membership probabilities for 848 stars (each having ⩾3 RV measurements).

We have plotted our completeness as a function of radius on the left panel of Figure 6 and as a function of V magnitude on the right panel of Figure 6. The top plots display our completeness in the stars with non-zero P03 PM membership probabilities. Our observations near completeness in both radius and magnitude for this sample. There is a slight bias toward the cluster center as a result of having prioritized these stars in our observations. Our completeness drops for magnitudes brighter than V = 12 as we have not observed these stars at the WIYN 3.5 m; the data for these stars comes from the DAO. There are only six stars with 10.5 ⩽ V < 12 and non-zero P03 PM membership probabilities. Additionally our completeness amongst stars observed ⩾3 times begins to drop at V ≳ 16. These stars are more difficult to observe, as they require clear skies and large separations from the moon. The bottom plots display our completeness for our entire stellar sample. The drop in completeness toward larger radii is due to a similar decrease in frequency of PM members at increasing radius from the cluster center. The incompleteness in V > 12 within our full stellar sample is again due to a decreasing frequency of PM members among fainter stars.

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In order to calculate a given star's RV membership probability, we first fit separate Gaussian functions for both the cluster and field-star populations to a histogram of the RV measurements for all single stars. We then define the membership probability of a given star by the usual equation:

$$\begin{equation} P_{\rm RV}(v) = \frac{F_{\rm cluster}(v)}{F_{\rm field}(v)+F_{\rm cluster}(v)} \end{equation} \tag{ 7 }$$

where F_cluster(v) is the cluster-fitted value for RV v, and F_field(v) is the field-fitted value for the same RV (Vasilevskis et al. 1958). We use only single stars in computing the cluster and field fits; we then re-normalized the Gaussians to the full sample size including velocity-variable stars. Figure 7(a) shows the RV histogram as well as the fits to the field and cluster stars used in Equation (7). The cluster is best fitted by a Gaussian centered on −42.43 km s⁻¹ with σ = 0.76 km s⁻¹. The cluster fit indicates a total of 485 stars integrated over the Gaussian fit. The field is best fitted by a Gaussian centered on −35.51 km s⁻¹ with σ = 38.01 km s⁻¹ and indicates a total of 314 stars.

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As mentioned above, Monte Carlo simulations have indicated that we require three observations covering a baseline of at least one year in order to be 95% confident that a star is either constant or variable in velocity (i.e., a single star or a binary- or multiple-star system). If this criterion is not met, we classify the star's membership as unknown (U). We have calculated RV membership probabilities for the 848 stars that do meet this criterion. For single stars we use the average RV to perform this calculation. For binary stars with orbit solutions we use the center-of-mass velocity (A. M. Geller et al. 2008, in preparation); for binary stars without orbit solutions, we use the average RV. For single stars and binaries without orbit solutions that have both WIYN and DAO RV measurements, we use a weighted average (weighted by the precision of each observatory; see Section 6) to calculate this average RV. We then round the resulting RV membership probability to the nearest integer value and quote this as the P_RV. Upon review of the histogram of membership probabilities (Figure 7(b)), we choose to define our RV members as those with a non-zero P_RV. We note that there are only 31 stars with 0% <P_RV< 80%. NGC 188 is out of the plane, and well separated from the field in RV space. As such, in the range of RVs that yield P_RV > 0% we would only expect to find a ∼6% contamination by field stars. This contamination will arise from field stars whose RVs are similar to those of the cluster members. We have expanded this portion of Figure 7(a) and plot the region in the right side of the figure.

We can securely compute RV membership probabilities for single stars as well as binaries with orbit solutions. For all stars, we also use the P03 PM membership probabilities as a further constraint on cluster membership, including only those with non-zero P_PM as cluster members. In addition to our RV membership probabilities, we classify single stars as either SMs or single non-members (SNs), based on the RV membership probability. Binary stars with orbit solutions are classified as either binary members (BMs) or binary non-members (BNs), based on the RV membership probability derived using their center-of-mass velocity (A. M. Geller et al. 2008, in preparation). For velocity-variable stars without orbit solutions, our RV membership probabilities are less certain; therefore we simply provide the classification as a distinction. Velocity-variable stars without orbit solutions are classified as binary likely members (BLM) if $P_{\rm RV}(\overline{\rm RV})>$ 0% (where $\overline{\rm RV}$ is the star's average RV), binary unknown (BU) if the range in RV measurements includes the cluster mean velocity but $P_{\rm RV}(\overline{\rm RV})=$ 0%, or BLN if all RV measurements fall outside of the cluster distribution. In addition to our stars that have been observed <3 times, we also classify certain stars as unknown (U) that do not have sufficient measurements to yield a reliable class (e.g., rapid rotators with few observations). Note that in Table 2, we have only included our P_RV measurements for the SM, SN, BM, and BN stars, as the RV memberships derived for the remaining classes of stars are uncertain. Table 3 displays the number of stars within each classification.

We define our sample of cluster members as the SM, BM and BLM stars; in total, we find 473 likely member stars. Including only the SM and BM cluster members, we find a mean cluster RV of −42.36 ± 0.04 km s⁻¹, with a standard deviation about the mean of 0.64 km s⁻¹.

7.1.1. Comparison of RV and PM Memberships

RV variable stars are evident from the standard deviations of their measurements. As described in Section 5.2 and shown in Figure 2, we have chosen a cutoff value of 4 times our precision (e/i > 4) to distinguish single stars from velocity-variable stars. For stars with both WIYN and DAO measurements, we take a weighted average of the two e/i measurements resulting from the different precision values¹⁰. If a star should have multiple observations from one observatory and only one observation from the other, we exclude that single observation and use the other observations with the appropriate precision value in calculating the e/i measurement. Note that we have not included e/i measurements in Table 2 for RR stars or SB2 binaries, as the uncertainties on these measurements are not well defined. In total, we find 124 velocity-variable cluster members. We assume that our velocity variables are binary- or multiple-star systems, and seek to derive orbit solutions for all of our detected binaries. To date, we have determined orbit solutions for 93 binaries, 78 of which are probable members; 16 are double-lined and 62 are single-lined (A. M. Geller et al. 2008, in preparation). See Table 3 for the number of stars within all of our velocity-variable classes.

We have also examined known photometric variable stars and X-ray sources for membership and RV variability. Table 4 presents the previously studied photometric variable stars (with prefix "V" from Zhang et al. 2002 and "WV" from Kafka & Honeycutt 2003) that are within our magnitude limits and have ⩾3 observations. Table 5 displays similar information for the X-ray sources (with prefix "X" from Belloni et al. 1998 and "GX" from Gondoin 2005). For both tables, the column headings describe the same measurements as in Table 2, with the exception of the first column that provides the respective photometric variable or X-ray ID. There are a few photometric variable stars that we have observed, but have not included in Table 4; each are W UMa stars (Zhang et al. 2002). We have observed star 5337 (V4) five times at the WIYN 3.5 m and find an average RV of −45.7 km s⁻¹ with a standard deviation of 1.19 km s⁻¹. We do not observe this star to be a rapid rotator; however the correlation function is unusually shaped requiring more detailed analysis. We require more observations of this star to derive a reliable membership and classification. We have also observed stars 5361 (V3), 4989 (V5), and 5209 (V13) multiple times, but have been unable to derive reliable RV measurements from their spectra, most likely due to their rapid rotation.

Without any membership selection, the CMD of NGC 188, within our magnitude range and spatial coverage, reveals little more than a poorly defined main sequence. Plot (a) of Figure 8 shows the locations on a CMD of each of the 1498 stars within our sample of NGC 188. Plot (b) shows only our RV member stars. The power of our combined RV and PM kinematic membership selection is clearly shown in plot (c), which displays our cleaned CMD plotting only our 473 3D kinematic member stars (P_PM > 0% and P_RV > 0%), and highlighting our 124 velocity-variable member stars. The main sequence is clearly visible in Figure 8(c) along with the main-sequence turnoff at about V∼ 14.9 and a well-defined giant branch.

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We have defined our BS stars as being bluer than the dashed line drawn in Figure 8(c) and having a non-zero PM and RV membership probability. These stars are both bluer and brighter than the main sequence and main-sequence turnoff. We have excluded stars near the turnoff so as to be sure not to include equal-brightness main-sequence binaries. The blue straggler stars are noted as BS in the comment field of Table 2. We have identified 21 BS stars, 16 of which are velocity variables with e/i > 4. Two of our BS stars (1366 and 8104) are fainter than the traditional BS region and are noted below. All but two of our BS stars were identified by P03; star 1947 has P_PM = 2%, and star 1366 has P_PM = 3%.

Interestingly, we note the very large frequency of detected binaries amongst the BS stars (16 out of 21). We will discuss these binaries in detail in our next paper (A. M. Geller et al. 2008, in preparation). Here we simply note that this large frequency is secure as we have derived reliable orbit solutions for all but three of our BS velocity variables; two of these remaining BLM stars have preliminary orbit solutions and are certainly binary stars.

We find 66 red-giant stars,¹¹ 16 of which are velocity variables. We also note the large scatter amongst the giant-branch stars in our cleaned CMD. Indeed, the NGC 188 giant branch is much broader and less well defined than that of M67, a similarly aged old open cluster (4.5 Gyr) (e.g., Janes & Smith 1984). This scatter in the NGC 188 giant branch was previously noted in the photometric studies by McClure (1974), McClure & Twarog (1977), and Twarog (1978), and is confirmed after our careful membership selection. The origin is still uncertain.

8.1.1. Stars of Note

The currently accepted model predicts that as two-body interactions lead toward equipartition of kinetic energy, we should, in a relaxed cluster, expect the more massive stars and star systems to have fallen to the center of the cluster potential dominated by the less massive stars. Mass segregation has been observed in relaxed clusters of a variety of ages, from the young open clusters Pleiades (100 Myr; Raboud & Mermilliod 1998a) and Preaesepe (800 Myr; Raboud & Mermilliod 1998b) to the old open cluster M67 (4.5 Gyr; Mathieu & Latham 1986). Harris & McClure (1985a) derived a relaxation time for NGC 188 of ∼10⁸ years, which suggests that NGC 188 may have lived through at least tens of relaxation times. Thus we would expect NGC 188 to be mass segregated. Indeed Sarajedini et al. (1999) found evidence for mass segregation in their photometric study of NGC 188. In addition, a study by Kafka & Honeycutt (2003) found their NGC 188 photometric variable star sample to be centrally concentrated, affirming the presence of mass segregation if their photometric variables are binary stars. Conversely Dinescu et al. (1996) did not find their BS population to show significantly more central concentration than the giant stars. If we assume that BS stars are formed through mergers of two (or more) stars (e.g., Bailyn 1995; Stryker 1993), we would expect the BS stars to be more massive than the giants. Hence this finding suggests a lack of mass segregation amongst the BS population, which would appear to disagree with Sarajedini et al. (1999) and Kafka & Honeycutt (2003). Therefore, we reanalyze NGC 188 for the presence of mass segregation in light of the precise membership determination presented in this paper. We compare the spatial distributions of the giant, velocity variable, and BS stars to that of the single stars (i.e., stars with constant velocity) within our NGC 188 sample, and plot the results in Figure 9.

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We first investigate the velocity variables by comparing their spatial distribution to the single stars. A Kolmogorov–Smirnov (K-S) test shows that the velocity variables are derived from a different parent spatial distribution than the single stars with 92% confidence. Presuming that the velocity variables are binary- or multiple-star systems, then the central concentration of the more massive binary-star population offers further evidence for the presence of mass segregation in NGC 188.

Next, we investigate the spatial distribution of the NGC 188 BS population by again comparing to the single-star population. A K-S test shows a low distinction of 63% between the two populations. We therefore do not find the full population of BS stars to be centrally concentrated with respect to the single stars, agreeing with Dinescu et al. (1996). The majority of our BS stars are found in binary systems; additionally, as stated above, the formation of BS stars is likely the result of mergers of two (or more) stars. Hence their combined mass is likely much greater than that of a single star in our sample. As such, this lack of central concentration is surprising and may provide a significant insight into the formation mechanisms and subsequent dynamical evolution of the NGC 188 BS stars.

Interestingly, in the similarly aged open cluster M67 (4.5 Gyr), Mathieu & Latham (1986) find secure evidence for mass segregation of the BS population. Perhaps importantly, their sample only extended to ∼five core radii,¹² whereas our NGC 188 sample extends to ∼13 core radii. If we limit our NGC 188 sample to five core radii and again compare the spatial distribution of the BS and single-star populations, we find 95% distinction between the two populations. Thus, this restricted population of BS stars does show evidence for central concentration.

We also note the seemingly bimodal spatial distribution amongst the NGC 188 BS stars. We suggest that we may be observing two populations of BS stars, with a centrally concentrated population, within ∼five core radii, and a halo population extending to the edge of the cluster. Similar bimodal BS spatial distributions have been observed in some globular clusters (e.g., M55, Lanzoni et al. 2007; 47 Tuc, Ferraro et al. 2004; NGC 6752, Sabbi et al. 2004; and M3, Ferraro et al. 1997). These bimodal spatial distributions are likely the result of the formation mechanism(s) and dynamical evolution of the BS populations. Scattering experiments performed by Leonard & Linnell (1992) suggest that BS stars formed through collisions within a population of primarily short-period binaries may be given enough momentum to achieve long radial orbits, spending the majority of their time in the outer cluster regions where the relaxation time is longer. Thus these halo BS stars may not attain significant central concentration. This result agrees with Sigurdsson et al. (1994) who suggest that the BS stars are likely formed in the core and ejected to varying cluster radii depending on the recoil of the dynamical interaction. BS stars that are only ejected to a few core radii quickly fall back to the cluster center through dynamical friction and mass-segregation processes, while those ejected farther may remain as a halo population. Conversely Ferraro et al. (1997, 2004) suggest that such a bimodal distribution is the result of two BS formation mechanisms, with the outer BS stars formed through mass transfer in primordial binaries, and the inner BS stars formed from stellar interactions which lead to mergers.

We can use Equation (5) and our detailed analysis of the precision of our measurements to solve for the combined RV dispersion of NGC 188: σ²_cb = σ²_c + σ²_b = σ²_obs − σ²_i. Using the average values derived above, the resulting global value of the combined velocity dispersion is σ_cb = 0.64 ± 0.04 km s⁻¹. This value is likely inflated due to the presence of undetected binaries; we attempt to evaluate this overestimate below.

Mathieu (1983, 1985) studied the effect of such undetected binaries in the old open cluster M67 through a Monte Carlo analysis. This cluster has a similar observed binary frequency and combined velocity dispersion to NGC 188. The Monte Carlo simulations used an Abt & Levy (1976) binary distribution. The results from that study suggest that for a combined velocity dispersion of 0.5 km s⁻¹ and a total binary fraction of 50%, one would expect the combined velocity dispersion to be inflated by 0.23 km s⁻¹. Given our observed hard-binary (P< 10⁴ days) fraction of roughly 30%, we expect a contribution from undetected binaries to be similar to the values derived by Mathieu (1983, 1985).

Given the mean velocity dispersion, we can estimate the virial mass of NGC 188. We use the equation given by Spitzer (1987), in projection:

$$\begin{equation} M = \frac{10\big\langle \sigma _c^2\big\rangle R_h}{G}, \end{equation} \tag{ 8 }$$

where σ_c is the line-of-sight (radial) velocity dispersion, and R_h is the projected half-mass radius. We use the P03 PM members (i.e., stars with P_PM > 0%) to derive R_h = 5.8 ± 0.3 pc. We estimate σ_c = 0.41 ± 0.04 km s⁻¹, using the correction value derived by Mathieu (1983, 1985) to recover σ_c from our derived σ_cb given above, and assuming an error equal to that of σ_cb. The resulting virial mass is 2300 ± 460 M_☉ . This value agrees well with the total mass one gets by counting the P03 PM members; if we assume a mass-to-light relationship of M ∝ L^1/3 and use the proper bolometric correction factors, we find a total mass of 2350 M_☉ . Our mass estimate also lies within the 90% confidence range of 230 to 2600 M_☉  derived by Harris & McClure (1985a) with an isotropic King model fit (King 1966). Additionally, our mass estimate agrees with the total mass of 3800 ± 1600 M_☉  derived by Bonatto et al. (2005) from integrating an extrapolated mass function.

We have also looked for a variation in the combined RV dispersion with the radial distance from the cluster center, again using only our SM stars (P_RV > 0%, P_PM > 0% and e/i < 4). We have chosen to plot the upper limits on the combined RV dispersion without attempting to correct for undetected binaries, and have binned the data in equal steps in log radius. We only use the WIYN RV data for this analysis to avoid unnecessary complications from the difference in precision of the two data sets. The results are shown in Figure 10. The combined RV dispersion ranges from ∼0.8 km s⁻¹ in the radial bins closest to the cluster center to ∼0.6 km s⁻¹ in the bins farthest away from the cluster center. We note that the central rise in combined RV dispersion is likely inflated due to undetected binaries, as the binaries are centrally concentrated in NGC 188 (see Section 8.2). Indeed, applying the correction described above to the central bins would lower these values to approximately that of our farthest bins, where we would expect the least inflation by undetected binaries. Moreover, this correction results in an RV dispersion distribution that is indistinguishable from an isothermal distribution.

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Harris & McClure (1985a) also performed a study of the radial dependence of the combined RV dispersion of NGC 188, though with a smaller sample size than our study. They found a higher combined dispersion in the inner cluster region of 1.16 (with a range of 0.88–1.81) km s⁻¹, but the values in their remaining radial bins, of 0.51 (0.22–1.06) km s⁻¹ at 5.6–13 arcmin and 0.20 (0.12–0.40) km s⁻¹ at 13–20 arcmin, are similar to the values we have obtained here. Undoubtedly, the combined dispersion measured by Harris & McClure in the inner bin was inflated due to undetected binaries, as that paper also concluded based on Monte Carlo models of observations of binaries.