WIYN OPEN CLUSTER STUDY LII: WIDE-FIELD CCD PHOTOMETRY OF THE OLD OPEN CLUSTER NGC 6819

We begin our analysis of NGC 6819 by determining the position of the cluster center. Locating an accurate cluster center is an important initial step for deriving robust physical parameters of star clusters. For instance, faulty centering would artificially flatten the inner part of the surface density profile of a star cluster, and this can cause significant systematic errors in any structural parameters such as core/tidal radii and total mass, and as a result important information on the dynamical evolution of the cluster could be compromised.

The centering of Galactic open clusters is generally less straightforward than Galactic globular clusters because the former tends to be more asymmetric than the latter. In order to avoid significant biases in the centering, we employed three independent centering algorithms and took the average of these results to derive an accurate center for the cluster.

First, we used well-photometered stars which passed our stellarity criteria using the image-shape statistics provided by DAOPHOT II (i.e., 0.2 < SHARP < 0.5, −0.2 < ROUND < 0.2, χ² < 2.0). We counted these stars within narrow bands (80 pixels wide) in both the horizontal (X) and vertical (Y) directions across the images. These counts are shown in Figure 2. We adopted the location at which the counts reached a maximum as the (X, Y) position of the cluster center.

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The second centering algorithm we employed is called the "four quadrants method" (i.e., Section 3 of K01). We adopted the (X, Y) location from the previous method as an initial estimate of the cluster center. Then we counted the number of stars in four equal quadrants around this center. We repeated the star counting by randomly moving the (X, Y) position of the center within a small circular area around the initial center until the number of stars in all four quadrants agreed.

Finally, we implemented a Monte Carlo centering method similar to the algorithm employed by Mackey & Gilmore (2003). We first placed a boxed area with a size (1200 × 1200 pixels) large enough to cover the central region of the cluster. Next, we randomly selected an (X, Y) position as a test center. We calculated the surface density in a circular area with a radius of 600 pixels about this test center. We repeated this random selection of the test center location 10,000 times. We consider the (X, Y) position that yields the highest surface density as the cluster center. The results of these three independent centering algorithms agree reasonably well with each other. The average value of the (X, Y) positions provides our final estimate of the cluster center; we find (X, Y) = (3297.86, 4770.56) or (R.A.(J2000) = 19:41:16.43, decl.(J2000) = +40:11:48.88). Our estimate of the centering errors in the X and Y directions is ∼25 pixels (106) each.

Our estimation of the cluster center shows excellent agreement with the recent calculation by Hole et al. (2009; R.A. = 19:41:17.5, decl. = +40:11:47; hereafter H09) obtained using stars with moderate-to-high membership probability (p > 50%); however, it differs slightly from the center published by K01 (i.e., R.A. = 19:41:17.7, decl. = +40:11:17) by ∼30''. Our tests have revealed that this difference is not significant enough to cause serious biases in the structural parameters of the cluster determined herein. Therefore, we will adopt our value for the cluster center in the remainder of our analysis.

Figure 3 presents deep VI CMDs in the region of NGC 6819. The two upper panels show CMDs for the cluster field defined by its visual extent (R < 13') and the control field acquired from the outskirts (27' < R < 30') of the cluster with an area equal to that of the cluster field. These two CMDs provide useful insight to understand the degree of field star contamination (see Section 3.5.1 for detail). In order to see the characteristic features of the cluster more clearly in the CMD, we extract stars from a central area (R < 6') centered around the cluster as shown in the lower left panel of Figure 3; this shows a clean RGB extending ∼1.5 mag brightward from a well-populated helium burning red clump (RC) at V ∼ 13 mag. There is a sparsely populated subgiant branch, a well-developed main sequence (MS) from V ∼ 14.5 mag down to our photometric limit of V ∼ 21 mag, and the signature of equal-mass binary stars (0.75 mag brighter than the MS). We note, however, that the observed equal-mass binary sequence in our VI CMD is not as prominent as that of K01. The relative sparseness of our binary sequence combined with the degree of field star contamination made obtaining a robust estimate of the cluster binary fraction challenging. Therefore, we have decided to adopt the result of K01 (binary fraction ∼11%) for the fraction of equal-mass binaries in NGC 6819.

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We also see the presence of a number of blue straggler stars (BSSs), which are more luminous and bluer than the MS turnoff. BSSs are rejuvenated massive stars which likely form via stellar mergers either due to dynamical collisions of two MS stars or through the coalescence of close binaries, and mass transfer of an evolved star onto the less massive MS partner in binary systems (Glebbeek et al. 2008). Some BSSs exhibit pulsation properties which obey period–luminosity and period–luminosity–color relations similar to those of classical Cepheids (Mateo 1993). The existence and pulsation properties of the BSS population in old Galactic open clusters might shed some light on their formation mechanism and role in the dynamical evolution of the host cluster. We will discuss the membership and physical properties of the BSS population of NGC 6819 in Section 3.6.

In order to derive the cluster age accurately by fitting theoretical isochrones to the observed CMD, we need well-determined cluster parameters such as reddening and metallicity. The early measurements of the interstellar reddening toward NGC 6819 in the 1970s spanned a substantial range between high (E(B − V) ∼ 0.30: Lindoff 1972; Auner 1974) and low (E(B − V) = 0.12: Burkhead 1971) values. However, as more observations of NGC 6819 have been accumulated, the reddening measurements from various independent studies have converged to the relatively low value of E(B − V) ∼ 0.15. The main reason for the high reddening measurements in the early studies is likely attributed to the photographic observations in the U passband, which was known to be of low photometric quality.

For the present study, we have adopted the reddening and metallicity values (E(B − V) = 0.14 ± 0.04; [Fe/H] = +0.09 ± 0.03 dex) of NGC 6819 from the measurements of Bragaglia et al. (2001, hereafter B01). Using effective temperatures of three RC stars in NGC 6819 derived from line excitation, which is a reddening-free index, B01 measured the interstellar reddening toward the cluster. The reddening was calculated by comparing the observed (B − V) colors with the intrinsic colors computed using the measured temperatures. Any possible zero-point offset in the intrinsic colors derived by model atmospheres was also corrected using a sample of 12 bright RC stars with accurate Hipparcos parallaxes. They selected this comparison sample to be similar to the stars of NGC 6819 in temperature and metallicity. B01's measurement of the interstellar reddening toward NGC 6819 agrees very well with other independent studies (Burkhead 1971; RV98; K01; Kang & Ann 2002). In addition, their study appears to be the lone high-dispersion spectroscopic analysis of this cluster. It is for this reason that we consider the reddening and metallicity values of B01 to be the most robust measurements for NGC 6819 to date.

Applying the adopted values of the cluster reddening and metallicity in the previous section, we generated a set of isochrones using the Dartmouth Stellar Evolution Database (Dotter et al. 2007, 2008(D08)). The database program handles convection by following the general principles of the standard mixing length theory and a solar-calibrated mixing length, α_ML = 1.938. Convective core overshooting is treated in the same fashion as the Y² models developed by Demarque et al. (2004); the amount of convective core overshooting is parameterized as a multiple of the pressure scale height (λ_P), and is a function of stellar mass and composition. The database program assigns 0.05λ_P of overshooting for the minimum stellar mass of 0.1 M_☉; then the amount of overshooting is linearly ramped from this value (0.05λ_P) with an increment of 0.1λ_P as the stellar mass increases by 0.1 M_☉; models with masses greater than or equal to the minimum plus 0.2 M_☉ receive 0.2 λ_P of overshooting.

The database uses a grid of PHOENIX model atmospheres (Hauschildt et al. 1999a, 1999b) for the range of T_eff = 2000 to 10,000 K and log g = −0.5 to 5.5 to derive the surface boundary condition (BC). The computation of PHOENIX model atmosphere grids covers the wide range of metallicity and α-element ratio (−2.5 < [Fe/H] < +0.5; −0.2 < [α/Fe] < +0.8). The effects of Helium enhancement to the physical structure and the stellar spectrum are also considered in the model calculation. We adopted the solar values of the α-element ratio and He content ([α/Fe] = 0.0; Y = 0.27) for the calculation of the isochrones.

A series of theoretical isochrones was calculated for a fixed metallicity of [Fe/H] = +0.09 dex and ages ranging from 1 Gyr to 4 Gyr. The interpolation routine in the database produces isochrones with a fine age spacing of 0.1 Gyr. Doing so allows for more precise age constraints. These theoretical isochrones were transformed into the Johnson–Cousins photometric system using the filter transmission curves defined by Bessell (1990). Adopting the reddening, E(B − V) = 0.14, and the initial estimate of distance modulus, (m − M)_V = 12.30 taken from K01, we shifted the theoretical isochrones in the absolute magnitude–intrinsic-color plane into the observed color–magnitude plane. Then, we varied the age and distance until a satisfactory fit was achieved between the observed CMD and the isochrones. Figure 4 shows our isochrone fits to the observed data of NGC 6819. We see that the 2.6 Gyr Darthmouth isochrone (red) with a refined distance modulus, (m − M)_V = 12.36, yields the best description of the observed data. For comparison we also plot a best-fitting Padova isochrone (cyan) calculated from their latest stellar evolution code, PARSEC (Bressan et al. 2012; B12), for the given reddening, distance, and age of the cluster. The Padova isochrone agrees well with the main features of the observed cluster CMD, such as the RGB, RC, and MSTO. However, it starts to show offsets in the lower MS (V > 18.5 mag) in the sense that the isochrone becomes systematically bluer than the observed sequence. The two model sequences also show a marginal offset at the subgiant branch, and upper-MS hook which is a characteristic feature among the old Galactic open clusters partly due to the effects of convective core overshooting. This offset is likely caused by the different treatment of convective core overshooting in each of the stellar evolution codes.

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The given amount of error (±0.04 mag) in the adopted reddening value yields a change in the derived distance modulus of ±0.1 mag and a change in the derived age of ±0.6 Gyr for a fixed [Fe/H] value of 0.09 dex. A change in [Fe/H] of ±0.1 dex results in a change in the derived age of ±0.3 Gyr and distance modulus of ±0.08 mag for the adopted E(B − V) value of 0.14 mag. This best-fitting Darthmouth isochrone seems to reproduce most of the characteristic features of the cluster CMD from the tip of the RGB down to the faintest low-mass MS stars.

Our estimates for the age and distance modulus of NGC 6819 agree well with those determined from the recent analysis of Basu et al. (2011). Just like in this study, Basu et al. (2011) adopted the same reddening and metallicity values from Bragaglia et al. (2001). They measured a cluster age of 2.4 Gyr and a distance modulus of (m − M)_V = 12.32 by employing a model-independent analysis of the astroseismic quantity, Δν using data obtained by NASA's Kepler mission. Our result is also consistent with previous estimates in the literature ((m − M)_V = 12.35, RV98; 12.30, K01; 12.11, Kang & Ann 2002; 12.30, H09).

The CMD in Figure 5 shows a direct comparison of our photometry for NGC 6819 and that of M67 (Montgomery et al. 1993), which is one of the most-studied old open clusters. This provides an alternative test of our adopted reddening and distance as well as the derived age for NGC 6819. In order to perform this direct CMD comparison, we need to adopt a reddening and distance for M67, for which we rely on the results of Sarajedini et al. (1999), who found E(B − V) = 0.04 and (m − M)_V = 9.69. These values along with the reddening and distance of NGC 6819 (E(B − V) = 0.14 and (m − M)_V = 12.36) allow us to produce the CMD comparison shown in Figure 5. We see that there is a nearly exact match of the unevolved MSs of these two clusters, and a reasonable separation of the MS turnoffs predicted by the age difference (Δt_age ∼ 2 Gyr) between them. The RGBs are also very well aligned, indicating that the two clusters have similar metallicities within roughly 0.1–0.2 dex. Thus, this CMD comparison strengthens the validness of our estimates of the distance modulus and age for NGC 6819.

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Now we direct our attention to the dynamical properties of the cluster. These have previously been investigated by K01 and Kang & Ann (2002). K01 conducted a wide-field (42' × 28') BV imaging survey of NGC 6819 using the 3.6 m Canada–France–Hawaii Telescope (CFHT) equipped with the CFH12K mosaic CCD. The structural parameters such as the core and tidal radii (r_c = 1.75 pc; r_t = 17 pc) of the cluster were derived through a King model fit to the observed density profile constructed with star counts. They found an almost flat present-day luminosity function (LF), and the clear signature of mass segregation from the analysis of radial CMDs and mass functions (MFs); their results indicate that NGC 6819 has dynamically evolved, and now contains mostly high-mass stars in the inner regions and lower mass stars at larger radii. Subsequently, Kang & Ann (2002) presented VI CCD photometry of the cluster. Their observational data covered an area of 18' × 18' on the sky, which is smaller than K01's field but still covered the visual extent of the cluster. By analyzing the derived LF and radial density profile, they confirmed K01's result that the cluster clearly exhibits mass segregation as a result of its dynamical evolution.

The NGC 6819 photometry presented herein represents the widest field coverage of this cluster to date. As such, it covers virtually the entire dynamical extent of the cluster. This provides an excellent opportunity to investigate the dynamical structure of the cluster in great detail. In the following two subsections, we describe our analysis of the surface density profile, LF/MF, and mass segregation in the cluster.

3.5.1. Surface Density Profile and Cluster Mass

In order to construct the surface density profile of the cluster, we used well-photometered stars that had passed our stellarity criteria of the image-shape statistics (i.e., 0.2 < SHARP < 0.5, −0.2 < ROUND < 0.2, χ² < 2.0; see Section 3.1), and fall in the magnitude range, 15 <V < 20 (i.e., the corresponding mass range is 0.73–1.52 M_☉). A series of concentric circles around the cluster center (R.A.(J2000) = 19:41:16.43, decl.(J2000) = +40:11:48.88, see Section 3.1) was used to construct the surface density profile shown in the upper panel of Figure 6. The stellar density appears to flatten out at ∼13' and becomes indistinguishable from the background field beyond this radius. We consider this region to constitute the visual extent of the cluster, "r_cl," which is the largest yet identified for this cluster (e.g., K01 measured a cluster extent of ∼95). The surface density profile of the cluster shown in the lower panel of Figure 6 includes corrections for photometric incompleteness (see the next section for detail) and for the density of the background field. The average background density was calculated in the outskirts (25' < R < 30') of the cluster.

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Once the background-subtracted photometrically complete surface density profile was constructed, we fitted the following empirical King model (1962) to this profile:

$$\begin{equation} f(r) = k \Bigg [\frac{1}{[1+(r/r_c)^2]^{1/2}} - \frac{1}{[1+(r_t/r_c)^2]^{1/2}} \Bigg ]^2, \end{equation} \tag{ 3 }$$

where f(r) is the number density as a function of r, r_c is the core radius, r_t is the tidal radius, and k is a constant related to the central surface density. We employed a Maximum likelihood method powered by a genetic algorithm PIKAIA (Charbonneau 1995) to derive the best-fit structural parameters of NGC 6819. We obtained r_c = 280 (∼2 ± 0.10 pc), r_t = 382 (∼27 ± 1.2 pc), and log (r_t/r_c) = 1.13. The quoted errors in r_c and r_t reflect the amount of errors propagated from the expected error of 0.10 mag in our distance modulus measurement.

Next we can calculate the mass of the cluster within its tidal radius. Accurate measurements of the current masses of star clusters are crucial for understanding cluster survival and destruction processes in the Galactic disk. According to the classical theory of Galactic dynamics, the tidal force exerted on star clusters reaches its maximum when the clusters are passing through their perigalactic positions. The tidal radius of a cluster is defined as the inner Lagrangian point where the Galactic gravitational potential exactly equals the gravitational potential of the cluster. Using this definition of the tidal radius of a star cluster and the point-mass Galactic model, King (1962) derived a relation between the tidal radius and the mass of a star cluster as follows:

$$\begin{equation} r_t = R_p \Big [\frac{M_c}{(3+e)M_g}\Big ]^{1/3}, \end{equation} \tag{ 4 }$$

where R_p is the perigalactic distance of the cluster from the center of the Milky Way, e is the eccentricity of the cluster orbit, and M_g is the mass of the Milky Way within R_p. This simple version of the formula has been used in the calculation of the total cluster mass using the tidal radius obtained from fitting a King model to the observed surface density profile, or alternatively, the tidal radius by assuming the total mass of the cluster. However, both theoretical considerations and numerical computations (Innanen et al. 1983; Keenan 1981) call for caution with regard to the use of this equation. This is because the tidal radius in this formula is not the same as the tidal radius derived from King model fitting to the observed surface density profile. The tidal radius calculated from the point-mass Galactic model (Equation (4)) refers to the semi-minor axis of the elongated tidal surface of the cluster, while the tidal radius obtained from the King model fitting represents the radius of the isothermal sphere at which the cluster stellar density vanishes into the Galactic field. This difference means that the semi-minor axis of the elongated tidal surface is two-thirds of the radius (r_{t, King}) of the isothermal sphere (i.e., the distance from the cluster center to the inner Lagrangian point). Therefore, the corrected tidal radius formula should be

$$\begin{equation} r_{t,{\rm King}} = \frac{3}{2} R_p \Big [\frac{M_c}{(3+e)M_g}\Big ]^{1/3}. \end{equation} \tag{ 5 }$$

By re-arranging the above equation, we obtain

$$\begin{equation} M_c = (3+e)M_g\Big [\frac{2}{3} \frac{r_{t,{\rm King}}}{R_p}\Big ]^3. \end{equation} \tag{ 6 }$$

In order to calculate the tidal mass of NGC 6819, we adopted R_p = 7.6 kpc as its perigalactic distance and e = 0.03 as the orbital eccentricity of the cluster from Wu et al. (2009). The Galactic mass M_g within R_p can be calculated using M_g = V²₀R_p/G = 8.8 × 10¹⁰ M_☉ where V₀ = 220 km s⁻1 is the circular rotation speed of the Milky Way. Using these inputs into Equation (6), we obtained the tidal mass of NGC 6819 to be M_{c, tid} = 3542.4^+536.8_{− 451.6} M_☉. We also estimated the total mass of the cluster by integrating the global MF which is derived from the stellar-mass–luminosity relation of the best-fitting stellar evolution model (see Section 3.5.3). Since the MF was obtained from the cluster LF with the help of theoretical models of stellar evolution, the total mass derived by this method is referred to as a "photometric mass," M_phot. We obtained M_{c, phot} = 2010 M_☉ which agrees fairly well with K01's calculation (∼2600 M_☉) of the photometric mass of NGC 6819. The estimates of the total mass of NGC 6819 from these three values yield the mean mass of NGC 6819 of 〈M_c〉 ∼ 2700 M_☉.

3.5.2. Luminosity Function

We constructed the present-day LF using the cluster MS stars with V magnitudes in the range 15 < V < 20. The number of cluster MS stars was statistically estimated by the following method. We defined the MS envelopes in the VI CMD obtained from the central region (R < 6') of the cluster with two loci aligned with the blue and red edges of the MS. Using this MS envelope, we drew stars from both the cluster region (R < 13') and an off-cluster region (25' < R < 30') with an area equal to that of the cluster field.

We need to correct for photometric incompleteness in order to properly interpret the cluster LF. This was done through a series of artificial star tests. We employed the ADDSTAR routine in DAOPHOT II to generate ∼2000 artificial stars with known magnitude and position. These were evenly distributed on each V and I program image. These test images were reduced using the same procedure as the original images. Our artificial star tests revealed that both the cluster and off-cluster fields show very similar photometric completeness profiles, and have a very similar overall recovery rate of ∼80%. The selection of the cluster MS stars and the photometric completeness as a function of V magnitude are illustrated in Figure 7. The photometric completeness correction factors are summarized in Table 2.

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Table 2. Photometric Completeness for the Cluster and Background Fields

Range	Correction Factor
	1/f = Nin/Nout
V (mag)	1/fcl	1/fbg
14.0–14.5	1.00	1.00
14.5–15.0	1.00	1.00
15.0–15.5	1.00	1.00
15.5–16.0	1.00	1.00
16.0–16.5	1.01	1.00
16.5–17.0	1.05	1.00
17.0–17.5	1.09	1.04
17.5–18.0	1.03	1.00
18.0–18.5	1.06	1.00
18.5–19.0	1.18	1.29
19.0–19.5	1.46	1.53
19.5–20.0	1.65	1.65
20.0–20.5	2.32	2.95
20.5–21.0	2.22	2.13

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The upper panel of Figure 8 shows a comparison between our LF (solid line) and that of K01 (dot-dashed line). The zero-point offset between these two LFs seems to be caused by the different size of the MS envelopes used for the selection of MS stars in each study. The error bars on each point reflect the uncertainties due to Poisson statistics and the photometric incompleteness correction. We followed K01's recipes for the estimation of the errors in the global luminosity and mass functions. The standard deviation in N, σ_N is derived from the error propagation equation (Bevington & Robinson 1992, Equation (3.14), p. 43) as follows:

$$\begin{equation} \sigma _{N}^2=\frac{N_{{\rm obs}}}{f^2}+\frac{(1+f)N_{{\rm obs}}^2}{N_{{\rm in}}f^3}, \end{equation} \tag{ 7 }$$

where N is incompleteness-corrected count defined by N = N_obs/f; f is photometric incompleteness given by f = N_out/N_in (i.e., therefore the completeness correction factor should be the inverse of the incompleteness, 1/f).

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These two LFs generally agree in the sense that both exhibit almost flat slopes, though K01's LF tends to decrease slightly toward faint magnitudes. The slope of the present-day LF of a given star cluster can serve as an excellent diagnostic tool for examining the dynamical state of clusters. This flat or very weak slope of the present-day LF of NGC 6819 indicates that the cluster is old enough to have lost a significant portion of its low-mass stars through evaporation as it has dynamically evolved.

3.5.3. Mass Function and Mass Segregation

Now we attempt to derive the present-day MF of NGC 6819 to further investigate the dynamical evolution of the cluster. The MF is defined as the number of stars per unit logarithmic mass interval, ϕ(log m) = dN/dlog m, and often expressed by the power law, ϕ(log m) ∝ $m^\Gamma$, where the slope of the MF is given as Γ = dlog ϕ(log m)/dlog m. The traditional value of Γ for the Salpeter IMF is Γ = −1.35. The present-day LF of NGC 6819, obtained in the previous section, was converted into the MF using the stellar-mass–luminosity relation defined by the theoretical stellar evolutionary tracks of Dotter et al. (2007, 2008). The lower panel of Figure 8 shows the present-day MF of the cluster MS stars drawn from the cluster region (R < 13'). The value of the cluster MF slope, Γ = −0.65, was computed using linear least-squares fitting as shown as a dashed line. The Salpeter IMF (Γ = −1.35) is also plotted in the same plane for comparison. We see that the present-day MF of NGC 6819 is shallower than the Salpeter IMF. Again we interpret this flattened cluster MF as being mainly due to the evaporation of low-mass stars during the dynamical evolution of the cluster. In Figure 9, we plot radial MFs. The slope of the MFs tends to be flatter in the central regions and appears to be steeper at larger radii, indicative of the presence of mass segregation in the sense that high-mass stars preferentially lie near the cluster center. The resultant luminosity and mass functions with detailed statistics are summarized in Table 3.

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Table 3. The Present-day Luminosity and Mass Functions for NGC 6819

V	Range	〈M〉	Count	σN	LF	MF
(mag)	Mass (M☉)	(M☉)	N		log N	log ϕ
15.0–15.5	1.5204–1.3509	1.4357	179.00	38.22	2.2529	3.5424
15.5–16.0	1.3509–1.2384	1.2946	183.00	42.16	2.2625	3.6851
16.0–16.5	1.2384–1.1434	1.1909	207.05	26.49	2.3161	3.7765
16.5–17.0	1.1434–1.0578	1.1006	223.65	26.64	2.3496	3.8204
17.0–17.5	1.0578–0.9795	1.0187	249.61	30.44	2.3973	3.8739
17.5–18.0	0.9795–0.9078	0.9437	181.28	25.07	2.2583	3.7396
18.0–18.5	0.9078–0.8441	0.8760	152.64	21.49	2.1837	3.6837
18.5–19.0	0.8441–0.7934	0.8187	238.36	31.26	2.3772	3.9477
19.0–19.5	0.7934–0.7436	0.7685	154.76	22.72	2.1897	3.7398
19.5–20.0	0.7436–0.6940	0.7188	265.65	37.72	2.4243	3.9479

Notes. N represents the number of cluster main-sequence stars after correcting the photometric incompleteness and field star contamination.

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To further evaluate this assertion quantitatively we calculated the half-mass relaxation time for the cluster using the following equation derived by Spitzer (1987):

$$\begin{equation} t_{{\rm rh}}= \frac{0.183M^{1/2}r_h^{3/2}}{mG^{1/2}{\rm ln}(0.4N)}, \end{equation} \tag{ 8 }$$

where M is the total mass of the cluster, m is the mean stellar mass, r_h is the radius containing half of the cluster mass, and N is the number of cluster members. We determined the half-mass radius using the cumulative mass distribution of the cluster stars. Figure 10 indicates that 50% of the cluster mass is contained within ∼7' (∼4.95 pc), which is very similar to half of the cluster extent. The total number of cluster stars was calculated by dividing the total mass of the cluster (M = 2700 M_☉, Section 3.5.1) by the mean stellar mass (m ∼ 1 M_☉ obtained from the global MF). By inserting these numbers into the equation above, we obtained the half-mass relaxation time of t_rh ∼ 200 Myr. Compared to the age of the cluster (2.6 Gyr), this relaxation time is about 10 times smaller, again indicating that NGC 6819 is old enough to be dynamically evolved and that the observed mass segregation is the result of its dynamical evolution.

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Searching for pulsating variable BSSs in open clusters is an important subject in stellar astrophysics because the pulsational properties of BSSs can be used to derive their basic physical properties such as mass and radius; they also shed light on the formation mechanism of BSSs in the relatively sparse environments of open clusters. However, our current knowledge of the periods and amplitudes of pulsating variable BSSs in open clusters is still highly uncertain. In his comprehensive review of BSSs, Mateo (1993) categorized variable BSS populations into two types: pulsating BSSs (δ Scuti or SX Phe type depending on its age) and eclipsing BSSs (mostly W UMa type). In particular, the pulsating BSSs are considered to be a class of Dwarf Cepheids (DCs) and appear to follow period–luminosity and period–luminosity–color relations similar to those of classical Cepheids. They pulsate with either fundamental or first-overtone mode periods ranging from ∼0.03 to 0.25 days, and exhibit very small amplitudes of 0.005–0.8 mag.

Previously, Kaluzny & Shara (1988, hereafter KS88) found three variable star candidates that are likely short-period contact binaries in a 4' × 4' field centered on NGC 6819. Among these three, there is one bright star (i.e., V2 in their list) included in the photometry of our study. Street et al. (2002, 2005) also detected a number of eclipsing binaries from a much wider cluster field (∼11' × 22'). They confined their search to variable stars in a magnitude range fainter than the MS turnoff. Most recently, Talamentes et al. (2010; hereafter T10) presented a detailed variability study of this cluster. They reported the detection of three bright blue straggler variables. The brightest one appears to be a δ Scuti-type pulsator, however, its color is slightly bluer than the blue edge of the instability strip of δ Scuti variables. Star A494 (from their table) turned out to be the same star as V2 discovered by KS88, while the other one is likely a newly detected variable BSS probably in the process of merging.

We carefully investigated the variability of our BSS candidates using the Welch–Stetson variability index (I_WS; Welch & Stetson 1993). We selected probable cluster BSS candidates from the VI CMD using our color–magnitude criteria (11.5 <V < 15.5; 0.2 <V − I < 0.7). This selection of cluster BSS candidates was restricted to within the half-mass radius (r_h = 7') of the cluster in order to increase the chances of including true cluster members. Then, we calculated I_WS values for these BSS candidates.

Figure 11 presents the I_WS index as a function of the V magnitude. Filled circles represent the probable cluster BSSs, while the variable BSSs confirmed in the previous studies (KS88; T10) are marked by filled diamonds. A probable δ Scuti-type variable candidate newly detected in this study is shown as a red triangle (see the following paragraphs). According to Figure 11, the photometric errors are less than 0.02 mag (see the upper panel) for all of our BSS candidates. We also see that the majority of our BSS candidates appear to be non-variable stars (I_WS < 1.0), while some show moderate variability (1.0 <I_WS < 2.0). There are two bright stars with large values of I_WS. They may not be bona fide variables because spuriously large values of I_WS could be the result of the small number of observations (i.e., n_obs < 6 for these stars). Figure 12 is a zoomed-in VI CMD showing these BSS candidates. We marked with crosses those BSSs whose I_WS values are greater than 1.0. We adopted the pulsational instability strip from T10, originally derived by Pamyatnkh (2000), and overlaid it on the CMD in order to isolate the pulsationally variable BSS candidates. We see that many BSSs with moderate variability (i.e., I_WS > 1.0) are located within this pulsational instability strip indicating that some of these could possibly be δ Scuti-type, short-period pulsators.

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We attempt to uncover possible δ Scuti-type pulsators by running our template light curve fitting routine, RRFIT (Yang & Sarajedini 2012), on all of the BSSs with moderate variability (I_WS > 1.0) and 30 individual photometric measurements (15 Vs + 15 Is). Figure 13 presents the best result from this exercise. We see what appears to be a probable δ Scuti-type BSS detected by our template fitting analysis. According to its best-fit light curve, this star appears to be a fundamental mode pulsator with a short period and a small amplitude. The derived period and amplitude fall in the period–amplitude range for typical δ Scuti-type pulsators (0.03 <P < 0.25 days; 0.005  <  Amp < 0.8 mag) consistent with the pulsational properties of the short-period pulsating BSSs found in M67 (Gilland & Brown 1992; P ∼ 0.06 days, Amp(V) ∼ 0.02 mag). However, we must stress that our estimates of the period and amplitude should be treated with caution because the observational data in this study are not optimized for the study of variable stars. The list of our BSS candidates in NGC 6819 is summarized in Table 4.

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