POPULATION PARAMETERS OF INTERMEDIATE-AGE STAR CLUSTERS IN THE LARGE MAGELLANIC CLOUD. II. NEW INSIGHTS FROM EXTENDED MAIN-SEQUENCE TURNOFFS IN SEVEN STAR CLUSTERS^*

Stellar photometry was conducted using point-spread function (PSF) fitting on the flat-fielded (flt) files produced by the HST calibration pipeline, using the spatially variable "effective PSF" (ePSF) method described in Anderson & King (2000) and tailored for ACS/WFC data by Anderson & King (2006). A detailed description of the application of the ePSF method to ACS/WFC data is given in Anderson et al. (2008b, 2008a). We selected all stars with the ePSF parameter "PSF fit quality" q < 0.5 and "isolation index" of 5. The latter parameter selects stars that have no brighter neighbors within a radius of 5 pixels. To further weed out hot pixels, cosmic rays, and spurious detections along diffraction spikes, the geometrically corrected positions among the three images per filter were compared. We selected objects with coordinates matching within a tolerance of 0.2 pixels in either axis, which eliminated the hot pixels and cosmic ray hits effectively.

Corrections for imperfect charge transfer efficiency (CTE) of the ACS/WFC CCDs were made specifically for the case of photometry from flt files featuring varying exposure times following Kozhurina-Platais et al. (2007). The accuracy of our CTE corrections is such that the rms scatter of the magnitude residuals is 0.02 mag at an instrumental magnitude of −2 (corresponding to m_F435W = 23.8, m_F555W = 24.4, and m_F814W = 23.5 in magnitude units relative to Vega). This uncertainty is smaller than the photometric measurement errors at those magnitudes.

Photometric incompleteness as functions of stellar brightness, color, and position in the cluster was quantified by repeatedly adding small numbers of artificial ePSFs to the images, covering the magnitude and color ranges found in the CMDs. The overall radial distribution of the artificial stars inserted followed that of the stars in the image. We refer the reader to Paper I for further details of our photometric methods and calibration.

CMDs for the star clusters in our sample were created for both m_F435W versus m_F435W − m_F814W and m_F555W versus m_F555W − m_F814W. CMDs for all selected stars are plotted in the left panels of Figures 1 and 2. To decrease and assess contamination by field stars, we also plot CMDs for all stars within one core radius based on fits to a single-mass King (1962) model from the cluster centers (derived as described in Section 3.3 below) in the middle panels, and CMDs for stars located in regions near the corners of the HST/ACS image, which are presumably dominated by non-cluster stars, in the right panels. The latter regions were chosen to cover the same surface area as those in the middle panels. In Figure 1 we identify and label the following evolved phases of stellar evolution: the MS, MSTO, red clump (hereafter RC⁵), the RGB, and the AGB.

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3.2.1. General Comments

3.2.2. Differential Reddening

The CMDs of most star clusters in our sample reveal compact RCs and/or well-defined RGBs and AGBs that have widths which are comparable to the photometric uncertainties. Since the reddening vector is approximately perpendicular to the RGB and AGB, this suggests that spatial variations in reddening are negligible for most clusters in our sample. There are, however, two exceptions: NGC 1751 and NGC 2108, both of which have a "fuzzy" RC and RGB. To correct their CMDs for differential reddening, we follow a method similar to that used by Sarajedini et al. (2007). We divide the cluster field into several subareas (the number of subareas depends on the total number of stars found in the cluster). After defining a grid of magnitude and color intervals along the MS below the MSTO region, a fiducial ridge line for the MS is derived for all stars located within the King core radius from running medians of star magnitudes and colors. We then measure the weighted mean m_F435W, m_F555W, and m_F814W magnitudes of stars within those grids for all subareas within the cluster. These weighted mean magnitudes define the local reddening in each subarea relative to the mean reddening within the King core radius of the cluster. These reddening values are then used to correct the magnitudes of stars in all subareas to a uniform reddening value. The effect of the correction for differential reddening is shown in Figures 3 and 4. The improvement is particularly significant for NGC 1751 where the RC becomes more compact than before the correction. The total amplitude of differential reddening within the cluster was found to be Δ E(m_F435W − m_F814W) = 0.14 and 0.10 for NGC 1751 and NGC 2108, respectively.

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We analyze the projected surface number density of stars in the sample star clusters for two main reasons: (1) to determine regions in the CMD that are strongly dominated by stars belonging to the cluster rather than to the underlying field in the LMC and (2) to allow an evaluation of dynamical properties of the clusters, which can help shed light on the origin of the eMSTOs. The cluster centers were determined by fitting a two-dimensional Gaussian to an image constructed from the completeness-corrected number density of stars brighter than the magnitude for which the completeness is 50% in the central regions in a given star cluster. This image was constructed using a bin size of 50 × 50 pixels (i.e., 25 × 25). Note that using the number density instead of surface brightness avoids biases that can arise because of a few bright stars near the center. The typical uncertainty of the centering procedure was ±5 pixels in either axis. The ellipticities of the clusters were derived by running the task ellipse within iraf/stsdas⁶ on the number density images mentioned above. Derived ellipticities stayed constant with radius to within the uncertainties. The area sampled by the ACS image was then divided into a number of centered, concentric elliptical annuli. The number of such annuli was chosen in an adaptive manner so as to include a minimum of 100 stars per annulus. The surface number density was corrected for incompleteness by dividing the number of stars by the average completeness fraction in each annulus. For annuli with radii larger than ∼850 pixels ($\hat{=}$ 425), care was taken to account for the limited azimuthal coverage of the cluster by the ACS/WFC image. Specifically, we first constructed a parallelogram whose edges stay 50 pixels within the area exposed by all (three) F555W exposures of a given star cluster. We then constructed "elliptical pie slices" that subtend angle intervals which are radius dependent in a way such that the angle subtended by the outer end of the pie slice fits fully within the parallelogram mentioned above. Stars were then counted within those pie slices, and the areas of each pie slice were evaluated to measure surface number densities. Error values were derived from Poisson statistics of the star number counts. Radii are expressed in terms of the "equivalent" radius of the ellipse, $r = a\, \sqrt{1-\epsilon }$ where a is the semimajor axis of the ellipse and its ellipticity. The resulting radial surface number density profiles were fitted with a King (1962) model combined with a constant background level:

$$\begin{equation} n(r) = n_0 \: \left(\frac{1}{\sqrt{1 + (r/r_c)^2}} - \frac{1}{\sqrt{1+c^2}} \right)^2 \; + \; {\rm bkg}, \end{equation} \tag{ 1 }$$

where n₀ is the central surface number density, r_c is the core radius, and c ≡ (r_t/r_c) is the concentration index (r_t being the tidal radius). The best-fit King models were selected using a χ² minimization routine that involves varying values of c. Figure 5 shows the best-fit models along with the individual surface number density values for each star cluster in our sample (except NGC 1846 for which we refer the reader to Paper I). Radius values in arcsec were converted to parsecs using the distance moduli listed in Table 3.

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3.3.1. Selection of Cluster-Dominated Regions on the CMD

The isochrone fitting was performed using the method described in detail in Paper I. We provide a less comprehensive description here, and concentrate most on parts of the procedure that are additions to the steps described in Paper I. We start the isochrone fitting by using parameters that involve pairs of fiducial points on the CMD that are (1) relatively easy to measure or determine from both the data and the isochrone tables, (2) sensitive to at least one population parameter such as age or metallicity, and (3) independent of the distance and foreground reddening of the cluster.

In the case of clusters with a well-defined RGB bump (i.e., NGC 1751, NGC 1783, NGC 1806, and NGC 1846), we use the following parameters (cf. Paper I).

• 1.  
  The difference in magnitude between the MSTO and the RGB bump,¹⁰ called ΔB^MSTO_RGBB and ΔV^MSTO_RGBB in the B and V filters, respectively. The MSTO is defined as the point where a polynomial fit to the stars (or the isochrones) near the turnoff is vertical in the CMD. We define the location of the RGB bump in the isochrones as the average magnitude and color of isochrone RGB entries between the two masses at which the magnitudes and colors "turn around" in direction on the CMD with increasing stellar mass.
• 2.  
  The difference in color between the MSTO and the RGB bump, referred to as Δ(B − I)^MSTO_RGBB and Δ(V − I)^MSTO_RGBB.
• 3.  
  The slope of the RGB. This was evaluated using the (mean) color of the RGB stars at two fiducial magnitudes, namely at m_RGBB + 1 and m_RGBB − 0.75. The former magnitude was chosen to represent a point intermediate between the RGB bump and the lower end of the RGB; the latter magnitude was chosen to avoid issues related to confusing RGB with AGB stars on the CMD. The mean colors of the RGB stars were derived from the CMD by means of a polynomial fit to the RGB star positions in the CMD. The predicted colors were derived from a linear interpolation between isochrone table entries.

The main reason for using the RGB bump as a prime parameter in this context is that all three isochrone families can be used this way. However, in the case of clusters for which the location of the RGB bump is not well constrained from the observations (i.e., NGC 1987, NGC 2108, and LW 431), we replace parameters (1) and (2) above with the following ones.

• 1a.  
  The difference in magnitude between the MSTO and the red clump (RC), named ΔB^MSTO_RC and ΔV^MSTO_RC in the B and V filters, respectively. We simply calculate the mean observed magnitude and color of stars in a box centered on the RC by eye. For the isochrones, we define the "mean" location of the RC as follows. After identifying the start of the RC in the Padova and Teramo isochrone tables, isochrone magnitudes and colors are recorded up to the point where the difference in color between two subsequent isochrone entries becomes ⩾3σ larger (redder) than the average color accumulated from the isochrone entries in the RC recorded up to that point. This procedure was empirically verified to yield the appropriate end point of the RC. Weighted magnitudes and colors for the RC are then derived from the recorded isochrone entries. To simulate the distribution of stars in the RC of a star cluster, weight factors are assigned during the latter operation by using a Salpeter (1955) mass function.
• 2a.  
  The difference in color between the MSTO and the RC, named Δ(B − I)^MSTO_RC and Δ(V − I)^MSTO_RC.

The sensitivity of parameters (1), (2), and (3) mentioned above to population parameters in the age range 1–3 Gyr was illustrated in Figures 9–11 of Paper I for all three isochrone families used here. Since we are using parameters (1a) and (2a) instead of (1) and (2) for some clusters in this paper, we now show a comparison of the sensitivity of parameters (1), (1a), (2), (2a), and (3) to age and [Fe/H] for the Padova and Teramo isochrone families in Figures 6 and 7 for the age range 1–3 Gyr. These plots show that while details of the dependences of ΔB^MSTO_RC and Δ(B − I)^MSTO_RC on age and [Fe/H] are different from those of ΔB^MSTO_RGBB and Δ(B − I)^MSTO_RGBB, both sets of parameters do yield mutually consistent results when compared with the observed values. As already mentioned in Paper I, the RGB slope is highly sensitive to metallicity and almost independent of age in the range studied here.

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We then select all isochrones (within each family) for which the values of the three parameters mentioned above lie within 2σ of the measurement uncertainty of those parameters on the CMDs. This yielded 6–15 isochrones depending on the isochrone family. For these isochrones, we then find the best-fit values for distance modulus (m − M)₀ and foreground reddening A_V by means of a least-squares fitting program. For the filter-dependent reddening we use A_F435W = 1.351 A_V, A_F555W = 1.026 A_V, and A_F814W = 0.586 A_V (cf. Paper I).

Finally, the isochrones were overplotted onto the CMDs for visual examination. As mentioned in Paper I for NGC 1846, this revealed that there was a small but systematic offset in [Fe/H] between best-fit isochrones for m_F435W versus m_F435W − m_F814W  and m_F555W versus m_F555W − m_F814W in the sense that the derived value of [Fe/H] was always higher for the isochrone fit to m_F555W versus m_F555W − m_F814W than to m_F435W versus m_F435W − m_F814W. This effect was most significant for the Padova and Teramo isochrones, and we have suggested (see Paper I) that the cause is related to the fact that those two isochrone families derive their T_eff–color relations from the ATLAS9 stellar atmosphere models of R. L. Kurucz (e.g., Castelli & Kurucz 2003) which have been shown to contain more flux in the range ∼5000–6500 Å (including much of the V band) than empirical star spectra from the Pickles (1998) library at the same stellar type (Maraston et al. 2008). Model SEDs therefore have bluer V − I colors than observed for RGB stars, consistent with what we see. Conversely, the Dartmouth isochrones are based on the Phoenix model atmospheres (e.g., Hauschildt et al. 1999) which include hundreds of millions more molecular transitions than the ATLAS9 models and hence a more accurate opacity modeling. Because of this effect, we focus on the m_F435W versus m_F435W − m_F814W CMDs for deriving population parameters.

The best-fit isochrones and their population parameters are listed for each isochrone family in Table 2 and shown in Figures 8 and 9, superposed onto the CMDs. The best-fit isochrones of each model family generally match the various stellar sequences well.¹¹ However, one significant difference among the isochrones is seen on the upper RGB for clusters NGC 1783 and NGC 1806 (the same was seen for NGC 1846; see Paper I). The best-fit Dartmouth isochrones typically provide a better fit to the upper RGB than the Padova and Teramo isochrones, both of which appear bluer than the observed stars. This difference is briefly discussed in Section 4.2 below.

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In order to find the spread in age that can explain the extended morphology of the MSTO regions of the clusters in our sample, we first note that the width of the RGB in all clusters is consistent with photometric errors and that the slope of the RGB in each cluster does not show evidence for an intrinsic spread in [Fe/H]. Hence we fix [Fe/H], A_V, and (m − M)₀ and vary the isochrone age using steps of 0.05 Gyr until one reaches the extremes of the eMSTO region populated by the cluster stars. These age spreads are listed in Table 3, along with the final adopted parameters for the clusters in our sample, including their uncertainties that reflect both errors associated with the isochrone fitting and systematic uncertainties related to the use of the different stellar models. The determination of the population parameters adopted and their uncertainties are discussed further in Section 4.3 below.

Table 3. Adopted Population Parameters of the Star Clusters Studied in This Paper

Cluster	Age	Age Range	[Fe/H]a	(m − M)0	AV	log ${\cal {M}}_{\rm cl}$
(1)	(2)	(3)	(4)	(5)	(6)	(7)
NGC 1751	1.40 ± 0.10	1.15 – 1.65	−0.50 ± 0.10	18.50 ± 0.03	0.40 ± 0.01	4.82 ± 0.09
NGC 1783	1.70 ± 0.10	1.50 – 1.90	−0.50 ± 0.10	18.46 ± 0.04	0.02 ± 0.02	5.25 ± 0.09
NGC 1806	1.67 ± 0.10	1.57 – 1.92	−0.50 ± 0.10	18.46 ± 0.04	0.05 ± 0.01	5.03 ± 0.09
NGC 1846	1.73 ± 0.10	1.53 – 1.93	−0.50 ± 0.10	18.45 ± 0.05	0.08 ± 0.01	5.17 ± 0.09
NGC 1987	1.05 ± 0.05	0.95 – 1.20	−0.50 ± 0.10	18.38 ± 0.02	0.16 ± 0.04	4.49 ± 0.09
NGC 2108	1.00 ± 0.05	0.90 – 1.10	−0.50 ± 0.10	18.45 ± 0.03	0.50 ± 0.03	4.41 ± 0.09
LW 431	1.73 ± 0.10	1.53 – 1.93	−0.50 ± 0.10	18.43 ± 0.03	0.15 ± 0.01	4.00 ± 0.09

Notes. Column 1: name of star cluster. Column 2: adopted age in Gyr. Column 3: age range in Gyr associated with the width of the observed MSTO region. Column 4: adopted [Fe/H] in dex. Column 5: adopted distance modulus in mag. Column 6: adopted foreground V-band reddening in mag. Column 7: logarithm of photometric mass (in M_☉). ^a[α/Fe] = +0.2 ± 0.1 is adopted for all clusters.

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Table 3 also includes present-day cluster masses which are estimated from the total V magnitudes listed in Table 1 and the A_V, (m − M)₀, [Fe/H], and mean age listed in Table 3. The masses use the ${\cal {M}}/L_V$ predicted by the SSP models of Bruzual & Charlot (2003), assuming a Salpeter (1955) initial mass function (IMF). The latter models were recently found to provide the best fit (among popular SSP models) to observed integrated-light photometry of LMC clusters with ages and metallicities measured from CMDs and spectroscopy of individual RGB stars in the 1–2 Gyr age range (Pessev et al. 2008).

We assess the influence that non-solar [α/Fe] abundances would have on the derived age and metallicity by comparing our CMD with Dartmouth isochrones for different values of [α/Fe]. The result is illustrated in Figures 10 and 11, which show the best-fit Dartmouth isochrones for [α/Fe] = 0.0, +0.2, and +0.4 superposed onto the m_F435W versus m_F435W − m_F814W and m_F555W vresus m_F555W − m_F814W CMDs of NGC 1783 and NGC 1806, the two star clusters in our sample (in addition to NGC 1846, cf. Paper I) that have RGB sequences sampled well enough to allow this exercise. Note that all three isochrones fit the MSTO and RGB bump locations well, which is likely (at least partly) due to the fitting method we used (see above in Section 4). However, the detailed fits along the RGB differ significantly from one [α/Fe] value to another. This is best seen in the m_F435W versus m_F435W − m_F814W CMD. In particular, there is a relation between the value of [α/Fe] and the curvature of the RGB in the sense that larger [α/Fe] yields stronger curvature for the RGB.

As already mentioned in Paper I for NGC 1846, larger values of [α/Fe] result in younger fitted ages. Hence there is a degeneracy between age and [α/Fe] if one does not take the detailed morphology of the RGB into account. From the results for NGC 1783, NGC 1806, and NGC 1846, the amplitude of this effect is a decrease of 9.3% (±1.0%) in age for an increase in [α/Fe] of 0.2 dex.

As Figures 10 and 11 show, the best fit to the full RGB is achieved using the isochrone with [α/Fe] = +0.2 for all clusters for which the extent and sampling of the RGB is sufficient to permit this comparison. Since the fit of the Dartmouth isochrones with [α/Fe] = +0.2 dex to the RGBs is clearly better than any isochrone that uses solar abundance ratios in any of the three families, we adopt [α/Fe] = +0.2 for NGC 1751, NGC 1783, NGC 1806, NGC 1846, and LW 431. Future spectroscopic determinations of [α/Fe] in RGB stars of these clusters and the surrounding field population would be very useful to confirm or deny the trends found here from photometry.

We evaluate "mean" ages of the star clusters in our sample as follows. As to the clusters for which we adopt [α/Fe] = +0.2 (i.e., all clusters except NGC 1987 and NGC 2108, the two youngest ones), we first consider the best-fit ages found for these clusters using the Dartmouth isochrones that employ [α/Fe] = 0.0. Mean values and standard deviations of the ages found from all three sets of isochrone families that include treatment of convective overshooting are then derived (at [α/Fe] = 0.0). Finally, those mean ages are converted to the equivalent for [α/Fe] = +0.2 using the relation between age and [α/Fe] mentioned in Section 4.2 above. As to NGC 1987 and NGC 2108, which are too young to exhibit an RGB bump, the isochrone fitting method employed here (see Section 4.1) does not work with Dartmouth isochrones. Hence we average the results from the Padova and Teramo isochrone families for those clusters. We note that the "by eye" Dartmouth isochrone fits shown in Figure 9 for those clusters do have ages and [Fe/H] values consistent with the Padova and Teramo isochrone fits.

We quantify systematic uncertainties in derived age, [Fe/H], distance, and reddening by comparing our best-fit results from each set of isochrones that includes treatment of convective overshooting, as compiled in Table 2. These results yield systematic uncertainties of ±7% in age, ±0.1 dex in [Fe/H], ±0.05 mag in (m − M)₀ (≃ 5% in linear distance), and ±0.02 mag in A_V (≃15%). We suggest that these values represent typical systematic uncertainties associated with the determination of population parameters of intermediate-age star clusters from m_F435W versus m_F435W − m_F814W CMD fitting by isochrones of any given stellar model.

We first simulate cluster CMDs where stars formed in two distinct epochs with an age difference Δτ. Each simulated cluster CMD is created by populating Dartmouth isochrones with stars randomly drawn from a Salpeter IMF between 0.1 M_☉ and the RGB-tip mass. A pair of ages is chosen for each synthetic CMD from an age grid that encompasses the age interval around the best-fit age implied by the range given in Column 3 of Table 3. The grid is populated using an age increment of 50 Myr, which is similar to the age of the youngest (most massive) types of stars that have been put forward as plausible donors of material from which a new stellar generation can be formed (i.e., the FRMS and massive binary stars). For each pair of ages, we vary the relative (mass) fraction in the younger population from 0.80 to 0.20 in 0.05 increments. The total number of simulated stars is normalized to the observed number of stars brighter than the 50% completeness limit. We add an unresolved binary companion to a fraction (see below) of the stars, drawn from the same mass function (i.e., using a flat primary-to-secondary mass ratio distribution). Finally, we add photometric errors to the artificial stars, modeled after the actual distribution of photometric uncertainties.

We use the width of the upper MS, i.e., the part brighter than the turnoff of the field stellar population and fainter than the MSTO region of the clusters, to determine the binary star fraction in our sample clusters. We estimate the internal systematic uncertainty in binary fraction to ±5% and defer a more detailed discussion of binary parameter degeneracies, in particular binarity versus mass fractions of stellar generations, to a future paper. For the purposes of this work the results do not change significantly within ∼10% of the binary fraction.

In order to compare the observed and simulated MSTO regions, we use a "pseudo-age" distribution. The pseudo-age distribution is determined by constructing a parallelogram in the CMD with (1) one axis approximately parallel to the isochrones, (2) the other axis approximately perpendicular to the isochrones, and (3) located in a region of the MSTO where the split between the isochrones is relatively evident. The (m_F435W − m_F814W, m_F435W) coordinates of the stars in the CMD are then transformed into the reference coordinate frame defined by the two axes of the parallelogram, and then considering the distributions of the coordinates of the stars in the direction perpendicular to the isochrones. To translate the latter coordinate to age, the same procedure is done for the Dartmouth isochrones for an age range that covers the observed extent of the MSTO region of the cluster in question, using an age increment of 0.05 Gyr. The relationship between age and the coordinate in the direction perpendicular to the isochrones is then determined using a polynomial least-squares fit. Since binary stars influence the distribution of stars in the MSTO region to some (albeit small) extent, we call the resulting age parameter "pseudo-age."

The procedure mentioned above is illustrated in Figures 14–17. The top panels show the simulated and observed CMDs and the parallelogram mentioned above (in blue, with the reference axis along the isochrones in red). The bottom panels show the corresponding pseudo-age distributions. These were calculated using the non-parametric Epanechnikov-kernel probability density function (Silverman 1986) for all objects in the parallelograms, in order to avoid potential biases that can arise if fixed bin widths are used. In the case of the observed CMD (i.e., panels (a) and (b)), this was done both for stars within a King core radius from the cluster center and for the "background region" for which the CMD was shown in the right panels of Figure 1. The panels (d) also list the fraction of stars in the two distinct SSPs (shown in black dots on panels (c)). The intrinsic probability density function of the pseudo-age distribution of the cluster was then derived by statistical subtraction of the background region (see panels (b)). The best-fit pair of two-SSP simulations was selected by conducting two-sample Kolmogorov–Smirnov (K–S) tests of the probability density functions of the simulations against those of the cluster data and picking the simulation with the highest p value. The properties of the selected simulations, including the best-fit binary fractions and the p values for each star cluster, are listed in Table 4.

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During the refereeing process of this manuscript, a paper by Yang et al. (2011) appeared, reporting on simulations of the impact of interactive binaries on eMSTO morphologies of intermediate-age star clusters. Their calculations, which assume that all cluster stars are members of binary systems, show that the presence of interactive binaries would cause (1) a slight extension of the MSTO region toward the blue and (2) a "fan" of low stellar density in the CMD toward brighter magnitudes and bluer colors than the MSTO. The distribution of this "fan" on the CMD shows similarities to the presence of younger stars, although most "fan" stars are fainter than the MSTO region of younger isochrones. We believe that the actual impact of such interactive binaries on the eMSTO morphologies of the star clusters in our sample is insignificant for two main reasons. First, the extension of the MSTO region mentioned in point (1) above involves color changes that are significantly smaller than the observed color ranges encompassed by the eMSTOs shown in the current paper. Second, a comparison between the middle and right-hand panels in our Figures 1 and 2 reveals that the stars in our CMDs that could be interactive binary stars in the "fan" mentioned in point (2) above are most likely LMC field stars rather than cluster stars, since they are at least as abundant in the "field" CMD as in the "cluster" CMD.

How well does the method described above allow us to resolve two (or more) discrete star formation events separated in time by Δτ? To address this question we use our simulations for NGC 1783, the cluster with the highest present-day mass in our sample. We consider the simulations that involved an age of the first generation of 2.00 Gyr with Δτ values of 0 (i.e., a single SSP), 50, 100, 150, and 200 Myr, and two different mass fractions, namely, f_Y = 0.70 and f_Y = 0.50 for the younger generation. These mass fractions were chosen to bracket the values found for the clusters in our sample (cf. Table 4). Results of these simulations are shown in Figure 18. For f_Y = 0.50, the bimodality of the pseudo-age distribution shows up clearly for Δτ ≳ 150 Myr, while Δτ ≳ 100 Myr yields a significantly broader distribution than a single SSP. For f_Y = 0.70, the "hump" on the right side of the pseudo-age distribution (due to the older generation) is already readily recognizable at Δτ ≳ 100 Myr. Hence our method can recognize age bimodality in CMDs if Δτ ≳ 100–150 Myr at an age of 2.0 Gyr, i.e., 0.05 ≲ (Δτ/τ) ≲ 0.08, with the exact value depending on the mass fractions of the different generations.¹³

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As Figures 14–17 show, the distribution of observed stars in the parallelogram typically peaks near the "young" end of the age range and then declines more or less uniformly toward older as well as younger ages. A comparison of panels (b) with (d) for a given star cluster shows that the distribution of the observed stars is typically more continuous than those of the best-fit two-SSP simulations. Two-sample K–S tests confirm this visual impression. The p values of the K–S tests to compare the distributions of the simulated CMDs of two SSPs with the best-fit mass fractions of the younger population against the data do not exceed 20% for the clusters (see Table 4). NGC 1987, NGC 2108, and LW 431, the three lowest-mass clusters in our sample, do not follow this rule (see Figures 16–17); these three cases are briefly discussed below. Therefore, a main conclusion of this paper is that the more massive clusters in our sample are better explained by a population with a distribution of ages rather than by two discrete SSPs. This conclusion differs from that of Mackey et al. (2008), whose analysis favored a bimodal distribution of ages for the clusters NGC 1806 and NGC 1846 and that of Milone et al. (2009), whose analysis did so for the clusters NGC 1751, NGC 1783, NGC 1806, and NGC 1846. The smooth and extended nature of the pseudo-age distributions seems to suggest that star formation did not actually occur in discrete events. This is further discussed in Section 7 below.

As to the low-mass clusters NGC 1987, NGC 2108, and LW 431, panels (c) and (d) in Figures 16 and 17 compare pseudo-age distributions for the observations and the simulated bimodal formation histories. The K–S test comparing these distributions indicates that a bimodal age distribution results in formally acceptable fits (cf. Table 4). However, these results do not preclude a continuous distribution of ages within these clusters. We also compared the pseudo-age distributions for these three clusters with those resulting from a single-age population, as shown in panels (e) and (f) of Figures 16 and 17. The observed distributions are significantly broader than a single-age population, supporting our general conclusion that more than one "simple" population is required to explain the morphology of the MSTO regions of these low-mass clusters as well.