COMPACT STAR CLUSTERS IN THE M31 DISK

We have identified high probability star cluster candidates (hereafter star clusters) by visually inspecting the Suprime-Cam V-band mosaic image made from 5 × 2 minute individual exposures of the 175 × 285 field, centered at α_J2000 = 0^h409^m, δ_J2000 = +40°45'. The characteristic full width at half maximum (FWHM) of stellar images (point-spread function, PSF) is 07. See Kodaira et al. (2004) for a description of the Suprime-Cam observations and data reduction details.

We have produced the homogenized UBVRI photometric catalog of 285 selected star clusters (V ≲ 20.5 mag) used as an initial sample in this study, based on photometry performed on the Local Group Galaxies Survey (LGGS) mosaic images (Massey et al. 2006). The variable PSFs of four overlapping LGGS field (F6–F9) mosaic images were homogenized to the same FWHM = 15. The mosaics were photometrically calibrated by using individual CCD color equations and the stellar photometry catalog from Massey et al. (2006) as a local photometric standard for zero point calibration. Different aperture sizes and photometric background estimation areas were selected for individual clusters, as they are given in the catalog paper (Narbutis et al. 2008).

We note that a comparison of three published stellar photometry data sets in our survey field (Narbutis et al. 2006) makes one to be cautious when using tertiary standards as local photometric standards. However, we used a carefully reduced and calibrated data set by Massey et al. (2006), which passed an internal consistency check, making it the most accurately calibrated photometry catalog of the M31 galaxy to date.

The color images of clusters were produced from the LGGS (U, B, V, I, Hα), GALEX (FUV, NUV), 2MASS (J, H, K_s), Spitzer (24 μm), and Hi (21 cm) images. They served as references for object properties and environment study. Seventy-seven star clusters from our catalog are located in the HST archive frames. For object selection, photometry, and multiband image details, see Narbutis et al. (2008).

We used Suprime-Cam V-band mosaic image (FWHM_PSF ≈ 07, image scale 02/pixel) to derive structural parameters of star clusters. The LGGS field's F7 V-band mosaic image (FWHM_PSF ≈ 08, image scale of 027/pixel) was substituted for objects saturated or having defects in the Suprime-Cam frame. Since the variability of FWHM_PSF was only ∼5% across the mosaic (see Section 3.1 for details), we have applied the DAOPHOT package (Stetson 1987) from the IRAF program system (Tody 1993) to compute a single mosaic PSF. IRAF's program "seepsf" was used to create PSFs suitable for our structural parameter fitting program.

The following M31 parameters are adopted in this study: distance modulus of m − M = 24.47 (785 kpc, thus 1'' ≡ 3.8 pc; McConnachie et al. 2005); center coordinates α_J2000 = 0^h42^m443, δ_J2000 = 41°16'09'' (NASA Extragalactic Database); major axis position angle of 38° (de Vaucouleurs 1958); disk inclination angle to the line of sight of 75° (Gordon et al. 2006).

The star clusters were selected from the initial sample of 285 objects by considering the following criteria: (1) the derived half-light radius is of r_h ⩾ 015(⩾0.6 pc)—this limit arises due to the resolution of ground-based observations; (2) there are no nearby contaminants strongly influencing the accuracy of the derived structural and evolutionary parameters. Note however, that several genuine star clusters, judged from the HST images, did not meet these criteria.

A few dim, i.e., low central surface brightness, objects with an unreasonably large derived half-light radius or composed of several resolved stars were rejected because of low reliability of determined parameters. Although PÉGASE (Fioc & Rocca-Volmerange 1997) SSP models incorporate nebular emission lines for young age objects, we stress that structural and evolutionary parameters of clusters showing up Hα emissions or superposed on inhomogeneous diffuse Hα background should be considered with caution. Note however, that objects with strong Hα emission were cataloged separately by Kodaira et al. (2004), therefore, these objects were excluded from the analysis in this study. We note, that foreground starburst galaxies possessing 24 μm emission, which have been revealed during spectroscopic study by Caldwell et al. (2009), might be present in a sample of young (t ≲ 20 Myr) clusters (see Appendix for details on KW271 and KW279).

This cleaning yielded a sample of 238 star clusters. Their ID numbers and center coordinates from Narbutis et al. (2008) are supplemented with the structural and evolutionary parameters described in the following, and are presented in Table 1, the full form of which is available in the electronic edition of the Journal.

The studied star clusters, projected on the crowded disk of M31, are unresolved or semiresolved in the Suprime-Cam images. Background and/or clusters' bright stars superposed on clusters' surface brightness distribution strongly influence photometric (Narbutis et al. 2007b) and structural parameter accuracy. However, stars projecting nearby to the cluster cannot be simply masked out because of flux superposition in their extended wings. To avoid this, we have developed a method to derive structural parameters of star clusters by including resolved stars in the cluster model fitting procedure. The method is presented in D. Semionov et al. (2009, in preparation); here we provide its brief description and derived half-light radii of star clusters (Table 1).

The developed program tool models star clusters as a smooth analytical surface brightness distributions. Observational effects are taken into account by convolving cluster models with PSF. Individual resolved stars are modeled with PSF. Free parameters, which describe a shape and position of the cluster model, are fitted using a genetic algorithm, providing the global solution. Fine tuning of this solution is performed using Levenberg–Marquardt nonlinear least-square algorithm. A photometric background value, which is kept constant during model fit, is estimated within individually selected circular annulus centered on the object. The model fitting radius is chosen seeking to enclose extended wings of the surface brightness distribution. The residual images are analyzed to ensure that there are no visible systematic deviations.

We have fitted 285 clusters from the initial sample with circular King (King 1962) and EFF (Elson et al. 1987) models. Since these models have different sensitivity to the cluster's "core" and "wings," comparison of the half-light radii derived by their means provides the robustness of the model fit and r_h. Elliptical models were not used in this study due to stochastic distribution of bright cluster stars, which introduce fake ellipticity for the low-mass clusters (see Deveikis et al. 2008 for a discussion of stochastic effects). A number of contaminating stars used for the model fitting was: 0 in 40%, 1 in 30%, and 2 in 20% cases. The maximum number of fitted stars was 6.

Typical fitted values of structural parameters are: (1) the core radius, r_c ∼ 0.75 pc, the concentration parameter, c ∼ 1.5 (the King model); (2) the effective radius, r_e ∼ 0.75 pc, and the power-law index, n ∼ 1.3 (the EFF model). For ∼70% of clusters the derived c and n parameter values were confined to the reasonable limits of 0.5 ⩽ c ⩽ 2.5 (see boundaries for MW GCs in Harris 1996) and 1.05 ⩽ n ⩽ 3.5, respectively. For practical convenience, the remaining ∼30% of clusters, having determined parameters outside these limits, were re-fitted by setting particular parameters to the corresponding boundary values. We note however, that although this slightly affects derived r_c or r_e, it does not change the derived half-light radii nor introduces systematic deviations in the residual images.

Finally, intrinsic half-light radii of objects, based on the King, r^K_h, and the EFF, r^E_h, models were derived. They are plotted in Figure 1(a) and show a good correlation. The average half-light radii values, r_h = (r^K_h + r^E_h)/2, for 238 star clusters are provided in Table 1.

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To estimate both the robustness of the model fitting and the uncertainty of the derived half-light radius, $\sigma _{r_{h}}$, we have performed two tests: (1) varied photometric background value, μ_sky, which is kept constant during model fitting; and (2) decreased fitting radius, r_fit, from the adopted initial value. These tests indicate that r_h based on the King and the EFF models are in a good agreement, and the overall accuracy of the r_h value is better than 20%. The typical $\sigma _{r_{h}}$ is indicated by error-bars in Figure 1(a). Based on these results, we applied a conservative lower limit (which is mainly subject to the resolution of ground-based observations) of r_h = 015 (0.6 pc) for the sample selection. Note however, that some clusters below this limit have been identified in the HST images, which turned out to be of a very compact nature or dominated by a central bright star.

To test the influence of PSF variability on the derived half-light radii, we have divided survey field into 15 regions and constructed individual PSFs for these regions. These PSF models were treated as "virtual objects" and analyzed as star clusters in attempt to derive their structural parameters. The derived half-light radii of eight "virtual objects" are displayed with asterisks in Figure 1(a) and are confined in the range of r_h < 0.4 pc. Note that only eight of 15 "virtual objects" are visible, since others failed to be fitted due to FWHM smaller than that of the adopted mosaic PSF. The scatter of r_h < 0.4 pc corresponds to the r_h uncertainty of 20% for objects with r_h ∼ 2 pc. It is the upper limit of uncertainty arising due to PSF variability across the survey field. For some individual clusters, we have additionally derived structural parameters by using local and mosaic PSFs, and found results being in a good agreement.

In case of semiresolved star clusters, subtraction of contaminating stars has a strong effect on the accuracy of derived half-light radii. We have automatically included all stars within r_fit + FWHM_PSF from the cluster's center and having magnitude ⩽(m_cluster + 3) mag, i.e., up to 3 mag fainter than the cluster itself, into the fitting procedure. Also some individual strong contaminants were marked by hand. Conventional tests of varying the background level and the fitting radius indicate that ∼20% accuracy of r_h is achieved. Visual inspection of residual images proved that clusters and contaminating stars are subtracted well. We have estimated the goodness of model fitting by comparing the standard deviation of the sky background within the cluster's area after model subtraction and that of the nearby sky region. For ∼90% of cases, these values were equal, except for the cases with larger residuals visible in the cluster's core, which, most probably, arise due to semiresolved stars. They are the main source of r_h uncertainty for low-mass young star clusters (see Figure 3(c) in Deveikis et al. 2008).

We have compared r_h values of 50 bright objects from our sample with the ones derived by Šablevičiūtė et al. (2007). Despite different analysis methods employed, we have found a good agreement between these two studies. The maximum r_h difference of ∼30% is attributed to the influence of contaminating stars not properly accounted for in the previous study.

The UBVRI aperture CCD photometry data of 285 star clusters from the initial sample (Narbutis et al. 2008) were compared with SSP models (PÉGASE code: Fioc & Rocca-Volmerange 1997) assuming the standard initial mass function (IMF; Kroupa 2002) and various metallicities to yield the best fit model evolutionary parameters, i.e., simple SSP model fitting was performed. The interstellar extinction was accounted for assuming the standard extinction law (Cardelli et al. 1989). The adopted parameter quantification technique and intrinsic precision analysis, based on the UBVRI integrated photometry, was presented in detail by Narbutis et al. (2007a) and Bridžius et al. (2008). This technique, supplemented with a detailed investigation of multiband images, introduced to reduce the age–metallicity–extinction degeneracy, was used in the study.

The simple SSP model fitting results for 238 star clusters are provided in Table 1. The determined evolutionary parameters are: absolute V-band magnitude (corrected for aperture correction, see below for details), M_V; age in Myr, log(t/Myr); mass in solar-mass units, log(m/m_☉); metallicity, [M/H]; and color excess, E(B − V).

The standard deviation of age, σ_log(t/Myr), is plotted versus V-band magnitude and age, log(t/Myr), in Figures 2(a) & (b), respectively. The standard deviation of age increases typically from ∼0.1 dex for young to ∼0.3 dex for old objects. We note that systematic differences of the derived parameters could be expected if another SSP model bank would be used. The histograms of the standard deviation of color excess, σ_{E(B − V)}, and metallicity, σ_[M/H], are shown in Figures 2(c) & (d), respectively. The characteristic σ_{E(B − V)} is ∼0.05, reaching ∼0.15 for t ∼ 10 Gyr objects due to stronger age–extinction degeneracy. The typical σ_[M/H] is ∼0.3 dex, reaching up to ∼0.7 dex for t ∼ 200 Myr objects due to stronger age–metallicity degeneracy (see Narbutis et al. 2007a for parameter degeneracy maps).

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We note that in this study numerous SSP model simplifications were assumed: (1) a constant IMF; (2) a standard for MW and constant for all cluster stars extinction law; (3) solar element ratio. Additionally, in cases of unreliable photometric solutions for metallicity, we restricted [M/H] determination within lower and upper boundaries as a function of age taken from the chemical evolution model prediction. Referring to models of MW and M31 by Renda et al. (2005), we assumed that metallicity evolution in the M31 survey field is not significantly different from that of solar neighborhood, which has been modeled by Schönrich & Binney (2009). Therefore, the [M/H] boundaries roughly between −1 and +0.5 were considered, also taking into account possibility that oldest objects could be globular clusters in M31 halo with [M/H] as low as −2.

The mass of clusters in the solar-mass unit, m/m_☉, was calculated equating the mass-to-luminosity ratio of the best fit SSP model of age, t, and metallicity, [M/H], to the absolute V-band magnitude, M_V. The mass of clusters strongly depends on the derived age, therefore, the primary sources of mass uncertainty are the accuracy of age and extinction. Since cluster candidates in the crowded field were measured through small apertures, we have taken into account an individual aperture correction for each star cluster by applying the following procedure. Artificial clusters based on the best-fitted King model parameters were generated and convolved with the homogenized PSF of FWHM = 15; i.e., resolution of model cluster images was matched to the LGGS frame resolution employed for cluster photometry. Individual aperture used for a real cluster was applied to measure corresponding model cluster, the difference between measured and total flux was assumed as an aperture correction for a real cluster photometry. A typical V-band aperture correction is of ∼0.4 mag and translates into a correction of mass by ∼0.15 dex in log(m/m_☉) scale.

Analysis of star cluster population requires a proper assessment of selection effects and resulting completeness. Stochastic modeling of clusters (Deveikis et al. 2008) unfolded a broad variety of their possible appearances, indicating that visual selection of low-mass clusters could be biased. However, for older clusters, i.e., after the phase of supergiants (t ≳ 100 Myr), this problem appears to become less crucial.

The histogram of clusters' half-light radius, r_h, displayed in Figure 1(b), spans the range from ∼0.6 pc to ∼10 pc with one large object of r_h ∼ 14 pc (KW249), which is described in the Appendix; the distribution peaks at r_h ∼ 1.5 pc. For comparison, the peak of the half-light radius distribution of MW GCs from Harris (1996 February 2003 rev.⁵) catalog is at ∼2.5 pc. Therefore, the majority of clusters from our M31 sample are smaller (more compact) than typical MW GCs.

The turnover and a lack of small-size clusters close to the applied lower limit of r_h = 0.6 pc can be attributed to the visual selection of the cluster sample. The exponential decrease in the number of clusters in the larger half-light radii domain should be real. Note however, tests performed with "SimClust" (Deveikis et al. 2008) show that only extended clusters of low luminosity (M_V ∼ −4.5) projected on star-forming regions could be missed by visual inspection. The overplotted power-law slope η = −2.2 line, defined as N(r_h)dr_h ∝ r^η_hdr_h, was found as the best fit for M51 star clusters by Bastian et al. (2005), and agrees well with our star cluster size distribution in the range of r_h = 3–8 pc. However, a considerably steeper slope was found in the M51 cluster sample of Scheepmaker et al. (2007), which is five times more numerous than that of Bastian et al. (2005).

The half-light radius of clusters is plotted versus age, log(t/Myr), in Figure 3(a). No obvious half-light radius evolution with the age of cluster population, as has been found in LMC/SMC by Mackey & Gilmore (2003), can be judged from this diagram, since objects with large r_h are observed over a wide range of age. However, the lack of small-size clusters at age t ∼ 3 Gyr can be attributed to the selection effects discussed by Hunter et al. (2003) for the LMC/SMC case, such that with increasing age only bright and, therefore, massive extended clusters are selected in the magnitude limited sample. At t ∼ 10 Gyr, halo GCs likely dominate in our sample and show r_h distribution similar to that of the MW GCs population with r_h as small as ∼1 pc. They have high densities (see Figure 3(b)) and likely have survived tidal disturbances by the disk.

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The half-light radius versus mass, log(m/m_☉), distribution is shown in Figure 3(b), overplotted with five lines of constant half-mass density, ρ_h = 3m/8πr³_h, ranging from 1 to 10⁴ m_☉ pc⁻³. The most numerous clusters in the M31 southwestern field are compact clusters of small size r_h ∼ 1.5 pc, intermediate mass m ∼ 4000 m_☉, and of young t ∼ 100 Myr age. Largest clusters seem to be enveloped by the lines of ρ_h from 1 to 100 m_☉ pc⁻³, lending an impression that massive clusters tend to be bigger in size as far as clusters of relatively low specific density of ρ_h < 100 m_☉ pc⁻³ are concerned (Hunter et al. 2003).

Comparing the r_h versus mass distribution of our sample with a sample of M51 star clusters (Scheepmaker et al. 2007), we note a lack of objects with 4 ≲ log(m/m_☉) ≲ 5 and r_h ≲ 5 pc in our sample. The most massive, log(m/m_☉) ≳ 4.5, clusters are "branching" in Figure 3(b) at small size, r_h ≲ 3 pc, and high density, ρ_h ∼ 10⁴ m_☉ pc⁻³, which is characteristic to old GCs.

The oldest clusters, which have highest densities, are most likely halo GCs of small r_h. That the typical lifetime of 10⁴ m_☉ mass star cluster is estimated to be of ∼300 Myr in the survey field (see Section 4.7 for details) implies very low probability for disk star clusters to reach an age of ∼10 Gyr without disruption. However, radial velocity measurements are necessary to confirm whether these objects belong to halo.

The distribution of clusters in the mass versus age diagram is shown in Figure 4. The upper right domain of the diagram, t ≳ 3 Gyr and 4.0 ≲ log(m/m_☉) ≲ 7.0, is occupied by ∼30 classical GC candidates. The remaining ∼210 are most likely disk clusters, which reside in the domain of t ≲ 3 Gyr and log(m/m_☉) ≲ 4.5. The curve along the lower envelope of cluster distribution is due to the selection limit of V ≲ 20.5 mag, which was calculated with PÉGASE SSP models of [M/H] = −0.4 dex metallicity, assuming zero extinction. Note, that star clusters are located above the selection limit due to aperture corrections taken into account for mass calculation. However, the upper envelope of the distribution should be free of selection effects.

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Cerviño & Luridiana (2004) have shown that SSP models cannot be used in a straightforward deterministic way for small-mass cluster analysis, when integrated luminosity of a model is lower than that of the brightest star included in the used isochrone, i.e., "Lowest Luminosity Limit" (LLL) requirement. They defined the smallest mass, m^LLL, associated with the LLL, and the mass, m^σLL, corresponding to the 10% stochastic fluctuation of the V-band luminosity, $\sigma _{L_{V}} = 0.1 \times L_{V}$, which translates into ∼0.04 dex uncertainty of log(m/m_☉). We have overplotted both parameters as a function of age in Figure 4, taken from Cerviño & Luridiana (2004), noting that m^σLL(t) ∼ 10 × m^LLL(t). Note however, that SSP models of solar metallicity used by Cerviño & Luridiana (2004) do not significantly influence a qualitative assessment of mass versus age distribution.

Judging from Figure 4, m^LLL is log(m/m_☉) ∼ 4 for clusters younger than ∼10 Myr. At the age of ∼50 Myr, m^LLL drops to log(m/m_☉) ∼ 3, therefore, the majority of clusters from our sample satisfy the LLL requirement. At t ∼ 100 Myr, the m^LLL coincides with V ≲ 20.5 mag selection limit; older objects are well above LLL. Therefore, clusters with log(m/m_☉) ≳ 4.5 should have the most accurate evolutionary parameters.

The stochastic effects on the derived cluster mass can be estimated by referring to the dotted line in Figure 4, based on the Cerviño & Luridiana (2004) methods. The line displays 10% uncertainty of cluster's luminosity at a specific mass for a given age. Since we derived the mass from the V-band luminosity, the mass as a function of age, having 10% uncertainty of mass, is depicted by the dotted line in Figure 4. Formally, mass uncertainty increases to 100% at the LLL indicated by the dashed line. However, the uncertainty of mass is additionally affected by the uncertainty of the derived age, which can be evaluated from the slope of solid line in Figure 4, and also by the uncertainty of extinction shown in Figure 2. Therefore, the LLL provides only the lower limit for uncertainty arising due to stochastic effects, which primarily bias age and extinction, and consequently—mass, i.e., actual accuracy of evolutionary parameters is lower than it is shown in Figure 2.

The histogram of cluster age, displayed in Figure 3(c), spans the range from t ∼ 5 Myr to ∼12 Gyr. Since evolutionary parameters of the clusters with Hα emission are less reliable due to emission lines altering photometry and stochastic effects, we cannot properly estimate parameters of the clusters younger than ∼20 Myr. Therefore, we did not specifically target our survey for such star clusters.

The age distribution of clusters is displayed in Figure 5(a). We have calculated the number of clusters per linear age interval, dN/dt, by binning data in dlog(t/Myr) = 0.2, 0.3, and 0.5 age intervals, while also shifting them by a half-bin width. These results were averaged and smoothed. The symbol size indicates an uncertainty of the resulting distribution, which is plotted versus age in a log scale. The age distribution is reasonably well represented with two power-law lines of different slopes, and is to be interpreted as a result of star cluster formation/disruption processes (see Section 4.7 for discussion).

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The cluster number distribution dN/dt (Figure 5(a)) in general follows the "evolutionary fading" line. However, after subtracting the "evolutionary fading" number gradient, a gradual increase in the number of clusters per log age interval from t ∼ 20 Myr to t ∼ 100 Myr is obvious. A peak of the cluster number distribution is followed with a decrease in the number of clusters, which is a consequence of an evolutionary cluster fading effect, being always significant in the magnitude limited sample (see Figure 4), and star cluster disruption. We note that analysis of both Figures 3(c) & 5(a) points to the peak of the cluster age distribution at t ∼ 70 Myr, suggesting an enhanced cluster formation episode at that epoch. A smaller secondary peak at ∼10 Gyr is due to old GC candidates.

The histogram of cluster mass, displayed in Figure 3(d), spans the range from ∼10² to ∼5 × 10⁶ m_☉ and shows a prominent peak at m ∼ 4000 m_☉. The decrease in the number of clusters in the low-mass domain is apparently not physical, but arises due to selection effects. The estimated completeness of our cluster sample is of ∼50% at V ∼ 20.5 mag (Narbutis et al. 2008). Judging from these circumstances, the completeness at the age of ∼100 Myr and the mass of ∼4000 m_☉ (Figure 4) should be credible, even taking into account a stochastic scattering of cluster luminosity at this age and mass. Therefore, decline in the number of clusters per log mass interval in the high-mass domain should be real.

The mass distribution of clusters is displayed in Figure 5(b). The number of clusters per linear mass interval, normalized to the total number of N_tot = 238 clusters in our sample, dN/dm/N_tot, was calculated by binning data in dlog(m/m_☉) = 0.2, 0.3, and 0.5 mass intervals, and also shifting them by a half-bin width. These results were averaged and smoothed. The symbol size indicates the uncertainty of the resulting distribution. For a more detailed comparison of intermediate age (from ∼100 Myr to ∼3 Gyr) star clusters from our sample we have used the selected subsample (V < 19.0 mag, age from ∼100 Myr to an upper age limit of their sample ∼1 Gyr) of star clusters from the M31 disk study by Caldwell et al. (2009). The dN/dm distribution presented in Caldwell et al. (2009) subsample was multiplied by 0.15, taking into account that our survey field covers only ∼15% of the M31 disk. We note, that there are only eight objects in common between our sample and that of Caldwell et al. (2009).

Decline in the distribution of mass (Caldwell et al. 2009) less than log(m/m_☉) ∼ 4.3 (see Figure 5(b)) arises due to a sample incompleteness and resembles a shape of our intermediate age sample distribution for mass less than log(m/m_☉) ∼ 3.5. Although we have only few star clusters in the high-mass range, the good match with a dN/dm slope (Caldwell et al. 2009) and extension down to the mass as low as log(m/m_☉) ∼ 3.7 allows us to constrain a shape of the star cluster mass function for age in-between of ∼100 Myr and ∼1 Gyr. We overplotted Schechter's mass function from Gieles (2009, Equation (1)), adopting the turnover mass of m* = 2 × 10⁵ m_☉, which shows a good agreement with both our intermediate age sample and Caldwell et al. (2009) subsample data. We note that only vertical adjustment for the Schechter's function was applied to match data at m ∼ 10⁴ m_☉.

Although we lack low-mass star clusters in our sample due to the selection limit at V ∼ 20.5 mag (M_V ∼ −4.0), the detailed HST study of star clusters in M31 by Krienke & Hodge (2007, 2008) has revealed a numerous population of star cluster candidates as faint as V ∼ 23 mag (M_V ∼ −1.5). They are spatially well correlated with brighter counterparts of t ∼ 100 Myr from our sample, therefore, it is reasonable to assume that they could be of a similar age. Thus, their mass should be significantly smaller than ∼4000 m_☉, corresponding to the mass range of typical MW OCs in the solar neighborhood (Piskunov et al. 2008). Although the parameters of MW star clusters are derived basing on individual stars, note that masses of MW clusters are a subject to the accuracy of their derived tidal radii, as discussed by Piskunov et al. (2008).

The high-mass domain of clusters from our sample is found to be overlapping well with the sample of ∼140 young clusters, sharing the M31 disk rotation, studied using MMT spectra and HST images by Caldwell et al. (2009), who found a mass peak at log(m/m_☉) ∼ 4, significantly higher than for our cluster sample, probably, due to brighter cluster selection limit applied for spectroscopy. The mass range of clusters from our sample younger than ∼3 Gyr falls in-between of the MW GCs (McLaughlin & van der Marel 2005) and the MW OCs (Piskunov et al. 2008), indicating that we are mainly dealing with the disk clusters from the intermediate mass range. The mass distribution of clusters older than ∼3 Gyr overlaps well with the mass distribution of MW GCs (McLaughlin & van der Marel 2005).

Comparing the mass versus age diagram of our cluster sample (Figure 4) with those of LMC/SMC (Hunter et al. 2003) and M51 (Bastian et al. 2005), we see a prominent lack of massive, log(m/m_☉) ∼ 5, clusters younger than ∼3 Gyr in M31. Although our survey covers only ∼15% of the deprojected M31 disk, it contains a representative part of its star-forming ring (Gordon et al. 2006) in the vicinity of NGC 206. We also note that other studies of the M31 disk clusters, which covered the whole galaxy, did not detect a significant number of clusters more massive than log(m/m_☉) ∼ 4.5 in this age range either (Caldwell et al. 2009, and references therein).

A few massive, log(m/m_☉) ∼ 5, objects with an estimated age of t < 1 Gyr were reported by Caldwell et al. (2009), who found that most of their young clusters have a low concentration, typical to the MW OCs. Their spatial distribution is subject to patchy field selection of HST high-resolution imaging over the M31 disk. Although there are only eight objects in common in their and our samples, judging from the overlapping mass range and the reasonable agreement of the derived age, we may tentatively assume that both samples belong to the same category of star clusters.

The general lack of star clusters of log(m/m_☉) ∼ 5 mass in the M31 disk could be due to a low-average star formation rate in M31 (Barmby et al. 2006). However, an unfavorable inclination angle of M31 could hide star clusters embedded in the M31 disk, mimicking the situation in MW (Clark et al. 2005)—there could be massive star clusters hidden by dust clouds and yet to be detected in M31. This hypothesis might be supported by the molecular cloud mass distribution in M31 (Nieten et al. 2006; Rosolowsky 2007), where star clusters with a mass of up to log(m/m_☉) ∼ 5 are expected to form, if the typical star formation efficiency of ∼30% in the cores of molecular clouds is assumed.

A typical color excess of star clusters in our sample is E(B − V) ∼ 0.25. Clusters younger than ∼500 Myr have a range of E(B − V) ≲ 1.0, which is consistent with a large extinction of objects residing in the galaxy disk, e.g., four high extinction, E(B − V) ∼ 1.1, objects of t ∼ 100 Myr are projecting close to the 24 μm emission regions and prominent dust lanes. The range of E(B − V) ≲ 0.4 derived for clusters of intermediate age, 500 Myr ≲ t ≲ 3 Gyr, might be related to the evolutionary fading of clusters and the observed narrow mass interval in this age range (see Figure 4), therefore, objects possessing color excess higher than ∼0.4 are missing from our sample. We note that E(B − V) versus projected distance from the M31 center does not show any tight correlation. Moreover, E(B − V) of an individual cluster seems to be more strongly dependent on the local interstellar matter environment than on the radial distance from the M31 center.

The GC candidates of t ≳ 3 Gyr have color excess over the range of E(B − V) ≲ 0.6, consistent with their presence in the M31 halo on both sides of the disk. There are numerous objects with E(B − V) ∼ 0.04, which is in a good agreement with MW color excess in the direction of the survey field in M31, estimated from the MW extinction maps (Schlegel et al. 1998).

Note however, that for eight objects in common with Caldwell et al. (2009) sample, which have E(B − V) ∼ 0.25 derived in their study, we determine color excess from 0.05 to 0.65. Although Caldwell et al. (2009) determined extinction by examining the shape of spectra continuum, for those cases when late-type stars dominate they used a mean color excess for young clusters of E(B − V) ∼ 0.28, which is in agreement with the typical value of E(B − V) ∼ 0.25 derived in our study for presumably the same population of star clusters.

The metallicity, [M/H], of star clusters, plotted versus age and projected distance from the M31 center, R_M31, in Figures 6(a) & (b), respectively, spans a range from −2.0 to +0.5 dex. We divided clusters into two age groups of t ≲ 3 Gyr and t ≳ 3 Gyr, indicated by a different symbol size in Figure 6(b). The apparent narrowing of the metallicity scatter at an age of ∼200 Myr is attributed to the SSP model fitting artifact due to a strong age–metallicity degeneracy in this age domain (Narbutis et al. 2007a). Although the accuracy of photometrically derived metallicity is low (see Figure 2(d)), the evolutionary trend with time is apparent.

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A rather constant average metallicity of [M/H] ∼ −0.4 dex is observed for clusters younger than ∼1 Gyr, however, there is a slight tendency of the average [M/H] to decrease with increasing galactocentric distance. The radial profile of cluster metallicity matches well those of disk red giant stars from Worthey et al. (2005) and Bellazzini et al. (2003) shown by lines in Figure 6(b).

There are 20 clusters in our sample, which have [Fe/H] estimates from the William Herschel Telescope spectra by Perrett et al. (2002). By accounting for α-element overabundance typical to MW GCs, we see a reasonable agreement between photometric and spectroscopic metallicity estimates of old, t ∼ 10 Gyr, clusters. Two of those metal-rich massive clusters (KW221 and KW225), presumably similar to GCs, have metallicity of [M/H] ∼ −0.8 and 0.0 dex, and are found in our survey field area closest to the M31 bulge, located at the projected distance of R_M31 ∼ 5 kpc from the galaxy's center. Note however, that for several objects of ∼100 Myr age, the metallicity estimated by Perrett et al. (2002) is of [Fe/H] ∼ −2.2 dex, i.e., significantly lower than a photometrically estimated [M/H] ∼ −0.4 dex.

The elaborated M31 evolution models, such as "M31 model-b" by Renda et al. (2005), may well reproduce the metallicity gradient and the metallicity change in time, as they are presented in Figure 6. Detailed studies of the age-dependent profiles of the disk cluster population would provide further constraints on the M31 disk evolution models.

Positions of star clusters are indicated on the Spitzer (24 μm) image of the M31 southwestern field in Figure 7. Clusters are divided into three age groups: young (t ≲ 150 Myr); intermediate (150 Myr ≲ t ≲ 3 Gyr); and old (t ≳ 3 Gyr), shown by different symbols. The distribution of young clusters resembles that of warm dust, but is shifted outward to larger R_M31 by ∼1.5 kpc along the galaxy's major axis. The intermediate age clusters are smoothly distributed over the whole field. Old age clusters (GC candidates) scatter over the area, but show a higher concentration toward the M31 bulge.

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Star clusters with ultraviolet (GALEX) or 24 μm (Spitzer) emission are mainly distributed along the sites of active star formation. At the bottom of the survey field, young and intermediate age clusters appear to be well mixed. Around the position of [6', 16'], where a lack of warm dust is attributed to the "split" of the star forming ring, intermediate age clusters dominate. However, around [17', 11'] mainly young clusters are observed. The spatial density of young and intermediate age clusters decreases toward the bulge, which is consistent with a gas distribution in the M31 disk (Nieten et al. 2006).

To explain the M31 spiral structure and features of the prominent star-forming ring at R_M31 ∼ 10 kpc, two scenarios of the M32 galaxy having passed through the M31 disk were recently proposed: (1) passage trough the disk occurred ∼20 Myr ago (Gordon et al. 2006) and created a "split" of the ring around position [6', 16'] (see Figure 7; NGC 206 is located at the ring's "branching" point [15', 17']); (2) passage trough the M31 center occurred ∼210 Myr ago (Block et al. 2006). The star cluster population data could provide a reference to disentangle the more preferable scenario.

True locations of objects in the M31 disk are subject to a strong projection effect due to an unfavorable inclination angle of ∼75°. The projection of the galaxy and its rotation are such that the southwestern part of the disk approaches the observer (Henderson 1979); the corotation radius lies beyond R_M31 ∼ 20 kpc (Efremov 1980; Ivanov 1985); and the rotation period at R_M31 ∼ 10 kpc distance is of ∼250 Myr. Assuming that all clusters in our sample are residing in the M31 disk, we made a deprojection of cluster locations to face-on view, and referring to the disk rotation curve (Carignan et al. 2006), computed object positions as a function of look-back time.

Judging by eye, the clumps of young clusters, which are located at [9', 5'] and [17',10'], formed two compact spatial configurations ∼80 ± 20 Myr ago. This age coincides well with the cluster age peak (see Section 4.2) and suggests a possible evolutionary connection. There was no other compact spatial cluster configuration noted in the look-back time range from ∼20 Myr to ∼150 Myr, which is valid for this kind of study based on our cluster sample. The range of the age distribution peak, estimated from the SSP model fitting, is of ∼50–120 Myr. However, there is an age–metallicity degeneracy observed at the age of ∼100 Myr, which could slightly bias the age determination. Therefore, we note this interesting coincidence of cluster age peak and compact spatial cluster configurations. To discuss this finding more reliably, a global kinematic study of the M31 disk cluster population and a detailed dynamic model of the M31 and M32 encounter are needed. Note however, that Caldwell et al. (2009) did not find evidence for a peak in star formation between 20 Myr and 1 Gyr.

If star clusters are being formed continuously at a constant rate with a power-law cluster initial mass function, their number increases with time in the mass versus age diagram as described by Boutloukos & Lamers (2003). Consequently, the mass of the most massive cluster in the older log age bins rises due to a statistical effect of sample size. However, in our sample, we see a lack of massive log(m/m_☉) ≳ 4 clusters older than ∼100 Myr (Figure 4), probably, indicating a non-constant cluster formation history. Therefore, using the observed mass and age distributions of star clusters we attempt to derive their formation and disruption rates in the studied field by adopting the technique developed by Boutloukos & Lamers (2003).

For a simple estimate, first we selected clusters more massive than log(m/m_☉) ∼ 3.5. There are N₁ ∼ 30 and N₂ ∼ 70 objects in the age ranges Δt₁ ≡ 30 Myr ≲ t ≲ 100 Myr and Δt₂ ≡ 100 Myr ≲ t ≲ 1 Gyr, respectively. If a cluster formation rate (CFR) is constant and there is no cluster disruption, then a ratio of N₂/N₁ = Δt₂/Δt₁ ≈ 13 is expected. The actual ratio is N₂/N₁ ∼ 2, indicating that CFR is not constant and/or significant cluster disruption took place during the last Gyr in the M31 disk.

The number of clusters per linear age interval, dN/dt, plotted versus age in Figure 5(a) can be described by two power-law slopes, intersecting at t ∼ 300 Myr. The slope values were taken from the fading/disruption models by Boutloukos & Lamers (2003, Equation (14): α = 2.0, γ = 0.62, ζ = 0.69) and only vertical adjustment to the observed distribution was applied. In the age domain of t ≲ 300 Myr the slope line, expected for a cluster population forming continuously and fading in the course of evolution below the detection limit, provides a rather good match to the observed data. Around t ∼ 70 Myr, there is a sudden increase in the number of clusters by a factor of ∼2, which could be attributed to the increased CFR at that epoch. In the age domain older than ∼300 Myr, the observed distribution can be well described by the slope expected for a cluster disruption at some arbitrary time. An increase in the cluster number at t ∼ 10 Gyr is attributed to the GC candidates, prominent in our sample.

The number of clusters per linear mass interval, dN/dm, is displayed versus mass in Figure 5(b). In the mass range of log(m/m_☉) ≲ 3.3 the incompleteness of our cluster sample is obvious. Due to incompleteness, the mass distribution of our cluster sample cannot be used to constrain the typical lifetime of star clusters since the fading line cannot be adjusted properly. Note that the distribution of clusters in between ∼100 Myr and ∼3 Gyr and in the mass range of log(m/m_☉) ≳ 4.5 is steeper than that of the whole sample, which includes the old GC candidates.

Using the Boutloukos & Lamers (2003) technique and relying on the power-law slope line intersection point in Figure 5(a), we estimate a conservative value of typical lifetime of log(m/m_☉) = 4 mass cluster to be of t^dis₄ ∼ 300 Myr in the M31 survey field. For comparison, the following values in other galaxies were derived by Lamers et al. (2005): M51 central region—∼70 Myr; M33—∼630 Myr; MW solar neighborhood—∼560 Myr; and SMC—∼8 Gyr.

The cluster disruption analysis gives the disruption time of ∼500 Myr if only cluster age distribution of our sample is used. If the mass distribution is taken into account (although it is difficult to define intersection point of fading and disruption lines in Figure 5(b)), the cluster disruption time can be estimated to be as small as ∼100 Myr since it is sensitive to the "intersection" mass. Therefore, we choose t^dis₄ ∼ 300 Myr as a conservative value. We note that a similar inconsistency between age and mass distribution analysis was noticed recently for LMC star clusters by Parmentier & de Grijs (2008). However, they found an opposite effect for the LMC sample—the cluster mass distribution analysis tends to yield higher cluster disruption timescale than the age distribution analysis. Such discrepancy might be related to the assumptions applied in the Boutloukos & Lamers (2003) analysis used in our study: (1) a constant cluster formation rate; (2) a power-law cluster initial mass function, which should be of the Schechter's function type as shown in Section 4.3.

Lamers et al. (2005) found anticorrelation between the lifetime, t^dis₄, of log(m/m_☉) = 4 mass cluster and the ambient density of its environment, ρ_amb, in the galaxies. Following the reasoning by Lamers et al. (2005), we attempt to estimate ρ_amb in the M31 disk. We took the structural parameters of the M31 disk from the "best-fit model" of Geehan et al. (2006): a disk scale-length, R_d = 5.4 kpc; a central surface density, Σ₀ = 4.6 × 10⁸ m_☉ kpc⁻². Therefore, the average disk surface density at R_M31 ∼ 10 kpc, i.e., at the center of our survey field Σ = Σ₀exp(−R_M31/R_d) ∼ 70 m_☉ pc⁻².

van der Kruit (2002) has shown that for spiral galaxies the vertical scale height is of h_z ∼ 0.15 R_d; this implies h^stars_z ∼ 800 pc. Braun (1991) found h^gas_z ∼ 350 pc for a gas distribution at a galactocentric distance of R_M31 = 10 kpc. The scale height for dust distribution is assumed to be of h^dust_z ∼ 100 and ∼150 pc by Hatano et al. (1997) and Semionov et al. (2003), respectively. Depending on the adopted h_z, the estimated ambient density in our survey field, ρ_amb ∼ 0.5 Σ h⁻¹_z, is in the range of ρ_amb ∼ 0.05...0.35 m_☉ pc⁻³. This density is much lower than the typical half-mass density ρ_h ∼ 10² m_☉ pc⁻³ of star clusters, shown in Figure 3(b).

Referring to Figure 4 in Lamers et al. (2005), we see that values of t^dis₄ ∼ 300 Myr and ρ_amb ∼ 0.05...0.35 m_☉ pc⁻³ derived here, place the studied field of the M31 disk in the position between MW and M33 and M51, however, slightly below the predicted theoretical t^dis₄ = t^dis₄(ρ_amb) line. Gieles et al. (2006) have modeled encounters of star clusters with molecular clouds to explain short disruption time for star clusters in the M51 central region. Our survey field in M31 is centered on the star-forming ring possessing high-density gas (Nieten et al. 2006), therefore, enhanced star cluster disruption in respect to MW and M33 could apparently take place. Although Rosolowsky (2007, and references therein) discuss similarity of the molecular cloud properties in M31 and MW, enhanced star cluster disruption should apparently take place in the M31 molecular gas "ring" in respect to that of the solar neighborhood. The whole field survey of the M31 disk clusters and more detailed models of cluster and cloud kinematics, which are needed to estimate encounter cross sections in the disk, should be considered in future.

Finally, we attempt to estimate the CFR in the M31 disk. We select N = 51 clusters with age Δt ≡ 30 Myr ≲ t ≲ 130 Myr and mass 3.3 ≲ log(m/m_☉) ≲ 4.5 (see Figure 4). This subsample should have accurately estimated parameters and be relatively free of selection and cluster disruption effects. The total stellar mass of these clusters is Δm ∼ 3 × 10⁵ m_☉. Assuming the constant power-law cluster mass function (index −2.0; Gieles 2009) down to log(m/m_☉) = 2, mass correction factor of ∼2.5 was deduced. Therefore, the approximate rate of star formation in the clusters was CFR_field = Δm/Δt ∼ 0.008 m_☉ yr⁻¹. The deprojected area of survey field (Figure 7) is of ∼70 kpc², making ∼15% of the galaxy disk. From this, we estimate the lower limit of the average CFR at the epoch of ∼100 Myr to be of CFR_M31 ∼ 0.05 m_☉ yr⁻¹ over the M31 disk. This can be compared to the present-day star formation rate of 0.4 m_☉ yr⁻¹ (Barmby et al. 2006), indicating that ≳10% of formed stars could remain "locked" in star clusters during ∼100 Myr.

For comparison, Williams (2003) found the mean star formation rate for the disk ∼1 m_☉ yr⁻¹ over the last 60 Myr. Based on stellar population analysis, he suggests that the lowest star formation rate occurred ∼25 Myr ago being ∼2 times lower compared with the epoch from ∼50 Myr to ∼250 Myr. Note, a good coincidence of star formation history in M31 deduced by Williams (2003) and one inferred from our cluster population data (Figure 5(a)), which suggests an episode of a double increase in cluster formation ∼70 Myr ago. Also, Kodaira et al. (1999) have found that the star formation rate in the field around OB association A24 has decreased by ∼2.5 times from the epoch at ∼1 Gyr to the present day.

Although clusters in the intermediate mass range (3.5 ≲ log(m/m_☉) ≲ 4.5) are not well known in the MW, the compact clusters in M31 might represent a new class of star clusters of intermediate age 30 Myr ≲ t ≲ 3 Gyr, linking the age of GC formation and the present. It could be suspected that this class of disk-population clusters is not observed in the MW because of high extinction through the Galactic plane. This might be supported by the Clark et al. (2005) discovery that a Galactic star cluster Westerlund 1 is a Super Star Cluster with a mass of up to log(m/m_☉) ∼ 5, age of ∼3 Myr, and radius of 0.6 pc, which suffers strong interstellar extinction of A_V ∼ 11.5 mag being at the distance of ∼5 kpc. Therefore, a more numerous massive compact star cluster population also could be hidden by dust clouds in the M31 disk.