© The Authors 2025

1. Introduction

Based on current knowledge of how star formation works, the leading belief is that the majority of stars are born in star clusters and associations (Lada & Lada 2003), most of which then dissolve into the general stellar population of their host galaxy due to tidal and dynamical perturbations (Portegies Zwart et al. 2010). This makes studies of star clusters ideal for understanding the formation and evolution of galaxies, as well as for constraining theories of star formation processes.

One of the most commonly applied methods of studying extragalactic star clusters involves fitting stochastic models to integrated cluster magnitudes and colour indices (CIs) (Deveikis et al. 2008; Fouesneau & Lançon 2010; Fouesneau et al. 2014; de Meulenaer et al. 2013; Krumholz et al. 2015). Compared to other widely used colour-magnitude diagram (CMD) fitting techniques (Weisz et al. 2015; Johnson et al. 2016; Wainer et al. 2022; Čeponis et al. 2024), aperture (or integrated) photometry methods are suitable for older, lower-mass, and more distant clusters, making them applicable for larger and more diverse samples. However, the accuracy of this approach strongly depends on the precision of aperture photometry measurements.

Previous studies (Fouesneau & Lançon 2010; Beerman et al. 2012; Anders et al. 2013; de Meulenaer et al. 2013, 2017) have shown that a major limiting factor for the accuracy of integrated photometry methods is their sensitivity to stochastic effects. This is particularly pronounced in young (<1 Gyr) low-mass (≲3000 M_⊙) star clusters, and it is mostly driven by the stochasticity of the stellar initial mass function (IMF). Such clusters tend to have under-sampled IMFs and exhibit sparsely populated upper main-sequence (MS) and post-main-sequence (PMS) evolutionary stages. Beerman et al. (2012) showed that a single bright star, or just a few PMS stars, can introduce significant stochastic variations in a cluster’s brightness and colours, which in turn severely affect the accuracy of cluster age and mass determination.

Another important factor limiting the accuracy of star cluster aperture photometry, especially in the local Universe, is the uncertainty in sky background estimation. Multiple Hubble Space Telescope (HST) studies of star clusters in the Andromeda galaxy (M 31) (Krienke & Hodge 2007, 2008; Hodge et al. 2009, 2010; Johnson et al. 2012) showed that the strongly variable background level of the M 31 disc, its estimation, and subtraction is one of the primary sources of photometric errors. Kodaira et al. (2004) and Narbutis et al. (2008) also reported that photometric uncertainties are predominantly driven by sky background variations and could hardly be reduced without interactive interventions. Moreover, de Meulenaer et al. (2017) show that the presence of bright field stars within the photometric aperture significantly affects the measurements, complicating the derivation of cluster parameters and often resulting in age–extinction degeneracies. Thus, to ensure the reliable interpretation of results derived from star cluster aperture photometry, it is crucial to quantify the limitations imposed by these effects.

In our previous study, Daugevičius et al. (2024, hereafter Paper I), we computed a large sample of star cluster models covering the parameter space representative of real M 31 clusters and generated their synthetic images mimicking observations from the Panchromatic Hubble Andromeda Treasury (PHAT; Dalcanton et al. 2012) survey. We performed aperture photometry of those artificial clusters and determined the best achievable accuracy and applicability limits of the aperture photometry method, imposed by the effects of the stochastic IMF. We showed that clusters with and without PMS stars have significant photometric differences, which at young ages (∼10 Myr) complicate the derivation of cluster parameters based on integrated photometry. Furthermore, we demonstrated that reliable physical cluster parameters can be derived when cluster CIs are measured using apertures with diameters larger than their doubled half-light radii.

In this study, we aim to expand on the results presented in Paper I by determining how the sky background affects the accuracy and applicability limits of the aperture photometry method for star cluster studies in the local Universe. We employed stochastic star cluster models and their realistic HST images, generated using the algorithm described in Paper I. To estimate background effects on aperture photometry limits, we placed simulated cluster images into five PHAT survey background fields of different stellar density and measured them using apertures of various sizes. Furthermore, we examined how selection effects may influence the capabilities of present and future star cluster surveys.

The structure of the paper is as follows. In Sect. 2, we briefly present our star cluster models. In Sect. 3, we show and discuss the selected background fields. In Sect. 4, we describe the aperture photometry procedures used to measure our mock observations. In Sect. 5, we present and discuss the results, and in Sect. 6 we provide a brief summary and conclusions.

2. Star cluster models

In this study, we employed artificial star clusters and their images, synthesised using the algorithm described in Paper I, to which we refer for a detailed description of the modelling procedure; a brief summary is provided here. Initial stellar masses were generated according to Kroupa (2001) IMF using fully stochastic sampling in the range from 0.1 M_⊙ to 100 M_⊙. Individual stars were sampled until the total mass within a sphere of 7.5″ radius (∼28 pc at the M 31 distance) reached the cluster’s target mass. For each individual star, PHAT magnitudes were assigned by interpolating the PARSEC-COLIBRI (Bressan et al. 2012; Marigo et al. 2017) isochrones¹ according to its mass and age. We used isochrones of solar metallicity and no interstellar extinction. To convert absolute magnitudes to apparent magnitudes, the M 31 distance modulus of (m−M)₀ = 24.47 (McConnachie et al. 2005) was applied. Then stars were spatially distributed within the cluster by stochastically sampling an empirical Elson-Fall-Freeman (EFF; Elson et al. 1987) profile, defined by the following radial density function:

$$\rho (r)={\rho }_0{\left(1+\frac{r^2}{r_0^2}\right)}^{-(\gamma +1)/2},$$(1)

where ρ₀ is the central cluster density, r the deprojected (3D) distance from the cluster’s centre, r₀ the scale factor, and γ the power-law slope.

To generate realistic mock observations of simulated star clusters, we used the TinyTim² package to model images of individual stars. TinyTim enables the modelling of point spread functions (PSFs) for various HST instrument and filter combinations. This representation of PSFs is appropriate for our study, since integrated cluster photometry is performed using apertures that are much larger than the full width at half maximum of PSFs. Each star was simulated by multiplying its total flux in a given passband by the corresponding TinyTim PSF. Images of clusters were generated within 15″×15″ frames (∼57×57 pc at the M 31 distance).

Following the algorithm described above, we generated a grid of star cluster models and their corresponding images, covering the parameter space representative of real clusters observed in the M 31 disc. Compared to Paper I, the grid was expanded from 500 to 1400 nodes by including more massive clusters and introducing additional ages. Each grid node consists of 100 independently generated artificial clusters. The parameters of the simulated star clusters are listed in Table 1.

Additionally, for results presented and discussed in Sect. 5.2, we used a separate set of artificial star clusters simulated using the same modelling approach. However, instead of a discrete grid, we generated a continuous set of artificial clusters in the F475W passband, containing 20 million clusters. The clusters span the following parameter ranges: log(M/M_⊙) = 2.4−4.1, log(t/yr) = 6.5−10.1, r_c = 0.05″−0.6″, and γ = 2.1−4.1 (Šablevičiūtė et al. 2006, 2007).

3. Background fields

To assess the impact of background effects on the aperture photometry of star clusters and on catalogue completeness limits in the context of the PHAT survey, we placed artificial clusters into five real background fields of different stellar densities in the M 31 disc (see Fig. 1). We used the same PHAT survey image data as in the studies by Naujalis et al. (2021) and Kriščiūnas et al. (2023). The full dataset consists of six broad-band filters from three HST instruments, covering a spectral range from 0.25 to 1.6 μm: the F275W and F336W passbands from the Wide Field Camera 3 UVIS channel (WFC3/UVIS); the F475W and F814W passbands from the Wide Field Channel of the Advanced Camera for Surveys (ACS/WFC); and the F110W and F160W passbands from the Wide Field Camera 3 IR channel (WFC3/IR). For information on the data processing, refer to Naujalis et al. (2021).

Fig. 1. Background fields in M 31. The top panel shows a grey-scale 24 μm map of M 31 (Multi-Band Imaging Photometer for Spitzer, Spitzer/MIPS) with the marked locations of background fields (B1–B5). The bottom panels show images of the artificial star cluster (log(M/M⊙) = 3.5, log(t/yr) = 7.0, rc = 0.2″, and γ = 2.8) placed in each of the background fields. B0: Background-less case. Magenta and cyan circles mark apertures with radii Rap = 1″ and 2″, respectively. The colour images (15″×15″) were produced by combining F336W, F475W, and F814W data.

Because only two repeated exposures are available for the F275W and F336W passbands, these frames are contaminated with numerous cosmic-ray artefacts. Since reliable automatic cleaning is challenging in such cases, we manually removed the most obvious and bright cosmic-ray defects from the selected background fields using the imedit task from PyRAF³.

For the background fields (Fig. 1), we used square cut-outs of 17″×17″ (∼65×65 pc at the M 31 distance). As these cut-outs are larger than the cluster images, we were able to place each cluster at a slightly different random location within a given background field, while preserving the same overall background features. This approach also helped mitigate potential biases that could be introduced by placing all artificial clusters in the same position, which might contain some peculiar features such as a very bright background star or a dusty patch.

The selected background fields differ not only in stellar density, which mostly depends on the radial distance from the galactic centre, but also in the type of environment they represent. As is shown in the Spitzer map (Fig. 1), two fields (B1 and B3) are located in the star-forming rings, while B2 and B4 lie in the inter-ring regions; meanwhile, B5 represents the dense inner regions of the M 31 disc. Fig. 2 shows CMDs of these fields. Due to their location in star-forming regions, fields B1 and B3 have prominent young MS populations, while the remaining fields are dominated by old stellar populations, such as red giant branch, asymptotic giant branch, and fainter MS stars. The dependence of the limiting magnitude on the stellar density is clearly visible. In the B5 field, a prominent MS is not visible, as the MS turn-off point lies below the limiting magnitude due to the crowdedness of old stellar populations. In addition, all background fields show some level of extinction, as is evidenced by observed MSs being shifted to the red relative to the isochrones’ MSs in the B1–B5 panels of Fig. 2. Another important aspect is the strong differential extinction effect in fields B1 and B3, which becomes particularly evident when comparing the red clumps of B1 with B2, and B3 with B4. Fields B1 and B3 exhibit strongly dispersed red clump populations, spanning multiple magnitudes in both brightness and colour. The general direction of this dispersion is similar to that of the extinction vector. This, combined with the fact that both fields are located in the star-forming rings, suggests that they are affected by differential extinction caused by dusty lanes and patches, which are prominent in star-forming regions. As the majority of young clusters reside in such regions, it is likely that at least some are affected by differential extinction. It would distort integrated photometric measurements and severely complicate the derivation of cluster parameters (Narbutis et al. 2007a; Bridžius et al. 2008). Particularly striking is the fact that such strong differential extinction effects are observed within a relatively small field of view.

Fig. 2. CMDs of the selected background fields. Each panel represents a different field (B0–B5). For the background-less field (B0), the CMD of the generated cluster (log(M/M⊙) = 4.5, log(t/yr) = 8.0), assuming photometry errors from Williams et al. (2023), is shown. Stellar photometry for fields B1–B5 was taken from Williams et al. (2023). The number of stars visible in the CMD is indicated below the field name. Coloured curves represent the PARSEC-COLIBRI solar metallicity isochrones for log(t/yr) = 8.0 (cyan), 8.5 (blue), 9.0 (orange), 9.5 (magenta), and 10.0 (red) ages. The black arrows mark the extinction vectors, calculated using the Fitzpatrick (1999) extinction law with AV = 1.

The B3 field is located within the 10 kpc star-forming ring, where the majority of clusters from the PHAT survey are found. For this reason, in Sect. 5, we use it (extensively) as the representative field when investigating various parameter dependencies.

4. Aperture photometry

In this section, we describe the aperture photometry procedures used in this study. As the first step, we determined the sky background level to be subtracted from the flux measured within a given aperture. We followed the background determination procedure described by Johnson et al. (2012), which we consider appropriate for this work, as our simulated observations mimic those of the PHAT survey. This method was also used for producing original photometric star cluster catalogues of the PHAT and PHATTER surveys (Johnson et al. 2012, 2015, 2022). A brief overview of the procedure is provided below.

Once an artificial star cluster was placed into a background field, a circular ring extending from 1.2×R_ap to 3.4×R_ap (Johnson et al. 2012) was defined and positioned at the centre of the cluster. In all cases, for sky background level determination we assumed R_ap = 2.0″. That large ring was then divided into 10 smaller annuli of equal area. Each of these had the same area as a circular aperture with R_ap = 2.0″.

Then we measured the integrated flux within each of the 10 annuli. An iterative 2σ-clipping procedure was subsequently applied to these flux values. This helped us to reject regions affected by bright background stars (or, in some cases, even artificial cluster bright stars) or other peculiar features. As the final sky background value, we used the mean of the non-rejected fluxes, while the standard deviation of these values was adopted as the background level determination uncertainty, σ_s. Since we measured artificial clusters using not only an aperture of radius R_ap = 2.0″, but also various smaller and larger apertures, we scaled the mean sky background estimate accordingly to match the corresponding aperture size. We emphasise that the sky background level was determined individually for each artificial cluster’s frame after the simulated cluster had already been placed into the background field. We consider this approach to be less idealised and more representative of real observational conditions, as the background level estimation can also be affected by cluster stars located in its outer regions.

Once the sky background level was determined, we performed aperture photometry of the artificial cluster placed in a given background field using multiple apertures of different sizes, ranging from R_ap = 0.5″ to 5.0″. Such procedures were performed on every generated star cluster, in all five background fields, with measurements carried out in three PHAT passbands – F336W, F475W, and F814W. Using the same apertures, we also performed measurements of these clusters in the background-less field (B0) as a reference for quantifying background effects. All aperture photometry procedures, including the sky background level estimation, were performed using tools provided in the Python package photutils⁴.

We note that in Sect. 5 we mostly used the cluster’s half-light radius (R₅₀) units to define the aperture size. For each individual artificial cluster, R₅₀ was determined by measuring its growth curve in the F475W passband up to 5″ and evaluating the radius at which 50% of the total integrated flux inside R_ap = 5.0″ aperture is reached. We emphasise that when R₅₀ was used to define aperture size and the resulting aperture radius was smaller than 0.5″, we ensured it was at least 0.5″. For example, if a cluster had R₅₀ = 0.3″ and the chosen aperture was R_ap = 1.5×R₅₀, the resulting aperture radius would be R_ap = 0.45″. However, in such cases, we increased it up to R_ap = 0.5″. This is justified in Paper I, in which we show that colour indices are strongly affected by stochastic effects if small apertures (R_ap<0.5″) are used. Additionally, we have demonstrated that such small apertures lead to unreliably determined physical star cluster parameters.

5. Results and discussion

5.1. Photometric uncertainties

In this study, we account for three different error sources in star cluster aperture photometry measurements. One of these is Poisson noise. However, since we used drizzled images and fluxes scaled to one-second exposures, it is problematic to determine these statistical errors correctly. Thus, to roughly estimate the impact of this error source, we multiplied one-second exposure fluxes by the PHAT exposure times provided by Williams et al. (2014). The Poisson uncertainty for the magnitude measurements was evaluated using the following formula:

$${\sigma }_p=2.5{log}_{10}\left(1+\frac{\sqrt{S_{\mathit{cl}}}}{S_{\mathit{cl}}}\right),$$(2)

where S_cl is the cluster signal (one-second flux, F_cl, multiplied by the exposure time) measured through the given aperture, in the background-less field (B0).

The second source of uncertainties we considered is the effect of possible aperture position bias. We estimated it following the approach described in Naujalis et al. (2021) and Kriščiūnas et al. (2023). For each simulated image, we performed aperture photometry by positioning the centre of the selected aperture at each of the nine central pixels of the frame (in the pixel scale of HST/ACS). This gave us nine cluster magnitude and CI measurements of each mock observation. Magnitude uncertainties due to this possible aperture position bias (σ_a) were estimated as a half of the 16th–84th percentile range. We emphasise that the σ_a uncertainty for CIs was determined from nine CI measurements, rather than by combining the uncertainties of the magnitudes in the individual passbands that comprise the CI.

The third source of errors we consider arises from the background level estimation procedure. It was evaluated as follows:

$${\sigma }_b=2.5{log}_{10}\left(1+\frac{{\sigma }_s·A_{\mathrm{ap}}}{F_{\mathit{cl}}}\right),$$(3)

where σ_s is the background level determination uncertainty, which was introduced in Sect. 4, A_ap is the area of the selected aperture in pixels, and F_cl is the integrated cluster flux within that aperture measured in the B0 field.

The aggregated photometric uncertainty of the magnitude in a given passband was calculated according to the following formula:

$${\sigma }_t=\sqrt{{\sigma }_p^2+{\sigma }_b^2+{\sigma }_a^2}.$$(4)

Meanwhile, the total photometric uncertainties for CIs were determined by adding squared σ_p and σ_b of the individual passbands that comprise the given CI, together with squared σ_a determined for that CI. Among the individual sources of uncertainty, we find that the background estimation error (σ_b) is the dominant contributor in the majority of cases, which is in agreement with previous HST studies of M 31 star clusters (Krienke & Hodge 2007, 2008; Hodge et al. 2009, 2010; Johnson et al. 2012). However, it is noteworthy that in some specific cases, the aperture centring error (σ_a) can exceed σ_b. This typically occurs when a bright cluster member or a background star is projected near the edge of the aperture. The uncertainty due to Poisson noise (σ_p) is relatively insignificant compared to the other two uncertainty sources. In the following analysis, we consider only the aggregated total uncertainties for magnitude and for CIs, both of which are referred to as σ_t in a general manner.

The distributions of cluster age and mass combinations that include at least 50% of models meeting indicated requirements (for F475W, F336W−F475W, and F475W−F814W) are shown in Fig. 3, as functions of background fields (B1, B3, B5), photometric error (σ_t) thresholds, and aperture sizes (R_ap). We applied three error thresholds (σ_t = 0.05,0.10,0.30) following de Meulenaer et al. (2015), who discussed how integrated photometry with ≤0.05 mag accuracy enables one to derive the cluster’s age, mass, and extinction, and also to estimate the metallicity of clusters older than 1 Gyr. Correspondingly, in cases of ≤0.10 mag photometric accuracy, we can expect to estimate cluster age, mass, and extinction for known metallicity; in cases of ≤0.30 mag photometric accuracy, we can expect to roughly estimate cluster age and mass, for known extinction and metallicity.

Fig. 3. Regions of star cluster parameter space where aperture photometry meets the required accuracy. The grey areas and regions with blue or red squares indicate the cluster mass and age combinations containing at least 50% (out of ∼1300) of clusters satisfying the photometric error threshold, σt. Top row: results for the F475W passband. Middle row: results for F336W−F475W. Bottom row: results for F475W−F814W. Left column: results for different background fields, using Rap = 1.5×R50 and the error threshold σt≤0.05 mag. Middle column: results depending on the σt threshold value for the B3 field, using Rap = 1.5×R50. Right column: results depending on the aperture size for the B3 field and the error threshold σt≤0.05 mag. Results are based on artificial clusters with half-light radii of R50 = 0.25″−1.00″.

The dependence on background environment is most clearly seen in the left column of Fig. 3, where results from measurements performed using the aperture of R_ap = 1.5×R₅₀ size in the fields B1, B3, and B5 are compared, assuming an error threshold of σ_t≤0.05 mag. Based on the results for F475W and F336W−F475W, we can state that the limits of the cluster age–mass parameter space, where photometric accuracy, required for physical parameter determination, can be achieved, barely depend on the background density in the UV-optical parts of the spectrum. However, a significant dependence is observed in the case of F475W−F814W. This is due to numerous bright red stars residing in the M 31 disc, whose number density is highly dependent on the radial distance from the galactic centre.

Therefore, the most limiting factor is our ability to perform precise photometry in the F814W passband due to numerous bright red background and/or foreground stars. More accurate photometric measurements in (semi-)resolved cases could be achieved using sophisticated background estimation techniques, such as interactive methods (Naujalis et al. 2021; Kriščiūnas et al. 2023) or advanced automated algorithms (Pandey & Saha 2025).

The middle column (Fig. 3) shows age and mass limits where at least 50% of models pass the given photometric accuracy thresholds (σ_t = 0.05,0.10,0.30). The results are provided for the B3 field, using an aperture of R_ap = 1.5×R₅₀. Taking into consideration results for both F336W−F475W and F475W−F814W CIs and interpolating linearly between mass limits derived for each of these CIs, we find that at 1 Gyr, photometric accuracy of σ_t = 0.05 can be achieved for ≳5000 M_⊙ clusters; σ_t = 0.10 for 1600≲M≲5000 M_⊙ clusters; and σ_t = 0.30 for 500≲M≲1600 M_⊙ clusters. However, at 100 Myr lower mass limits can be reached: σ_t = 0.05 is attainable for ≳1000 M_⊙ clusters; σ_t = 0.10 for 500≲M≲1000 M_⊙ clusters; and σ_t = 0.30 for 200≲M≲500 M_⊙ clusters.

The rightmost column (Fig. 3) shows how the age–mass parameter space that meets the photometric precision requirement σ_t≤0.05 mag depends on aperture size, R_ap. Here, we compare measurements performed in the B3 field using three different aperture sizes (R_ap): 1″,R₅₀,1.5×R₅₀. We notice that aperture sizes scaled by an individual cluster’s R₅₀ yields better results than a fixed one-for-all aperture (R_ap = 1″ in this case). We find that the best results are achieved using R_ap = 1.5×R₅₀ apertures, which is consistent with our findings in Paper I. It is noteworthy, however, that in some cases R_ap=R₅₀ performs better than R_ap = 1.5×R₅₀. Nevertheless, as is shown in Paper I, measurements with R_ap<R₅₀ are susceptible to significant stochastic effects, especially for small clusters (R₅₀≈0.5″), and can result in unreliable estimates of cluster physical parameters. Therefore, we recommend that such small apertures be used with caution, and only in certain special cases; for instance, to avoid contamination from bright field stars projecting near the cluster (Narbutis et al. 2007b).

To expand recommendations on appropriate aperture sizes for cluster colour measurements, we determined the optimal aperture radius (R_opt) for each mock observation – defined as the aperture that yields the smallest σ_t. In Fig. 4 we show how R_opt (expressed in units of R₅₀) depends on cluster luminosity for three cluster geometric parameter (r_c, γ) combinations. The green markers represent compact clusters, blue – typical M 31 clusters, and red – extended ones. The optimal aperture, R_opt, versus F475W within the 10-magnitude interval differs by a factor of approximately 1.5–2.0 between the faintest and brightest star clusters. The optimal aperture radius for measuring CIs of typical M 31 disc star clusters is (1–2)×R₅₀, which is consistent with our recommendations given in Paper I.

Fig. 4. Optimal aperture radii, Ropt, versus the F475W cluster magnitude measured in the B0 field. Top panel: Ropt for measuring the F475W magnitude. Middle panel: F336W−F475W. Bottom panel: F475W−F814W. Results are shown for the B3 field using three combinations of geometric parameters, rc and γ (see legend, where approximate median half-light radii, R50, are also included for reference). Markers show the Ropt median values. Error bars indicate the 16th–84th percentile ranges. Dashed lines represent second-order polynomial least-squares fits.

In Fig. 5, we show differences of Δ(F475W), Δ(F336W−F475W), and Δ(F475W−F814W), defined as the magnitudes and colour indices of artificial star clusters measured in the B1, B3, and B5 fields minus the ones obtained for the same clusters in the B0 field. We find that, beyond a certain magnitude limit (which depends on the background field), systematic deviations from the B0 case begin to appear, increasing with decreasing cluster brightness. Also, the measurements of faint clusters placed in the background fields tend to be systematically brighter compared to the magnitudes measured in the B0 field. Similarly, CI measurements are often biased towards redder values. However, a notable exception is observed in the B3 field, where F475W−F814W shows a slight bias towards bluer values. This may be explained by the presence of a significant population of relatively young, blue stars in that field. It is also noteworthy that background effects for F475W−F814W are stronger and the bias begins to appear at brighter magnitude compared to F475W or F336W−F475W. This is due to the brighter M 31 background in the F814W passband, resulting in a lower contrast between the cluster and the background signal than in F336W or F475W. Such biases can lead to inaccuracies in the determination of physical parameters: a biased higher luminosity may result in an overestimated mass, while redder CIs can lead to an overestimated age, metallicity, or extinction.

Fig. 5. Differences in F475W (top), F336W−F475W (middle), and F475W−F814W (bottom), calculated as measurements in the B1, B3, and B5 fields minus the ones in the B0 field, versus the F475W cluster magnitude measured in the B0 field. Markers show the median differences. Error bars indicate the 16th–84th percentile ranges. Blue and red markers are slightly offset along the X axis for clarity. Results are based on artificial clusters with half-light radii of R50 = 0.4″−0.6″.

In Fig. 6 we show the same analysis as in Fig. 5, but for the B3 field and three different cluster sizes. We find that, at a certain cluster magnitude, larger clusters (red dots) exhibit stronger biases and broader magnitude and CI difference spreads compared to more compact ones. Moreover, for larger clusters, systematic deviations begin to appear at brighter magnitudes. This is due to the use of larger apertures for extended clusters, which include more background signal, thereby reducing the cluster-to-background contrast. Additionally, the probability of a bright background and/or foreground star falling within the aperture increases with aperture size.

Fig. 6. Differences in F475W (top), F336W−F475W (middle), and F475W−F814W (bottom), calculated as measurements in the B3 field minus the ones in the B0 field, versus the F475W cluster magnitude measured in the B0 field. Results are shown for the same combinations of geometric parameters (rc and γ) as in Fig. 4 (see legend, where approximate median half-light radii R50 are also included for reference). Markers show the median differences; error bars indicate the 16th–84th percentile ranges. Blue and red markers are slightly offset along the X axis for clarity.

5.2. Completeness and selection effects

In this section, we explore two key aspects of background influence: its impact on completeness limits and on cluster detection quality. These factors define the accessible parameter space for star cluster studies and must be understood to avoid over-interpreting survey data.

To determine cluster detection completeness limits in the PHAT survey, the cluster-to-background signal ratio (CBR) was defined as:

$$\mathrm{CBR}=\frac{F_{\mathit{cl}}}{F_{\mathit{bg}}},$$(5)

where, F_cl is the cluster’s F475W flux measured in the background-less field (B0) using an aperture R_ap=R₅₀, and F_bg the background F475W flux measured within the same aperture in the selected M 31 fields (see Fig. 1). For CBR evaluation, we used the half-light radius aperture R₅₀, as it captures only the brightest central region of the cluster, providing the strongest contrast against the background, which is most significant for cluster detection. In cases where R₅₀<0.5″, we set R_ap = 0.5″ to avoid the extreme stochastic effects associated with small apertures, as is discussed in Sect. 4.

In Fig. 7, we show artificial star cluster images of various age and mass combinations placed in the B0 and B3 fields. For each cluster in the B3 field, the corresponding CBR value is indicated.

Fig. 7. Images of artificial star clusters of various ages and masses in the B0 (left) and B3 (right) fields. The cluster mass is indicated below each column, and the age is noted to the left of each row. All clusters were generated using rc = 0.2″ and γ = 2.8 parameters. The corresponding CBR values (see definition in Sect. 5.2) determined in the B3 field are indicated above the image pairs. Magenta circles mark apertures used to measure CBR values. The colour images (5″×5″) were produced by combining F336W, F475W, and F814W data.

Further, we used a set of CBR threshold values as cluster detection criteria: CBR≥2.5, CBR≥1.0, and CBR≥0.4. As is shown in Fig. 7, clusters with CBR≥1.0 can be visually detected with relative ease. We therefore adopted this threshold as a detection limit achievable through visual searches, such as the ones used in citizen science projects (Johnson et al. 2015). The threshold CBR≥2.5 corresponds to clusters that are ∼1 mag brighter, and CBR≥0.4 to those that are ∼1 mag fainter. We assumed that the threshold CBR≥0.4 represents the PHAT survey depth achievable with current machine-learning (ML) algorithms. Thilker et al. (2022) show that ML-produced star cluster catalogues in nearby galaxies can reach depths ∼1 mag fainter than the ones achieved through human detection.

In Fig. 8, we show the regions in the cluster age–mass parameter space where at least 50% completeness can be achieved (in the context of the PHAT survey), depending on the selected CBR threshold, the background environment, and the cluster’s geometric parameters. In the top panel of Fig. 8, we show results for the three aforementioned detection thresholds in the B3 field (clusters with R₅₀ = 0.25″−1.00″ are used). For the detection threshold CBR≥2.5 – corresponding to clusters that can be studied with high accuracy – the 50% completeness limits are reached at ∼1300 M_⊙, ∼2500 M_⊙, and ∼8000 M_⊙ (for ages of 100 Myr, 300 Myr, and 1 Gyr, respectively). At a lower threshold of CBR≥1.0 – corresponding to clusters detectable through visual searches – these limits shift to ∼500 M_⊙, ∼1000 M_⊙, and ∼3000 M_⊙ for the same ages. If the detection algorithm is capable of reaching CBR≥0.4, the resulting cluster catalogue becomes significantly deeper, with 50% completeness achieved at ∼250 M_⊙, ∼400 M_⊙, and ∼1300 M_⊙ (for ages 100 Myr, 300 Myr, and 1 Gyr, respectively). This represents an improvement of more than two times in the detectable mass range compared to the CBR≥1.0 cases. Interestingly, Thilker et al. (2022), in the context of the PHANGS-HST survey, reported a similar gain when comparing the performance of their advanced ML-based cluster detection algorithm with expert scientists’ visual identification.

Fig. 8. Cluster age–mass parameter space where at least 50% completeness can be achieved. Top panel: dependence on CBR, with results shown for the B3 field. Middle panel: completeness limits across three background fields, assuming the detection threshold of CBR≥0.4. Bottom panel: results for three different cluster geometric parameter combinations in the B3 field, also assuming CBR≥0.4. The results shown in the top and middle panels were determined using artificial clusters with R50 = 0.25″−1.00″.

The middle panel of Fig. 8 shows variations in the 50% completeness limits depending on the environment (radial distance from the M 31 centre). Here, results are presented assuming the detection threshold of CBR≥0.4 and using clusters of typical size (R₅₀ = 0.25″−1.00″). Across the full range of environments – from the sparse outer regions to the dense inner disc fields – we find that the achievable survey depth at a certain age generally changes by ∼1 dex in mass. For example, at 1 Gyr, the 50% completeness limit is ∼250 M_⊙ in the B1 field, ∼1300 M_⊙ in the B3 field, and ∼3000 M_⊙ in the B5 field. These findings highlight the importance of carefully accounting for selection effects when investigating radial or environmental trends, as the completeness of a cluster catalogue strongly depends on the local background.

The bottom panel of Fig. 8 illustrates the dependence of completeness limits on the geometric parameters of clusters, showing results for three r_c and γ combinations in the B3 field, assuming a detection threshold of CBR≥0.4. A significant selection bias toward more compact clusters is evident: for extended clusters (red area), the region of parameter space where at least 50% completeness is achieved is substantially shallower than typical M 31 (green) and compact (blue) clusters. For example, at an age of 1 Gyr, the 50% completeness limit is ∼400 M_⊙ for compact clusters, ∼800 M_⊙ for average ones, and only ∼2500 M_⊙ for extended clusters.

This highlights the importance of carefully accounting for selection effects when studying phenomena related to cluster size – such as the cluster size distribution, mass–radius relation, dynamical evolution, and tidal disruption (Portegies Zwart et al. 2010; Krumholz et al. 2019; Adamo et al. 2020; Brown & Gnedin 2021). In (semi-)resolved cases, the completeness of star cluster catalogues can be biased towards more compact objects, particularly in nearby galaxies like M 31 or M 33. On the other hand, in more distant galaxies, catalogue completeness could be biased towards larger clusters, as compact clusters are often unresolved and can be misclassified as point sources due to limited spatial resolution (Ryon et al. 2017; Brown & Gnedin 2021; Maschmann et al. 2024).

Johnson et al. (2015) report that the PHAT cluster catalogue is 50% complete down to ∼500 M_⊙ for ages younger than 100 Myr. Our results, for 100 Myr old clusters and a CBR≥1.0 detection limit, are in general agreement, at ∼500 M_⊙. Thus, our assumption that the CBR≥1.0 limit is similar to human-eye abilities appears to be reasonable. However, while Johnson et al. (2015) include all clusters younger than 100 Myr, our estimate is for an exact artificial cluster age of 100 Myr. The assumed CBR≥1.0 detection limit might be slightly looser than the one achieved by citizen science in Johnson et al. (2015). Therefore, it seems that the PHAT cluster catalogue could be expanded at the low-mass (≲500 M_⊙) end by employing advanced ML-based cluster detection methods (Bialopetravičius et al. 2019; Thilker et al. 2022).

Furthermore, we used CBR measurements to estimate the impact of selection effects on the cluster mass function (CMF) in the context of the PHAT survey (Fig. 9). As the initial CMF, we adopted the Schechter function with a power-law index of β=−2 (Schechter 1976; Gieles 2009) and truncation mass of M_C = 10⁴ M_⊙, consistent with results for M 31 (Johnson et al. 2017).

Fig. 9. Selection effects on CMFs due to detection incompleteness. The initial CMF is shown as a black line (Schechter function with power-law index β=−2 and truncation mass MC = 104 M⊙). Top panel: CMFs for artificial clusters of various ages (see legend) in the B3 field, showing results for CBR≥1.0 (dashed lines) and CBR≥0.4 (solid lines). Bottom panel: CMFs for clusters of log(t/yr) = 9 age, measured in the B1–B5 background fields at different radial distances from the M 31 centre, assuming the detection threshold CBR≥0.4. Results are based on artificial clusters with half-light radii of R50 = 0.25″−1.00″.

In the top panel of Fig. 9, we show CMFs for several cluster ages, each shown under two detection thresholds: CBR≥1.0 (dashed lines) and CBR≥0.4 (solid lines). In the case of the CBR≥0.4 threshold, for clusters younger than 100 Myr, the CMF can be recovered down to ∼300 M_⊙, and for clusters of 300 Myr ages the reachable mass limit could be ∼600 M_⊙.

The bottom panel of Fig. 9 shows the 1 Gyr age CMF dependence on the environment (radial distance from the M 31 centre), assuming a cluster detection threshold of CBR≥0.4. We find that the CMFs can be recovered down to mass of ∼500 M_⊙ in the outer disc regions (the B1 and B2 fields), and to ∼1500 M_⊙ in the intermediate disc areas or star-forming regions (the B3 and B4 fields). In contrast, in the dense inner disc environment (the B5 field), the CMF is only complete down to ∼3000 M_⊙. We note that the flat tail at ∼300 M_⊙ observed in the B5 field is statistical in nature and arises from the limited number of low-mass clusters measured.

It is important to note that interstellar extinction effects were not included in our modelling of completeness and selection effects. However, as is discussed by Maschmann et al. (2024), a significantly higher incompleteness can be expected in UV-optical-NIR observations of very young clusters (≲5 Myr), as they are often enshrouded in their natal gas and dust clouds. To avoid presenting potentially misleading information, in Figs. 8 and 9 we do not provide results for clusters younger than 10 Myr.

6. Summary and conclusions

In this study, we have investigated the accuracy and applicability limits of the aperture photometry method for star clusters, focusing on the effects of background and stochasticity. Here, we expand on our previous study Paper I, in which we mainly explored the impact of stochasticity. We used a large grid of artificial star clusters covering the parameter space of real objects observed in M 31. Cluster images were generated in six HST ACS+WFC3 passbands, mimicking star cluster observations from the PHAT survey (Dalcanton et al. 2012; Johnson et al. 2012, 2015). To create realistic mock observations of clusters, we placed them into five background fields selected from the PHAT data, each representing a different stellar density and environment. We performed aperture photometry on the resulting images, estimated photometric uncertainties, determined optimal aperture sizes, and estimated deviations in photometric measurements introduced by background effects. Furthermore, using artificial clusters, we investigated star cluster selection effects and completeness limits in the context of the PHAT survey.

For the PHAT survey, we identified regions of the age–mass parameter space of star clusters where photometric uncertainties are low enough to allow one to simultaneously determine the age, mass, extinction, and metallicity (de Meulenaer et al. 2014, 2015). We also demonstrated the dependence of these regions on both the environment (radial distance from the M 31 centre) and the aperture size used for photometry. Additionally, we identified which cluster age and mass combinations in a typical M 31 environment ensure a PHAT photometric accuracy that is sufficient for cluster parameter determination. We estimated that at 1 Gyr, photometry is accurate enough to determine the age, mass, and extinction (assuming known metallicity) for clusters ≳1600 M_⊙; however, only rough age and mass estimates are possible for clusters 500≲M≲1600 M_⊙. At an age of 100 Myr, the corresponding mass limits are: ≳500 M_⊙ and 200≲M≲500 M_⊙.

Based on our estimates of photometric uncertainties, we derived optimal aperture radii, R_opt. We demonstrated that R_opt depends on cluster brightness, with more luminous objects requiring larger apertures. Within the cluster magnitude range explored in this paper, we show that R_opt varies by a factor of ∼2, despite identical cluster geometric parameters. For CI measurements, we find that the optimal aperture radius for typical-size clusters, ∼(1–2)×R₅₀, is consistent with our recommendations from Paper I. After evaluating background effects, we reiterate that using small apertures (R_ap≲R₅₀) should be avoided due to strong stochastic and background-induced uncertainties.

Moreover, we investigated the biases in cluster magnitude and CI measurements introduced by background effects for M 31 star clusters. We find that, beyond a certain cluster brightness threshold, systematic biases begin to appear – clusters tend to be measured as brighter and redder than they truly are. These deviations increase as the cluster brightness decreases. Additionally, larger clusters are more susceptible to these effects. Such biases can lead to significant inaccuracies in the derived physical parameters of star clusters.

Furthermore, we investigated star cluster selection effects in the PHAT survey and identified the regions in the cluster age–mass parameter space where at least 50% completeness limits can be achieved. For the PHAT survey, we derived that the 50% completeness limits could be reached at ∼250 M_⊙ for clusters of 100 Myr, ∼400 M_⊙ at 300 Myr, and ∼1300 M_⊙ at 1 Gyr.

We show that within the context of the PHAT survey, a significant selection bias towards more compact clusters may exist. Thus, when studying various relations connected with cluster size – such as structural evolution, disruption, or scaling relations – it is necessary to account for these selection effects.

Finally, we estimated the impact of selection effects on the CMF in the M 31 disc. We derived that the CMF at 1 Gyr can be recovered down to: ∼500 M_⊙ in the outer regions of the disc; ∼1500 M_⊙ in dense or star-forming regions; ∼3000 M_⊙ in the inner disc regions. These results stress the need to carefully account for selection effects when studying radial or environmental dependencies of the CMF.

We anticipate that the results presented in this study will be valuable for planning future observations and for interpreting results from current and upcoming star cluster surveys in the local Universe. In the future, we aim to apply the findings from Paper I and this study to develop automated, ML-based cluster detection and aperture photometry algorithms. Such tools will be especially important in the era of big data astronomy, where large wide-field star cluster surveys will make manual, interactive approaches impractical.

Acknowledgments

This research has made use of: the NASA/IPAC Extragalactic Database (NED), which is funded by the National Aeronautics and Space Administration and operated by the California Institute of Technology; SAOImage DS9, developed by the Smithsonian Astrophysical Observatory (https://ds9.si.edu); Astropy (http://www.astropy.org), a community-developed core Python package for Astronomy (Astropy Collaboration 2013, 2018); APLpy, an open-source plotting package for Python (Robitaille & Bressert 2012); SciPy, an open-source scientific library for Python (Virtanen et al. 2020); Matplotlib, a plotting library for Python (Hunter 2007); TOPCAT (https://www.star.bristol.ac.uk/mbt/topcat). It is based on observations made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Centre (CADC/NRC/CSA). This project has received financial support from the Research Council of Lithuania (LMTLT), agreement No S-MIP-24-98. Computations were performed on the supercomputer GALAX of the Center for Physical Sciences and Technology, Lithuania.

References

All Tables

All Figures

Fig. 1. Background fields in M 31. The top panel shows a grey-scale 24 μm map of M 31 (Multi-Band Imaging Photometer for Spitzer, Spitzer/MIPS) with the marked locations of background fields (B1–B5). The bottom panels show images of the artificial star cluster (log(M/M⊙) = 3.5, log(t/yr) = 7.0, rc = 0.2″, and γ = 2.8) placed in each of the background fields. B0: Background-less case. Magenta and cyan circles mark apertures with radii Rap = 1″ and 2″, respectively. The colour images (15″×15″) were produced by combining F336W, F475W, and F814W data.

In the text

Fig. 2. CMDs of the selected background fields. Each panel represents a different field (B0–B5). For the background-less field (B0), the CMD of the generated cluster (log(M/M⊙) = 4.5, log(t/yr) = 8.0), assuming photometry errors from Williams et al. (2023), is shown. Stellar photometry for fields B1–B5 was taken from Williams et al. (2023). The number of stars visible in the CMD is indicated below the field name. Coloured curves represent the PARSEC-COLIBRI solar metallicity isochrones for log(t/yr) = 8.0 (cyan), 8.5 (blue), 9.0 (orange), 9.5 (magenta), and 10.0 (red) ages. The black arrows mark the extinction vectors, calculated using the Fitzpatrick (1999) extinction law with AV = 1.

In the text

Fig. 3. Regions of star cluster parameter space where aperture photometry meets the required accuracy. The grey areas and regions with blue or red squares indicate the cluster mass and age combinations containing at least 50% (out of ∼1300) of clusters satisfying the photometric error threshold, σt. Top row: results for the F475W passband. Middle row: results for F336W−F475W. Bottom row: results for F475W−F814W. Left column: results for different background fields, using Rap = 1.5×R50 and the error threshold σt≤0.05 mag. Middle column: results depending on the σt threshold value for the B3 field, using Rap = 1.5×R50. Right column: results depending on the aperture size for the B3 field and the error threshold σt≤0.05 mag. Results are based on artificial clusters with half-light radii of R50 = 0.25″−1.00″.

In the text

Fig. 4. Optimal aperture radii, Ropt, versus the F475W cluster magnitude measured in the B0 field. Top panel: Ropt for measuring the F475W magnitude. Middle panel: F336W−F475W. Bottom panel: F475W−F814W. Results are shown for the B3 field using three combinations of geometric parameters, rc and γ (see legend, where approximate median half-light radii, R50, are also included for reference). Markers show the Ropt median values. Error bars indicate the 16th–84th percentile ranges. Dashed lines represent second-order polynomial least-squares fits.

In the text

Fig. 5. Differences in F475W (top), F336W−F475W (middle), and F475W−F814W (bottom), calculated as measurements in the B1, B3, and B5 fields minus the ones in the B0 field, versus the F475W cluster magnitude measured in the B0 field. Markers show the median differences. Error bars indicate the 16th–84th percentile ranges. Blue and red markers are slightly offset along the X axis for clarity. Results are based on artificial clusters with half-light radii of R50 = 0.4″−0.6″.

In the text

Fig. 6. Differences in F475W (top), F336W−F475W (middle), and F475W−F814W (bottom), calculated as measurements in the B3 field minus the ones in the B0 field, versus the F475W cluster magnitude measured in the B0 field. Results are shown for the same combinations of geometric parameters (rc and γ) as in Fig. 4 (see legend, where approximate median half-light radii R50 are also included for reference). Markers show the median differences; error bars indicate the 16th–84th percentile ranges. Blue and red markers are slightly offset along the X axis for clarity.

In the text

Fig. 7. Images of artificial star clusters of various ages and masses in the B0 (left) and B3 (right) fields. The cluster mass is indicated below each column, and the age is noted to the left of each row. All clusters were generated using rc = 0.2″ and γ = 2.8 parameters. The corresponding CBR values (see definition in Sect. 5.2) determined in the B3 field are indicated above the image pairs. Magenta circles mark apertures used to measure CBR values. The colour images (5″×5″) were produced by combining F336W, F475W, and F814W data.

In the text

Fig. 8. Cluster age–mass parameter space where at least 50% completeness can be achieved. Top panel: dependence on CBR, with results shown for the B3 field. Middle panel: completeness limits across three background fields, assuming the detection threshold of CBR≥0.4. Bottom panel: results for three different cluster geometric parameter combinations in the B3 field, also assuming CBR≥0.4. The results shown in the top and middle panels were determined using artificial clusters with R50 = 0.25″−1.00″.

In the text

Fig. 9. Selection effects on CMFs due to detection incompleteness. The initial CMF is shown as a black line (Schechter function with power-law index β=−2 and truncation mass MC = 104 M⊙). Top panel: CMFs for artificial clusters of various ages (see legend) in the B3 field, showing results for CBR≥1.0 (dashed lines) and CBR≥0.4 (solid lines). Bottom panel: CMFs for clusters of log(t/yr) = 9 age, measured in the B1–B5 background fields at different radial distances from the M 31 centre, assuming the detection threshold CBR≥0.4. Results are based on artificial clusters with half-light radii of R50 = 0.25″−1.00″.

In the text