Luminosity functions of globular clusters in five nearby spiral galaxies using HST/ACS images

Before running SExtratctor it was necessary to astrometrize all HLA images. We performed this with the help of GAIA stars using Gaia Data Release 2 (Gaia DR2; Gaia Collaboration et al. 2016; Gaia Collaboration et al. 2018). We used the IRAF task ccmap with a second order polynomial in coordinates to achieve this. Minimum of 10 stars were used in each pointing (5 stars in one pointing of NGC 4258), resulting in mean rms astrometric accuracies of $\sim$0.095 arcsec (M101), $\sim$0.014 arcsec (NGC 4258), $\sim$0.048 arcsec (M51), and $\sim$0.042 arcsec (NGC 628). We took into account these rms errors to identify and eliminate duplicate sources in the overlap zones between two adjacent frames. In Figure 2, we zoom in on an image section in M101 to illustrate a typical field before and after astrometric correction. The left image shows the HST coordinate system for this field, whereas the image on the right shows the corrected coordinate system. The circles show the coordinates of GAIA (Gaia DR2) stars in this field of view.

In this work, we aim to obtain a sample of GCs with properties similar to those in the Milky Way, which are marginally resolved on the HST/ACS images at the distances of sample galaxies. We considered all objects with fwhm$>$2.4 pixel as GC candidates. This cut-off corresponds to 2.1 pc and 5.7 pc, in the nearest (M81) and farthest (NGC 628) galaxies of the sample. Thus, farther a galaxy is, lesser would be the number of compact objects we would detect. GCs do not show a mass-radius (or equivalently luminosity-radius) relationship (Gieles et al., 2010) and hence this bias is not expected to affect the LF of GCs. GCs are roundish objects, and hence their ellipticity parameter is expected to be close to zero. SExtractor-measured ellipticity even for roundish objects could be as high as 0.3, as it is measured at the isophote corresponding to the detection threshold on the background subtracted image. Another parameter that SExtractor calculates is area, with is the number of pixels enclosed by the isophote where ellipticity is measured. Both the ellipticity and area depend on the threshold used for the detection, and hence a cluster of same magnitude and fwhm can have different values of ellipticity and area, depending on the underlying background.

We have carried out Monte Carlo simulations to understand the behaviour of these parameters for clusters of different fwhm and magnitudes, appropriate to the background and crowding encountered in each galaxy in the $F814W$ band. In Section 4, we describe in detail these simulations. In Figure 3 (right), we show the simulated clusters in area vs fwhm diagram for each of our sample galaxies. Simulations were carried out for a range of magnitudes between 19 and 24 magnitudes, for fixed values of fwhm. As expected, brighter clusters have larger areas at a given fwhm, and the area increases quadratically with fwhm for a cluster of given magnitude. We show the dependence of area with fwhm for objects of 23 mag by the blue solid curve, which is defined by the same equation, ${\text{AREA}=52.5-1.0\ \text{FWHM}+0.50\ \text{FWHM}^{2}}$, for all our sample galaxies. On the left panel, we show all the detected sources in fwhm vs area diagram for our five sample galaxies. The parabola defined by the simulations is shown, which separates the bonafide cluster candidates (that are above the parabola) from contaminating sources (image blemishes, stellar asterisms, image borders etc.), which dominate the number of detected sources at every fwhm. A hard cut in magnitude or area can also eliminate the contaminating objects, but the use of parabola is the most effective way to eliminate these contaminating sources, without missing many genuine clusters. GCs at the distance of sample galaxies are expected to have fwhm close to the observational lower limit of 2.4 pixels. In the top panel of Figure 4, we show the observed distribution of fwhm for our final GC sample for one of our sample galaxies (M101), which indeed peaks at the first bin. However, we include objects up to fwhm=$10$ pixels to account for possible errors in the measurement of fwhm at detection limits. Simulations suggest that even the brightest clusters ($F814W=19$) do not occupy an area$>$500 pixels as long as the fwhm$\sim$5 pixels, and hence we eliminated sources with area$>$500 pixel.

The HLA catalogues include a parameter known as concentration index (CI), defined as the difference between magnitudes measured in 1 and 3 pixel radius apertures. Some studies have made use of this parameter to discriminate between point and extended sources (e.g., Whitmore et al. 2014, Simanton et al. 2015) CI $<1.15$ for stars. In our study, we have used the fwhm $=2.4$ pixel (0.12 arcsec), a direct discriminator between unresolved (stars) and resolved (extended) sources. In the bottom panel of Figure 4, we show CI against the fwhm for all our clusters for M101. As expected, the two parameters are correlated. The minimum CI (1.07) for our sample clusters is slightly less than 1.15, the value adopted in other studies. Forty-five ($\sim$4%) of our sources wouldn’t have been classified as stellar cluster if we had adopted the CI criterion. A visual inspection of these borderline objects suggests they are likely to be clusters, rather than stars. We identify these borderline cases (1.07$<$CI$<$1.15) by red dots surrounded by black circles in Figure 8. In NGC 4258, these include two of the brightest GCs.

We define GCs as old (age$>$10 Gyr) metal-poor ($Z\lesssim 0.001$) clusters, having properties similar to that for the sample of Galactic GCs. However, unlike the Galactic GCs, extragalactic GCs cannot be selected by their visual appearance even on the HST images. Our cluster sample contains GCs as well as relatively young clusters, such as SSCs. In Table 5, we list the source detection statistics in each galaxy. The second column contains all SExtractor-defined sources, with the column 3 containing the number of cluster sources. Cluster samples in spiral galaxies contain more SSCs than GCs, and hence quantitative criteria are required to discriminate between GCs and SSCs. Cluster colour is the most useful discriminator for achieving this. For example, metal-poor SSPs (Z$\leq$0.001) predict $B-I\geq$1.5 mag for populations older than $\sim$3 Gyr (Bruzual & Charlot, 2003).

In Figure 7, we plot the distribution of $(F435W-F814W)_{0}$ colours for all clusters (black histogram) and for clusters brighter than $M_{F814W}=-$9 mag (red histogram) in each galaxy. The latter is multiplied by a factor which is indicated in the figure annotation. For the bright cluster sample, four of the five galaxies studied here show a bimodality in the distribution, the exception being M101. The colour that separates the two distributions is nearly the same in all these galaxies, and corresponds to the $(F435W-F814W)_{0}$=1.5 mag bin. The full cluster sample also shows a minimum in the distribution close to this colour in four of the galaxies. In M101, although the evidence for bimodality is not strong for the bright cluster sample, the total sample indicates bimodality with the saddle point again corresponding to the $(F435W-F814W)_{0}$=1.5 mag. The blue and red distributions correspond to the SSCs, and GCs, respectively. Based on this, we use $(F435W-F814W)_{0}$=1.5 mag to separate GCs and SSCs. Column 4 of Table 5 lists the number of clusters classified as GCs using this colour criterion. Coordinates and photometry for each of the selected GCs are given in Table 6. The Table in the body of the text contains data only for three brightest objects in each galaxy. The entire Table is available in electronic version.

The red tail of the reddened SSC colours most likely extends beyond the colour cut of $(F435W-F814W)_{0}$=1.5 mag, and hence colour-selected GC sample in spiral galaxies is expected to have some contamination from the reddened SSCs. The SSC population has a long tail on the red side, which is either due to a spread in reddening, or/and due to a spread in ages of SSCs. In either case, the red tail of SSC distribution has very few clusters beyond $(F435W-F814W)_{0}>$1.5 mag. The estimation of contaminating objects in the GCs samples is discussed in Section 3.5.

In Figure 8, we plot all candidate clusters in a colour-magnitude diagram (CMD) formed using $M_{F814W_{0}}$ vs $(F435W-F814W)_{0}$, where SSCs and GCs are shown by blue and red points, respectively. The evolutionary loci of clusters for Simple Stellar Population (SSP) models from Bruzual & Charlot (2003) at typical metallicities of SSCs (Z=0.008$\sim$1/3 solar) and GCs (Z=0.001) are shown. These models correspond to synthetic clusters of mass $=1\times 10^{6}$ $\mathcal{M}_{\odot}$ obeying the Kroupa initial mass function (IMF) between masses 0.1 and 100 $\mathcal{M}_{\odot}$. If the GCs are as old as 12 Gyr, the range of magnitudes covered by the GCs corresponds to mass range shown by the dashed vertical line, and the range of colours corresponds to reddening equivalent to $A_{V}$=0 to $\sim$2 mag.

Figure 8 suggests that the reddened SSC colours overlap with the GC colours. The colour histogram (Figure 7) also suggests that the distribution of the SSC colours (the bluer peak) most likely has a long tail on the redder side, that overlaps with the GC colours. Thus, contamination to some degree from reddened SSCs is unavoidable when selecting GC samples in spiral galaxies using colour-cuts. We use the sub-sample of clusters with $U$-band photometry to estimate the fraction of contaminants in each galaxy.

Use of colour-colour diagrams involving ultraviolet and optical filters is known to break the age-reddening degeneracy (e.g., Georgiev et al. 2006; Bastian et al. 2011; Fedotov et al. 2011). In particular, $U-B$ colour separates clearly clusters younger (SSCs) and older (GCs) than $\sim$3 Gyr. Keeping this in mind, we searched the HST archives for images in the WFC3/$F336W$ filter. All our sample galaxies have at least one pointing in this filter (see Table 3). For M101, the $F336W$ images had good astrometry. For the fields in other galaxies, we carried out the astrometry following the same procedure as described in section 3.1.

WFC3/$F336W$ image for M81 is available for only one pointing, as compared to the 29 pointings with the ACS, with only 7 GC candidates (4%) falling in the FoV of the $F336W$ image. The contamination fraction obtained from such a small coverage of the FoV is not expected to be representative. On the other hand, Sloan Digital Sky Survey⁶⁶6https://dr12.sdss.org/fields (SDSS) image of this galaxy obtained from multiple pointings covers the FoV of all the 29 HST/ACS pointings. The relative nearness of this galaxy allows the detection and photometric analysis of 65% of clusters that occupy the relatively uncrowded fields. We hence used the SDSS $u-g$ colours for estimating the contamination fraction in M81. In column 5 of Table 5, we present the fraction of GC candidates with $U$-band images, i.e. SDSS-u for M81 and HST/WFC3 $F336W$ for the rest. We performed photometry using the phot task in iraf using the same photometric parameters as for the HST/ACS images.

In Figure 9, we plot all candidate clusters with $F336W$ coverage in $(F336W-F435W)_{0}$ vs $(F435W-F814W)_{0}$ colour-colour diagram. For M81, we show the SDSS $u-g$ colours (in the AB system) in the ordinate. The evolutionary loci of clusters in this diagram for theoretical SSPs from Bruzual & Charlot (2003) of different metallicities are shown. The $U-B$ colours of reddened young ($<10$ Myr) clusters are distinctly different from that of clusters older than $\sim$3 Gyr, which allows us to break the age-reddening degeneracy. Thus reddened young SSCs (contaminants) would lie below the SSP locus for age$>$3 Gyr. In other words, for a redder ($F435W-F814W_{0}>1.5$) cluster to be considered a genuine GC, its $U-B$ colour, after taking into account photometric errors, should correspond to a location above the SSP locus in the figure. As illustrated in Figure 6, the real errors in photometry are larger than the formal error bars, which limits the use of the colours for a precise determination of age. Nevertheless, the photometric quality is good enough to separate reddened SSCs from GCs. In the last column of Table 5, we give the fraction of contaminants in our GC samples. The values lie between 14–35% in the sample galaxies.

In Figure 10, we explore whether contaminant fraction in our GC samples depends on the colour and magnitude of the GCs. We carry out this analysis in bins of 0.5 mag in colours and magnitudes. In order to avoid fluctuations caused due to small number statistics in some of the bins, we plot only those values for which the number of contaminants in any bin was more than the Poisson error in that bin, taken as the square root of the total number of GC candidates in that bin. For the rest of the bins, we plot the global values. In general, all galaxies have higher contaminant fraction for $(F435W-F814W)_{0}>$2.2 mag, with as much as 50% of the GCs of $(F435W-F814W)_{0}=$2.5 mag being contaminants (reddened young SSCs) in M101 and M81. None of the galaxies show any significant dependence of the contaminating fraction with magnitude. We hence use the global values of the contaminant fraction to correct the LF obtained from our entire GC sample in Section 5 for all galaxies.

In four of our five sample galaxies, there exists a previous catalogue of GCs. We reiterate that none of these catalogues are as complete as our catalogues in terms of spatial and magnitude coverages, as well as in the estimation of contamination fraction from reddened SSCs. We here compare our catalogues, obtained using uniform selection criteria, with the catalogues from other authors, using different selection criteria.

M81: Nantais et al. (2010b) reported a sample of 233 GC candidates in M81, that had made use of data from HST/ACS and SDSS. They used a FWHM$>$3 ACS pixel (0.15$\,arcsec$) as the main discriminator. We find that 107 of our 154 GCs are present in the catalogue of Nantais et al. (2010b). Most of the remaining 49 GCs in our sample have 2.4$<$FWHM$<3$ pixel, which is the main reason for their exclusion in Nantais et al. (2010b). Lower number of GCs in our sample as compared to that of Nantais et al. (2010b) is due to the more stringent filtering that we imposed in the selection of GCs.

M101: Similarly, using data from the HST/ACS, Simanton et al. (2015) reported a sample of 326 star clusters in M101. The selection was made using the concentration index and a colour cut. Their sample includes extended objects, and hence they used a magnitude cut of M${}_{V}\leq-6.5$ to define GCs, resulting in a sample of 98 GC candidates. We find that 48 are present in their sample of GC candidates. Reasons for us missing 50 of their objects are that 25 of these have ELLIPTICITY$>$0.3, 19 do not meet the AREA criterion and the rest do not meet our FWHM criterion.

NGC 4258: González-Lópezlira et al. (2017) reported a sample of 39 GCs in NGC 4258 using near infrared (NIR) data from CFHT (Canada France Hawaii Telescope) over a large FoV ($1^{\circ}\times 1^{\circ}$), but based on seeing-limited ($\sim$0.7 arcsec) dataset. The selection of the GCs was carried out using the colour-colour diagram $u^{*}i^{\prime}K_{s}$ ($(i^{\prime}-K_{s})$ vs $(u^{*}-i^{\prime})$) (Muñoz et al., 2014). In comparison, our catalogue using HST/ACS images contains 226 GCs over a FoV of $\sim 12.3^{\prime}\times 17.2^{\prime}$. From a spectroscopic follow-up of the objects, González-Lópezlira et al. (2019) concluded that the selection based on colour-colour diagram has between 10 to 30% contaminants (e.g. Muñoz et al., 2014; González-Lópezlira et al., 2019). Our FoV includes 29 objects of González-Lópezlira et al. (2017), of which 23 are in our catalogue. Out of the 6 missing objects in our sample, two are confirmed to be GCs using spectroscopic data by González-Lópezlira et al. (2019), another two are dwarf galaxies, with the remaining one being a foreground star. Reason for the two of their GCs to be absent in our sample is that they do not satisfy our selection criteria.

M 51: Star clusters in M51 were catalogued by Hwang & Lee (2008) using HST/ACS images. They used the SExtractor parameters stellarity, FWHM and ellipticity for star cluster selection. They created two subsamples, one with 2.4$<$FWHM$<$20 and the other with 2.4$<$FWHM$<$40 pixel, resulting in a catalogue of 8400 stellar cluster candidates. If we applied their colour discriminator ($(B-V)>$0.5 and $(V-I)>$0.8) in their catalogue we obtained a sub-catalogue of 224 red clusters, 214 of which have FWHM$<$10 ACS ACS pixels. Only 85 of these objects are in our catalogue of 223 GCs. We searched our master SExtractor output catalogue to find out the reasons for we missing 139 of their objects, and found that 85 of these have ELLIPTICITY$>$0.3, 15 do not meet the AREA criterion and the rest do not meet our FWHM criterion.

NGC 628: The last galaxy of our sample, NGC 628, has also been a target for cluster searches using HST images by Ryon et al. (2017). However, they reported only young and intermediate-age clusters, and no GCs.

We generated mock clusters using the IRAF/DAOPHOT tasks $addstars$ and $mkobjects$. A cluster is defined by an intensity profile that follows a Gaussian function of a given fwhm and a total magnitude, with fwhm taking values of 2.0, 2.4, 3.0, 4.0, 5.0, 6.0, 7.0 and 8.0 pixels, and magnitudes varying between 19 and 24 magnitudes at interval of 0.5 mag. For a fixed fwhm, 1100 clusters were generated, 100 for each simulated magnitude.

Coordinates of these sources were randomly selected and were inserted onto an observed HST image. The faintest object that can be detected in an observation under uniform background conditions defines the detection limit of that observation. In real observations, the background is often non-uniform, and each non-point source has a different size. Further, crowding of objects in an image can lead to non-detection of objects that are brighter than the detection limit. Hence, for each galaxy, we chose HST images corresponding to two FoV, representing zones of (i) low background and low crowding, and (ii) high background and high crowding. The former simulates the conditions typical of clusters in the external parts of galaxies, whereas the latter represents the characteristics of clusters in the bulge and spiral arms. For M81, which has as many as 29 pointings, simulations were carried out on an additional frame containing the bulge and nucleus (R09; see Figure 1 in Santiago-Cortés et al., 2010, for the footprint of M81). The results of these latter simulations were used exclusively in the analysis of radial density distribution of clusters in the inner part of this galaxy. The same object detection and selection criteria (§3.2) we had used for real objects were applied on the mock-object added frames. In Figure 11, we zoom-in on a randomly selected region of M101, before (left) and after (right) inserting mock sources. In the image the positions of the mock sources, which cover a range of magnitudes, are indicated by blue circles.

We have also used these simulations to establish criteria to select cluster candidates from all sources catalogued by the SExtractor, and apply fwhm-dependent aperture corrections, described in Section 3.3. We used the results of the simulations for low background and low crowding for this purpose.

For a proper counting of a population, it is necessary to know completeness factor as a function of object magnitude. We used the results of our simulation to determine the completeness factors. This factor is expected to depend on the fwhm.

In the top panel of Figure 4, we plot the distribution of fwhm, which peaks at fwhm$<$3 pixel. Hence, we calculate the completeness factors for mock clusters of input fwhm=3 pixel. The completeness factor is defined as the ratio of the number of recovered objects ($N_{\rm out}$) to the inserted objects ($N_{\rm in}$) at each inserted magnitude.

In Figure 12, we show completeness curves in the $F814W$-band for all of our sample galaxies, for low (circles) and high (triangles) background frames. In order to quantitatively obtain the magnitude at which the sample is 50% complete, $m_{\textrm{50}}$, we fitted the points with the Pritchet function (e.g., McLaughlin et al., 1994; Alamo-Martínez et al., 2013; González-Lópezlira et al., 2017) given by:

where $\alpha$ is a fitting constant that determines curve’s slope. These fitted curves are shown by dotted and dashed lines, for low and high background frames, respectively. The GCs are distributed homogeneously across galaxies, including regions with a high and low crowding. To make the correction for incompleteness of the luminosity function, we decided to use the values of the mean curve, which is shown by the solid curve. In Table 7, the values for the 50% completeness for three curves are given for the $F814W$-band.

While the images obtained from HST observations offer the best spatial resolution, they do not cover the whole extent of galaxies even with the multiple pointings. Thus, outer halo GCs are often missing in catalogues selected from HST images. The following analysis was carried out to correct for the absence of GCs beyond the galaxy’s disk.

The number density obtained wide-field ground-based surveys underestimates the number of GCs inside $R_{\rm e}$ by more than an order of magnitude in both the galaxies with such data. In NGC4258, the slope of our fit in the external part agrees with the observed number densities from wide-field survey, with the absolute values marginally below our predictions. On the other hand, number densities obtained from wide-field survey in M81 remain constant over a large radial range, which are likely due to some selection biases in the survey. On the other hand, the wide-field survey suggests an excess of GCs in the external parts of this galaxy as compared to our predicted values. This excess could be due to contamination from foreground stars and unbound intergalactic GCs in the M81 group (see e.g. Jang et al., 2012; Ma et al., 2017).

Two of the most well-studied spiral galaxies, the Milky Way and M31, have $n=1.9$ and $R_{\textrm{e}}=4.41$ kpc and $n=1.6$ and $R_{\textrm{e}}=4.59$ kpc (Battistini et al., 1993), respectively. In comparison, for elliptical galaxies the calculated values are larger, e.g., NGC 720 (E5): $n=4.16\pm 1.21$, $R_{\textrm{e}}=13.7\pm 2.2$ kpc; NGC 1023 (S0): $n=3.15\pm 2.85$, $R_{\textrm{e}}=3.30\pm 0.9$ kpc; NGC 2768 (E): $n=3.09\pm 0.68$, $R_{\textrm{e}}=10.6\pm 1.8$ kpc (Kartha et al., 2014). The $n$ and $R_{\textrm{e}}$ for our sample galaxies are in the range of expected values in spiral galaxies. The $n$ values are $<1$ in four of our galaxies, suggesting flatter distribution in the inner parts of our galaxies, as compared to the Milky Way and M31.

For comparing colours of our GCs with those in the MW, we transformed all our photometry into the Johnson-Cousins photometric system using the transformation equation 7 and the corresponding coefficients in Table 13 in Appendix B, which were taken from Sirianni et al. (2005).

In Figure 14, we show the distributions of $(B-I)_{0}$, $(B-V)_{0}$ and $(V-I)_{0}$ colours of GC systems in our sample galaxies, where the sub-index 0 stands for Galactic reddening-corrected colours, using the $A_{V}$ values in Table 1. For comparison, we also plot the colour histograms of the GCs in the Milky Way using the data from Harris (1996). The black and blue histograms correspond to the observed and dereddened colours, respectively. In Table 9, we list the mean and mode of the distributions of the three colours $(B-I)_{0}$, $(B-V)_{0}$ and $(V-I)_{0}$ for our sample as well as for the MW sample from Harris (1996). We use only 97 Galactic GCs which had integrated colours available in Harris (1996). For comparison, we list the colour of a SSP of 13 Gyr with Z=0.001, the typical metallicity of old GCs, in the last row. The reddening-corrected colours of MW GCs agree within 0.1 mag with these SSP colours.

The mode is systematically bluer than the mean in all the three colours in our sample galaxies as well as in the MW. This difference is due to a noticeable skewness in the colour distributions, with the peak (mode) occurring on the bluest bin in $(B-I)_{0}$ and $(V-I)_{0}$, in all cases except in M81. The skewness is not due to a colour cut we have used to define GCs. In fact the choice of the cut-off colour between SSCs and GCs corresponds to the saddle point in the distribution of colours of all cluster candidates (see Figure 7), with the peak in the GC colours occurring in the first or second bin redward of the saddle point. We analyse the possible role of contaminants in our GC samples to the origin of skewness in colour distributions. For this, we use the fraction of contaminants as a function of colour (see left panel in Figure 10) to correct statistically the $(B-I)_{0}$ colour histograms, which is shown in Figure 14 in magenta. The contamination-corrected histograms maintain the blue skewness in the distribution. This suggests that the skewness in colour distribution is an intrinsic property of the GC systems.

The mean and mode colours of our GC systems should correspond to SSP colours of metal-poor old clusters, if reddening is negligible. We find that the mode is only marginally redder than the colours of the metal-poor SSPs at old ages, whereas the mean is much redder. The redward shift of the mode from the expected SSP colours and the long red tail of the colour distribution, even for the MW sample, illustrate that the GC colours are often reddened by dust. The typical reddening experienced by GCs in each of our sample galaxies can be determined by comparing the mode of the colour distribution with the colour of the metal-poor old SSPs. We made use of the $(B-I)_{0}$ colour to obtain the colour excess $A_{V}({\rm int})=((B-I)_{0}-(B-I)_{\rm SSP})/(E_{B}-E_{I})$, where $(B-I)_{\rm SSP}$ is the $(F435W-F814W)_{0}$ colour of the SSP, $E_{B}=1.33$, and $E_{I}=0.60$ are the values of the the Cardelli et al. (1989) extinction curve at the effective wavelengths of $B$ and $I$-bands. The resulting values of $A_{V}$(int) using mean and mode are given respectively in columns 8 and 9 of Table 9. The $A_{V}$(int) obtained from the mode for our galaxies range from 0.01 (NGC 628) to 0.32 mag (M81) which is comparable to the 0.17 mag for the MW. The $A_{V}$(int) values obtained from the mean colours vary between 0.60 (NGC4 258) to 1.23 (M101), which is also comparable to the 1.04 mag for the MW.

The difference between the mean and median $A_{V}$ values is $\sim$0.5–1.0 in our sample galaxies, which is comparable to that for the Milky Way (0.87). Thus, GC systems in spiral galaxies experience relatively large mean extinctions. In fact, this is one of the reasons for the slow progress in establishing the properties of the GC systems in spiral galaxies. Some of the reddening could be due to dust internal to the clusters, whereas the majority is expected due to the dust along the line of sight in the disk of the host galaxy. The colour histogram in M101 shows a second peak at $(B-I)_{0}\sim$3.5 mag after correction for contaminants, suggesting the presence of a distinct red population. The analysis of LF below also suggests the presence of a population fainter than GCs. As we will see in Section 5.4, this population of faint red clusters corresponds to reddened intermediate-age (1–10 Gyr) SSCs. Under such a scenario, the basic assumption we have made in determining $A_{V}$, namely the clusters are old ($>10$ Gyr) is not applicable to all clusters in M101 and hence the $A_{V}$(mean) would be an upper limit for this galaxy.

We chose the Galactic extinction-corrected magnitudes in the $F814W$ band to construct the GCLF. For this purpose, we constructed histograms in bins of 0.5 mag over the entire range of detected magnitudes for each galaxy. The resulting histograms are shown in Figure 15 with red solid lines. The magnitudes corresponding to 50% completeness, $m_{50}$, at low and high surface brightness parts of for each galaxy are shown by dotted and dashed vertical lines, respectively. The observed numbers are corrected for incompleteness using the mean function shown in Figure 12. The corrected histograms are shown by gray bars. As expected, the correction factor is negligible at the bright end ($F814W\lesssim 21$ mag), and gradually increases at fainter magnitudes.

The GCLFs in the five galaxies show very similar form, increasing smoothly up to reaching a peak value, and then again decreasing smoothly. In other words, all our sample galaxies show a turnover, the TO. The TO values lie on the brighter side of $m_{50}$ in all galaxies. The incompleteness-corrected histograms are fitted with a log-normal function given by:

where $N_{\textrm{0}}$ is a normalization factor, $M$ is the absolute magnitude of the fitted bin, $M_{\textrm{0}}$ is the absolute magnitude of TO and $\sigma_{M}$ is the dispersion. It can be seen that the log-normal functions are good fits to the observed LFs.

We obtained the TO magnitude after correcting the LF in each galaxy for a possible contribution from the contaminants. Following the discussions in Section 3.5, we used the magnitude-dependent contaminant fractions for NGC 4258 and constant contaminant fractions from the last column in Table 5 for the rest. The resulting plots for each galaxy are shown in the right panels of Figure 15 and the results are tabulated in Table 10. In order to determine the possible errors on the determined TO, we carried out 1000 Monte Carlo simulations, by adding a Gaussian noise to each bin of the histograms. For this purpose, we used the square root of the number in each bin as the $\sigma$ of the Gaussian. The histogram resulting from each experiment is fitted by Equation 3, each time finding TO and $\sigma_{\rm M}$. The rms values of these 1000 simulations are taken as the error on TO and $\sigma_{\rm M}$.

In order to investigate the universality of the function, we converted the apparent TO magnitudes to the absolute values using the best-available distances for the sample galaxies (see Table 1), and corrected these magnitudes for internal extinction using $A_{F814W}$(int)=0.58$A_{V}$(int), where $A_{V}$(int) is taken from the last column of Table 9. The resulting values are also given in Table 10. All galaxies with the exception of M101 are consistent with ${M_{I}}_{0}=-8.21\pm 0.06$ mag, with a mean ${M_{V}}_{0}=-7.41\pm 0.14$ mag. The TO of M101 is 1.16 mag fainter. We will discuss this case separately in next section.

How does the mean value for the galaxies compare with the values for elliptical galaxies and the MW? TO value for the MW is $M_{\rm V}=-7.40\pm 0.10$ mag (from Jordán et al. 2007), which is almost identical to the mean value of ${M_{V}}_{0}=-7.41\pm 0.14$ mag obtained for four of our galaxies. The individual differences of the TO magnitudes for our sample galaxies from that of the MW are given in column 8 of the Table 10. The differences are well within 1$\sigma$ value in all these four galaxies. Hence, the GCLFs in four of our sample galaxies have the same TO as that in the MW. In comparison, M31 value of ${M_{V}}_{0}=-7.60\pm 0.15$ obtained by Secker (1992) and Reed et al. (1994) is $\sim$0.2 mag brighter than our mean value, which is marginally larger than the 1-$\sigma$ error of our measurements. As pointed out earlier, TO for M101 is clearly different, which could be due to underestimation of distance, or an intrinsically different value of TO. This will be analysed in detail in the next sub-section.

Despite the near universality of the TO values, the $\sigma_{M}$ of the log-normal function varies widely in the sample galaxies. These differences in widths are illustrated in Figure 16, where we plot all the log-normal fits normalized to their peak values. The TO value along with its error for the Milky Way GCs is shown as a vertical band. In this figure we also show the positions of the brightest GC in each of these galaxies, as well as the position of the third brightest GC. It can be easily seen that the dispersion in the brightest and the third brightest clusters in these galaxies is much higher than the dispersion in the TO values, suggesting that the TO values are more universal than the magnitude of the brightest, or the third brightest GC, in galaxies. The brightest GC in our sample of five galaxies belongs to M81, which is discussed in detail in Mayya et al. (2013).

For obtaining the TO, we have used the distances tabulated in Table 1. The tabulated distances correspond to the most recent determination of distances using Cepheids for M81 and M101, and the MASER distance for NGC 4258. For M51 and NGC 628, SNII distances were used.

Using only the distances reported after the year 2000 in NED⁷⁷7https://ned.ipac.caltech.edu/, we found 56 independent measurements for M81⁸⁸8https://ned.ipac.caltech.edu/byname?objname=m81&hconst=67.8&omegam=0.308&omegav=0.692&wmap=4&corr_z=1, 79 for M101⁹⁹9https://ned.ipac.caltech.edu/byname?objname=m101&hconst=67.8&omegam=0.308&omegav=0.692&wmap=4&corr_z=1, 77 for NGC 4258¹⁰¹⁰10https://ned.ipac.caltech.edu/byname?objname=ngc4258&hconst=67.8&omegam=0.308&omegav=0.692&wmap=4&corr_z=1, 27 for M51¹¹¹¹11https://ned.ipac.caltech.edu/byname?objname=m51&hconst=67.8&omegam=0.308&omegav=0.692&wmap=4&corr_z=1 and 23 for NGC 628¹²¹²12https://ned.ipac.caltech.edu/byname?objname=ngc628&hconst=67.8&omegam=0.308&omegav=0.692&wmap=4&corr_z=1. The methods of distance measurements include those using Cepheids, MASER, tip of the red giant branch (TRGB), SNII-optical, and Tully-Fisher. We recalculated TO values for each of these measured distances, which are shown plotted against the corresponding distances in Figure 17. The value of TO for the best-determined distance (that in Table 1) is indicated by a thick circle, which is colour-coded according to the method that was used to obtain the distance. The error on this best measurement is highlighted by the gray band. Most of the distance measurements for the 4 galaxies (M81, NGC4258, M51 and NGC628) agree with the value obtained by our best distance estimate, after taking into consideration the errors on each of them. In M101 where we found TO value fainter by 1.16 mag, some of the Cepheid distances are not consistent with the Cepheid distance (6.95 Mpc) we have used. If we use the farthest distance reported for this galaxy (8.99 Mpc, Macri et al., 2001), the TO value is still 8$\sigma$ fainter than the universal value we have in other galaxies. Hence, we conclude that the TO value in M101 is intrinsically fainter compared to that in other galaxies of our sample, and investigate the possible physical reasons for this difference.

Among the galaxies analysed in this study, M101 stood out from the rest for having a fainter derived TO value, as well as for having a large population of very red clusters ($(F435W-F814W)_{0}>$2.2 mag), including a second peak at $(F435W-F814W)_{0}>$3.5 mag in its colour distribution (Figure 14). We here discuss whether these differences are related to its late morphological type (Scd). In Figure 18, we plot the TO values with respect to that in the MW as a function of the morphological type of the host galaxies. In this plot, we include the MW, M31 and the Large Magellanic Cloud (LMC). We estimated the TO in the LMC using data from Mackey & Gilmore (2003) for clusters with age $\geq$1 Gyr. TO values in all galaxies with morphological type Sc or earlier are in agreement with the value in the MW within the errors of measurements. The two galaxies later than Sc deviate from this trend. Thus, there is an indication that the different TO value found in M101 is indeed related to its late morphological type. In the following paragraphs, we discuss a possible scenario that naturally explains the fainter TO in late-type galaxies.

Classical GCs are among the oldest objects in galaxies, and their formation is related to the formation of halo and bulge (Brodie & Strader, 2006). By definition, the prominence of these spherical components in galaxies decreases along the Hubble sequence. For this very reason, most of the searches of GCs were traditionally carried out in elliptical and early-type galaxies. Such studies have found relatively larger population of classical GCs in ellipticals as compared to that in spiral galaxies (e.g., Harris & Harris 2011; Harris et al. 2013). On the other hand, M101 has a weak bulge, classified as a pseudobulge by Kormendy et al. (2010), and a very poor stellar halo population (Jang et al., 2020), as expected for its late morphological type. These late-type systems are not expected to contain a large population of classical GCs. However, searches of GCs on HST images, including this study, find as many GCs in late-type galaxies as in early-type spirals. It is likely that not all the inferred GCs in late-type galaxies are classical GCs. We analyse this issue below.

Spiral and irregular galaxies are known to contain old SSCs that were formed in their disks and have survived the tidal shocks of their host galaxies. For example, González-Lópezlira et al. (2019) found that some of the objects classified as GCs in NGC 4258, one of the galaxies analysed in our study, belong to its disk and share the same kinematics as the disk HI gas. Formation and growth of galaxies through hierarchical merging, expects formation of such clusters during every epoch of merger involving gas-rich galaxies (Mamikonyan et al., 2017). The selection filters used for GC searches on HST and ground-based images do not distinguish these old stellar clusters from the classical GCs. Thus generally speaking the sample of objects observationally classified as GCs contains two kinds of objects, classical GCs related to the spherical component of galaxies, and old disk star clusters, formed in-situ or accreted onto, during galaxy mergers as part of the hierarchical growth of galaxies. It is the former one that is expected to have universal TO. The fraction of old star clusters is expected to increase towards later Hubble types as they are more gas-rich than the early types. Hence, the TO obtained in early-type spirals represents that of the classical GCs, whereas in late-type galaxies it represents the population of old SSCs in the disk. This is found to be true in the LMC, where 50% of the old cluster population that is used to obtain the TO has ages in the 1–10 Gyr range.

In Section 3.5, we have estimated the contamination fraction from reddened SSCs in our sample of GCs in each galaxy, using the photometry in the $U$-band. The technique we have used helps to distinguish reddened SSCs from GCs only for SSCs younger than $\sim$3 Gyr. Thus, our GC samples, in principle, contain all clusters older than $\sim$3 Gyr. The universality of the TO in all our galaxies earlier than Sc, suggests that these early-type spirals did not form significant amount of SSCs after the formation of halo and bulge, or even if they are formed, they did not survive the gravitational shocks.

The GC population in M101 has been the subject of study by Simanton et al. (2015). They found a total of 326 candidate GCs, with their luminosity function matching very well the LF for the Galactic GCs, but starts increasing rapidly for M${}_{V}\geq-$6.5 mag. They identified these relatively fainter clusters as belonging to a second population, which is statistically more extended than the brighter population, the so-called faint fuzzies (e.g., Larsen & Brodie 2000b; Brodie & Larsen 2002; Peng et al. 2006; Liu et al. 2016). They discuss these red extended clusters as old stellar clusters formed in the disk. Only 98 (i.e. 30%) of their clusters belonged to the brighter population, the classical GCs. Our selection criteria, which selects objects 2.4$<$FWHM$<$10 pixel, naturally rejects extended objects that Simanton et al. (2015) have found, but contains compact objects fainter than their magnitude cut of M${}_{V}\geq-$6.5 mag. The fainter TO we have found suggests that our sample is dominated by a population of red faint compact clusters of ages between $\sim$1–10 Gyr. These properties correspond to relatively low-mass reddened SSCs in the disk of M101.

The TO magnitude of a GC system is controlled by four physical parameters, namely, the mass function (MF) at birth, age, metallicity and internal extinction. In addition to these, a dynamical process changes the form of the mass function as the clusters evolve under the gravitational potential of their parent galaxies. At ages greater than a few billion years, dynamical evolution is mass-dependent, which leads to selective destruction of low-mass clusters. In general, a cluster of mass $M_{\rm cl}$ loses all its mass over a time-scale, $t_{\rm dis}$, given by,

where $t_{4}$ is the destruction time for a cluster of $10^{4}M_{\odot}$ under a specific gravitational potential, and the power-law index $\gamma$ is found to have a value of 0.6 (Boutloukos & Lamers, 2003). The $t_{4}$ is expected to be shorter in strong gravitational field, such as massive early-type galaxies, and longer, in low-mass late-type galaxies. Fall & Zhang (2001) found that the net result of the dynamical evolution is to create a MF that is log-normal, like the one found in our study, almost independent of the initial form of the MF. The turnover of the resulting MF is shifted to higher masses, equivalently to brighter magnitudes, in early-type galaxies, as compared to the late-type galaxies. Thus, even if the four physical parameters are the same for all GC systems, dynamical evolution alone can introduce dispersion in the observed TOs in different galaxies. The TO is also shifted to higher masses (brighter magnitudes) at longer ages due to dynamical effects.

Using our TO absolute magnitude in the $F814W$ filter, corrected for internal and Galactic extinction ($M_{F814W_{0}}^{\rm TO}$), we can determine the TO mass of the cluster in our sample of galaxies using an SSP of age and metallicity typical of GCs.

where $M_{F814W}^{\rm SSP}$ is the $F814W$ magnitude of an SSP at 13 Gyr for a total mass of 1 M$\odot$. The resulting mass values for two of the lowest metallicities (Z=0.0004=1/50 solar and Z=0.001=1/20 solar) are given in Table 11. The derived masses are 3% higher for Z=0.0004 as compared to that obtained using Z=0.001. In the four galaxies with almost universal TO, the mass corresponds to $\sim 3\times 10^{5}$ M_⊙, whereas it is lower by a factor two in M101. The disk clusters of M101 are expected to be younger than the uniform age of 13 Gyr we have used. If they are as young as 8 Gyr, the masses will be further lower by $\sim$30%.

GC searches in external galaxies using HST data are subjected to two kinds of incompleteness: (i) magnitude incompleteness arising due to the missing low-luminosity GCs beyond the detection limit, and (ii) area incompleteness due to non-coverage of outer-halo GCs because of the limited FoV of HST. The alternative strategy of searching GCs in wide-field ground-based surveys also suffers from these two effects, with the second effect arising due to confusion with stellar sources in the inner regions due to lack of spatial resolution. Characterization of the LF and the radial number density distributions with analytical functions, allows us to calculate the total number of GCs by integrating over the entire range of these functions. We carry out these integrations in two steps, as explained below.

First, we integrate over the fitted GCLF to obtain the number corrected for the missing low-luminosity GCs, which are given in column 3 of Table 10. We define the missing factor, $f_{\rm LF}$=$N_{\textrm{GC}}$($\int{LF})$/$N_{\textrm{GC}}$(obs). In order to correct for the GCs outside our FoV, we counted the number of GCs within $R_{\rm e}$. By definition, total number of detectable GCs is twice this number. The total number of GCs, corrected for contamination from reddened SSCs, and incompleteness both in luminosity and volume is: