NEW 2MASS NEAR-INFRARED PHOTOMETRY FOR GLOBULAR CLUSTERS IN M31

We selected our sample GCs and candidates from RBC V.5, which contains precise cluster coordinates, classifications, metallicities, reddening values, radial velocities, galactocentric distances, structure parameters, and photometry, including GALEX (FUV and NUV), optical broadband, and 2MASS NIR magnitudes for 2060 objects. However, RBC V.5 provided JHK_s magnitudes in only three bands for ∼1100 objects. In addition, there were no magnitude errors for JHK_s bands in RBC V.5. Therefore, homogeneous photometric data and precise magnitude errors for JHK_s bands are urgently needed. We selected classes 1, 2, 3, and 8 from column "f" in RBC V.5, which include GCs, candidate GCs, controversial objects, and extended clusters. This resulted in an initial selection of 991 objects. We first employed iraf/daofind to find the sources in the images, then we matched them to the coordinates of the objects given by RBC V.5, and checked each object visually in the images to confirm the positions for photometry. During this process, we found that there are 68 clusters for which accurate photometric measurements cannot be obtained for various reasons. Some clusters (B024D, B053D, B061D, B523, B537, BH16, M023, SK068B, and SK110A) are superimposed onto a bright or strongly variable background. Some clusters (B102D, HEC2, and MCEC4) are very faint and the signal-to-noise ratio (S/N) is low. Some clusters (B189D, B246, B274, B538, BH09, BH25, C011, C041, KHM31-77, M050, M065, M075, M078, NB89, SK059B, SK061A, SK063C, SK097B, SK104B, and SK153C) are very close to other objects. Some clusters (B016D, B088D, B530, BH01, BH03, BH04, BH28, DAO93, H20, HEC6, HEC9, HEC13, KHM31-345, M055, M079, NB17, SH06, SK049A, SK222C, SK216B, SK223B, and SK242B) are indiscernible with the coordinates presented by RBC V.5 in the 2MASS images. There are one or two nearby objects within 2 ∼ 4 pixels for five clusters (B016D, B530, M055, SH06, and SK223B); however, after careful comparison with the images from the Local Group Survey (LGS; Massey et al. 2006) and from the Digitized Sky Survey (DSS), these objects cannot be confirmed due to their low S/N values. In addition, we found that some clusters (B196D, B257D, B318, B368, B453, B475, G099, KHM31-85, M056, M068, M088, M089, SK166B, and V031) have one or more nearby bright objects in the images from LGS or from the Wide Field Planetary Camera on the Hubble Space Telescope, indicating that they are "adhered" in 2MASS images because of the poor resolution. Ma (2012b) presented JHK_s photometry, ages, and masses for 10 newly discovered halo GCs in M31, so we will not re-estimate JHK_s magnitudes for these GCs. Finally, here we will derive NIR photometry for the remaining 913 star clusters and cluster candidates. The position of cluster H26 presented by RBC V.5 may be of insufficient accuracy, with R.A. and decl. being 00:59:27.37 and +37:41:34.1 (Huxor et al. 2008). We checked the images from 2MASS and from DSS, and corrected the coordinates to 00:59:27.48 and +37:41:31.37. The updated position was then used for the following photometry. Figure 1 shows the spatial distribution of the sample clusters in M31.

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We used 2MASS archival images of M31 in the JHK_s bands to perform photometry. The images were retrieved using the 2MASS Batch Image Service.³ Considering that the 2MASS-6X images, which were observed with six times the normal exposure of 7.8 s, have a deeper magnitude limit and a higher S/N than the All-Sky Release Survey images, the 2MASS-6X data were preferred. In this paper, photometry for 627 objects was performed using the 2MASS-6X data, while photometry for the remaining 286 was performed using the All-Sky Release Survey data. We used the uncompressed atlas images provided, with a resampled spatial resolution of ∼1'' pixel⁻¹. We performed aperture photometry of the 913 M31 star clusters in all of the JHK_s bands to provide a comprehensive and homogeneous photometric catalog for them. The photometry routine we used is iraf/daophot (Stetson 1987).

To determine the total luminosity of each object, we produced a curve of growth from J-band photometry obtained through apertures with radii in the range of 4–16 pixel with 1 pixel increments. The most appropriate photometric radius from the curve that encloses the total cluster light but excludes lights from extraneous field stars was adopted. In addition, we checked the images of all three bands for each cluster by visual examination to make sure that the photometric aperture was not so large as to be contaminated by other sources. The local sky background was measured in an annulus with an inner radius 1 pixel larger than the photometric radius and 5 pixels wide. The sky fitting algorithm was set to "mode." We abandoned derived magnitudes with large uncertainties (>1 mag) to keep high photometric precision. The instrumental magnitudes were then calibrated using the relevant zero points obtained from the photometric header keywords of each image. Figure 2 shows the aperture distribution as a function of J-band magnitude, with the highest peak at 4''. In general, magnitudes of brighter objects are measured with larger apertures.

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Finally, except for 9 objects in the J band, 31 objects in the H band, and 51 objects in the K_s band, we obtained photometry for 913 star clusters in the individual JHK_s bands. Table 1 lists our new JHK_s magnitudes and the aperture radii used, in conjunction with the magnitude uncertainties given by daophot. Figure 3 shows the uncertainty distribution as a function of magnitude for the JHK_s bands. Filled circles represent photometry performed on the 2MASS-6X images, while red pluses represent photometry performed on the 2MASS All-Sky images. It is clear that the photometry from 2MASS-6X data show smaller magnitude uncertainties, which are due to high S/N with longer exposures. Objects in these dashed boxes are bright sources with slightly large magnitude uncertainties (>0.2 mag). Six clusters (AU008, AU010, B132, NB21, NB39, and NB41) have large magnitude uncertainties in all three bands, two clusters (B070 and B119) have large magnitude uncertainties in the K_s band, two clusters (B104 and NB108) have large magnitude uncertainties in the JH bands, and three clusters (B264, B324, and SK050A) have large magnitude uncertainties in the HK_s bands. All these clusters are close to the galaxy center and suffer from a bright background contamination.

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To examine the quality and reliability of our photometry, we compared the aperture magnitudes of the 913 objects obtained here with previous results, in the sense of values from this paper minus values from the others. Figures 4 and 5 plot the comparison of our obtained 2MASS photometry with the aperture photometry from the 2MASS catalog, with r = 4'' for the Point Source Catalog (PSC) and r = 5'' for the Extended Source Catalog (XSC), following Galleti et al. (2004). The 2MASS PSC was derived by combining the 2MASS All-Sky PSC and the 2MASS-6X Point Source Working Database (6X-PSWDB), with the 6X-PSWDB being preferred. The 2MASS XSC was derived by combining the 2MASS All-Sky XSC and the 2MASS-6X Extended Source Working Database (6X-XSWDB), with the 6X-XSWDB being preferred. Each panel displays the mean difference (Δ), rms scatter (σ), and the number points (N) for comparison. There are nine objects with large offsets (|Δm| > 1 mag) compared with PSC in Table 2. It was surprising that the magnitudes of eight of these clusters were measured with an aperture of r = 4'' here, which is the same as adopted in the PSC. We checked the images for these clusters and found that some clusters (SK049C, SK054C, and SK086C) are located in a strongly variable background, while some clusters (B038D, B104, and SK047B) lie close to the M31 center. In addition, there is one very bright source near SK115B. For objects in the PSC, the sky background is measured in an annulus with an inner radius of 14'' and an outer radius of 20''. To check whether the discrepancy is caused by a different choice of annulus for the background, we used the same annulus with PSC and re-estimated magnitudes for the eight clusters. We found that most of the newly derived magnitudes agree well with those from PSC, except SK115B and SK136C. For SK115B, about a quarter of the area of the annulus (14''–20'') is affected by the nearby bright source, which would lead to a high estimate of the sky background. For SK136C, the J-band magnitude given by PSC is 13.419, much brighter than the H-band magnitude of 16.700, which is suspicious. We think that the annulus set in this paper is more reasonable for these clusters. In addition, for cluster B072D, a larger radius would include lights from outer sources. In this paper, photometry was performed with an aperture radius larger than 4'' for more than 400 clusters, which may lead to brighter magnitudes than those in 2MASS PSC, on average. We concluded that this is the main reason for the offset of the JHK_s magnitudes (−0.029, −0.051, and −0.060). There is no big difference between the magnitudes measured here and those from the XSC.

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Barmby et al. (2000) measured JK magnitudes for a sample of M31 GCs, with most of the new NIR data taken with the STELIRCam on the 1.2 m telescope at Fred Lawrence Whipple Observatory (FLWO), and the observation data for four objects from the 2MASS (Skrutskie et al. 1997) scans of M31. Barmby et al. (2001) obtained NIR photometry for 38 GCs and candidates from JHK' observations using Gemini on the Lick Observatory 3 m telescope and JK observations using the STELIRCam on the FLWO 1.2 m, and updated magnitudes for some clusters studied in Barmby et al. (2000). Figure 6 plots the comparison of our obtained 2MASS photometry with the magnitudes from Barmby et al. (2000, 2001). The average differences for the J and K_s bands are 0.024 and −0.032, which may be caused by different photometry methods, with the rms scatters around the mean being 0.193 and 0.231. There is only one cluster (DAO69) with larger magnitude offsets, as listed in Table 2. A larger radius would include lights from several outer sources, so we think our photometry result is more reasonable for DAO69.

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Figure 7 displays the comparison of our obtained 2MASS photometry with the magnitudes from RBC V.5. The 2MASS photometry in RBC V.5 (Galleti et al. 2004) was derived from the 2MASS All-Sky PSC and XSC, by correcting the systematic shifts between the magnitudes from the 2MASS catalogs and previous NIR photometry. In RBC V.5, the 2MASS JHK_s magnitudes were transformed to the CIT photometric system (Galleti et al. 2004). However, we needed the original 2MASS JHK_s data to compare with our results, so we reversed the transformation using the equations given by Carpenter (2001). The average differences for the three bands are 0.059, 0.050, and 0.032, with rms scatters around the mean of 0.309, 0.264, and 0.327. There is an obvious offset between their estimated J magnitudes and ours, which is mainly caused by the large scatters listed in Table 2. If these clusters with larger offsets (|Δm| > 1 mag) are not included, the systematic offset in the J band can be reduced to 0.035 mag. The clusters with the largest discrepancy are B041, B090, and SK054C. There are several bright sources beyond B041, and a larger radius would include lights from these sources. The K_s magnitude of B090 in RBC V.5 is 17.881, which is from the 2MASS All-Sky PSC. However, in the 2MASS-6X-PSWDB, the magnitude is 15.308. We checked the images of the 2MASS All-Sky Survey, and found that the low S/N is the main reason for the K_s magnitude inaccuracy.

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The four objects with magnitude scatters larger than 2.0 mag are plotted in Figure 8, in which the circles are the photometric apertures adopted here. As discussed above, the inaccuracy of the K_s photometry for B090 is mainly caused by the low S/N. B104 lies close to the M31 center, while SK054C is located in a variable background. A different annulus set for the sky estimation results in a discrepancy of the magnitudes, and we think that the photometry method in this paper is more reasonable for these clusters. For SK136C, the J magnitude in 2MASS PSC (13.419), which is much brighter than the H magnitude (16.700), may be a typing error. In all comparisons, the average absolute difference is ⩽0.06 mag, and the rms scatter around the mean varies between 0.1 and 0.4. No system offset between our magnitudes and previous determinations can be seen.

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Caldwell et al. (2009, 2011) published reddening values for a large sample of clusters in M31 by comparing the observed spectra with model ones. Considering the high accuracy of their spectral results, we preferentially adopted Caldwell et al. (2009, 2011) reddening values. However, the mode value of 0.13 and values from Barmby et al. (2000) in Caldwell et al. (2009, 2011) were not adopted. For star clusters with no reddening value given by Caldwell et al. (2009, 2011), we adopted the results from Barmby et al. (2000), Fan et al. (2008), and Fan et al. (2010). Barmby et al. (2000) and Fan et al. (2008) determined the reddening for M31 clusters using correlations between optical and infrared colors and metallicity by defining various "reddening-free" parameters. Because the reddening values from Fan et al. (2008) comprise a homogeneous data set and the number of GCs included is greater than that of Barmby et al. (2000), we preferentially adopted Fan et al. (2008) reddening values, followed by those of Barmby et al. (2000). Finally, the results from Fan et al. (2010), which derived reddening values from spectral-energy distribution fitting, were adopted.

There are 378 star clusters with no available reddening values in the literature, and we would calculate reddening values for them in a way similar to that described in Kang et al. (2012). For clusters with a deprojected galactocentric distance <22 kpc, the reddening values were measured as the mean reddening values of star clusters located within an annulus at every 2 kpc radius from the center of M31. For clusters beyond a deprojected distance of 22 kpc, most of which are located in halo regions, a reddening value of E(B − V) = 0.13 mag was adopted as suggested by Kang et al. (2012).

Caldwell et al. (2011) provided new homogeneous estimates of metallicity for more than 300 GCs in M31, using high-quality spectra obtained with the Hectospec multifiber spectrograph on the 6.5 m MMT. Kang et al. (2012) derived mean metallicity values for a catalog of clusters from previous studies (Barmby et al. 2000; Perrett et al. 2002; Galleti et al. 2009; Caldwell et al. 2011). Metallicities for 312 and 289 clusters in our sample were given by Caldwell et al. (2011) and Kang et al. (2012), respectively. Table 1 lists the metallicities from Caldwell et al. (2011) and Kang et al. (2012) for the sample clusters.

To obtain the M31 GCLF peaks, we used the least square and maximum likelihood methods to fit a Gaussian and Student's t distribution to the dereddened JHK_s data. The Gaussian distribution is given as

$$\begin{equation} f(x)={\frac{1}{\sqrt{2\pi }\sigma _{G}}}{\rm exp}\left({-}{\frac{(x-\mu _{G})^2}{2\sigma _{G}^2}}\right). \end{equation} \tag{ 1 }$$

The Student's t distribution is defined by

$$\begin{equation} f(x)={\frac{\Gamma ((n+1)/2)}{\sqrt{\pi n}\Gamma (n/2)\sigma _{t}}} \left(1+{\frac{(x-\mu _{t})^2}{n\sigma _{t}^2}}\right)^{-(n+1)/2}, \end{equation} \tag{ 2 }$$

where n is the degree of freedom (DOF). Secker (1992) and Secker & Harris (1993) reported that the Student's t distribution with DOF of 5 (hereafter "t₅"), which presents a power-law falloff in its wings, is more robust than the Gaussian in describing the outlying data points and estimating the GCLF peak. The "t₅" function is evaluated as

$$\begin{equation} f(x)={\frac{8}{3\sqrt{5}\pi \sigma _{t}}} \left(1+{\frac{(x-\mu _{t})^2}{5\sigma _{t}^2}}\right)^{-3}. \end{equation} \tag{ 3 }$$

Some other distributions (Baum et al. 1995; Larsen et al. 2001) were also investigated for the GCLF. Baum et al. (1995) found that a composite of two exponentials is a better fit over the Gaussian or Student's t distribution to the combined LF of the Galactic and M31 GCs, since those two distributions fail to deal with the asymmetry and with the sharpness of the peak of the histogram.

Figure 9 displays the luminosity histograms and the best fitting lines for the sample clusters. The top panels show the fitting with the least square technique, while the bottom panels show the fitting with the maximum likelihood method. The black lines show the fitting with a t₅ distribution, while the red lines show the fitting with a Gaussian distribution. The maximum likelihood method is used to estimate the most likely parameter values from the sample data (Secker 1992), which would not suffer from the effect of binning. As shown in Figure 9, the fitting with a t₅ distribution shows a more extended wing than that with a Gaussian distribution (Secker 1992). Table 3 lists the LF parameters for the sample clusters, the confirmed GCs, and those confirmed GCs with metallicities available from Kang et al. (2012). The LF peaks for confirmed GCs are much brighter than those for all clusters, indicating that a clean sample of GCs is critical to obtain accurate LF parameters.

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Table 3. M31 GCLF Parameters

Bandpass	Least Square				Maximum Likelihood				N
	μG	σG	μt	σt	μG	σG	μt	σt
	(mag)	(mag)	(mag)	(mag)	(mag)	(mag)	(mag)	(mag)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
All sample clusters
J0	$15.873_{-0.198}^{+0.190}$	$1.406_{-0.142}^{+0.176}$	$15.893_{-0.192}^{+0.184}$	$1.355_{-0.138}^{+0.174}$	$15.726_{-0.198}^{+0.190}$	$1.393_{-0.142}^{+0.178}$	$15.726_{-0.190}^{+0.182}$	$1.393_{-0.132}^{+0.170}$	914
H0	$15.224_{-0.186}^{+0.178}$	$1.340_{-0.136}^{+0.170}$	$15.245_{-0.184}^{+0.174}$	$1.290_{-0.132}^{+0.168}$	$15.095_{-0.186}^{+0.178}$	$1.377_{-0.136}^{+0.170}$	$15.095_{-0.182}^{+0.174}$	$1.377_{-0.130}^{+0.166}$	892
Ks0	$14.887_{-0.144}^{+0.140}$	$1.237_{-0.108}^{+0.134}$	$14.903_{-0.146}^{+0.140}$	$1.198_{-0.106}^{+0.134}$	$14.816_{-0.144}^{+0.140}$	$1.324_{-0.108}^{+0.136}$	$14.816_{-0.144}^{+0.140}$	$1.324_{-0.104}^{+0.132}$	872
Confirmed GCs
J0	$15.405_{-0.222}^{+0.218}$	$1.505_{-0.160}^{+0.196}$	$15.409_{-0.212}^{+0.208}$	$1.459_{-0.148}^{+0.192}$	$15.348_{-0.220}^{+0.218}$	$1.424_{-0.158}^{+0.196}$	$15.348_{-0.208}^{+0.206}$	$1.423_{-0.144}^{+0.184}$	579
H0	$14.816_{-0.192}^{+0.186}$	$1.431_{-0.134}^{+0.170}$	$14.829_{-0.182}^{+0.178}$	$1.387_{-0.128}^{+0.164}$	$14.703_{-0.192}^{+0.186}$	$1.367_{-0.134}^{+0.168}$	$14.703_{-0.180}^{+0.176}$	$1.367_{-0.124}^{+0.158}$	565
Ks0	$14.621_{-0.150}^{+0.144}$	$1.354_{-0.108}^{+0.136}$	$14.640_{-0.148}^{+0.142}$	$1.310_{-0.106}^{+0.134}$	$14.534_{-0.150}^{+0.144}$	$1.357_{-0.108}^{+0.136}$	$14.534_{-0.146}^{+0.142}$	$1.357_{-0.102}^{+0.132}$	559
Confirmed GCs (metallicity available from Kang et al.)
J0	$14.583_{-0.148}^{+0.144}$	$1.192_{-0.110}^{+0.134}$	$14.596_{-0.144}^{+0.140}$	$1.141_{-0.108}^{+0.132}$	$14.534_{-0.146}^{+0.146}$	$1.180_{-0.110}^{+0.134}$	$14.534_{-0.144}^{+0.138}$	$1.180_{-0.104}^{+0.130}$	287
H0	$14.031_{-0.132}^{+0.130}$	$1.214_{-0.098}^{+0.120}$	$14.051_{-0.130}^{+0.126}$	$1.162_{-0.096}^{+0.122}$	$13.982_{-0.134}^{+0.130}$	$1.221_{-0.098}^{+0.122}$	$13.982_{-0.130}^{+0.126}$	$1.214_{-0.094}^{+0.120}$	287
Ks0	$13.902_{-0.100}^{+0.098}$	$1.214_{-0.074}^{+0.088}$	$13.927_{-0.098}^{+0.096}$	$1.150_{-0.076}^{+0.090}$	$13.862_{-0.100}^{+0.098}$	$1.225_{-0.074}^{+0.088}$	$13.861_{-0.098}^{+0.094}$	$1.161_{-0.074}^{+0.088}$	288

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The GCLF peaks derived by Barmby et al. (2001) are J₀ = 15.26 and K₀ = 14.45, by fitting a t₅ distribution using the MAXIMUM program written by J. Secker (see Secker 1992, for details). Nantais et al. (2006) derived similar results for the JHK_s bands using same methods: J₀ = 15.31, H₀ = 14.76, and K_s0 = 14.51, and these authors concluded that the little fainter peaks than those from Barmby et al. (2001) could be real, due to a larger and possibly deeper sample. In this paper, the LF peaks for the confirmed GCs, derived by fitting a t₅ distribution using the maximum likelihood method, are $J_0 = 15.348_{-0.208}^{+0.206}$, $H_0 = 14.703_{-0.180}^{+0.176}$, and ${K_{\rm s}}_0 = 14.534_{-0.146}^{+0.142}$, all of which agree well with previous results. The σ parameters derived here are slightly larger than in previous studies.

It has long been known that the peak of the GCLF is nearly constant in different galaxies, providing a standard indicator for the cosmological distance measurement (Racine 1968; Hanes 1977; Ferrarese et al. 2000). However, several studies (Crampton et al. 1985; Gnedin 1997; Barmby et al. 2001) found that GCLF varies between MR and MP, and between inner and outer subsamples within a galaxy, and the GCLF peak becomes fainter as the local density of galaxies increases (Blakeslee & Tonry 1996). Larsen et al. (2001) found that the V-band turnover of the blue GCs is brighter than that of the red ones by about 0.3 mag on average, with a study of GCs in 17 nearby early-type galaxies. Barmby et al. (2001) found that MR clusters are brighter than MP ones in M31, and inner clusters are brighter on average than outer clusters, indicating that the luminosity function is different among these subpopulations. Goudfrooij et al. (2004) confirmed that the GC system in the early-type galaxy NGC 1316, which is an intermediate-age merger remnant, can be divided into a blue GC subpopulation, consistent with a Gaussian LF, and a red GC component with a power-law LF. These authors also found that the LF of the inner half of the MR population differs significantly from that of the outer half. The difference in GCLF between dwarf and giant ellipticals has been studied by many authors (Harris 1991; Durrell et al. 1996; Strader et al. 2006; Jordán et al. 2006; Miller & Lotz 2007); however, whether there is a trend between GCLF peak and galaxy luminosity is still under debate.

To investigate the effects of metallicity and galactocentric distance on the LF, we divided the confirmed GCs into four subsamples as in Barmby et al. (2001). The MR and MP subpopulations are divided at [Fe/H] =−1.1, using metallicities from Kang et al. (2012), while the inner and outer subsamples are divided at R_gc = 10 kpc. Figure 10 displays the J₀ luminosity histograms for the four subsamples, together with the best-fit t₅ distribution using the maximum likelihood method. Table 4 lists the GCLF fit results in the JHK_s bands for the four subsamples. It is evident that the MP clusters are fainter than their MR counterparts, and the inner clusters are brighter than the outer ones. Gnedin (1997) reported that the destruction of the inner faint low-density clusters due to strong tidal shocks may lead to the difference between the LF of the inner and outer populations. Kavelaars & Hanes (1997) asserted that the difference in the GCLF of the inner and outer halo populations is due to dynamical evolution and/or a dependence of GCLF shape on the environment. Ostriker & Gnedin (1997) found that the predicted differences between the peaks of the inner and outer cluster populations, due to the tidal shocks and dynamical friction, agree quantitatively with the observed differences within the errors in MW, M31, and M87. However, Barmby et al. (2001) concluded that metallicity, age, and cluster IMF may also be important for the variation in the M31 GCLF, besides the dynamical destruction.

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Table 4. M31 GCLF Parameters for Four Groups of Confirmed GCs

Subsample	J0			H0			Ks0
	μt	σt	N	μt	σt	N	μt	σt	N
	(mag)	(mag)		(mag)	(mag)		(mag)	(mag)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
MR inner	$14.140_{-0.120}^{+0.118}$	$1.157_{-0.084}^{+0.104}$	139	$13.548_{-0.098}^{+0.096}$	$1.171_{-0.070}^{+0.084}$	139	$13.400_{-0.110}^{+0.106}$	$1.130_{-0.076}^{+0.092}$	139
MP inner	$14.904_{-0.064}^{+0.064}$	$1.067_{-0.054}^{+0.064}$	83	$14.413_{-0.064}^{+0.062}$	$1.123_{-0.056}^{+0.066}$	84	$14.294_{-0.050}^{+0.050}$	$1.022_{-0.044}^{+0.048}$	84
MR outer	$14.672_{-0.044}^{+0.044}$	$0.874_{-0.032}^{+0.036}$	25	$14.118_{-0.062}^{+0.064}$	$0.980_{-0.042}^{+0.050}$	25	$13.910_{-0.066}^{+0.064}$	$0.996_{-0.046}^{+0.054}$	25
MR outer	$15.033_{-0.080}^{+0.084}$	$1.079_{-0.058}^{+0.068}$	43	$14.484_{-0.052}^{+0.052}$	$1.041_{-0.036}^{+0.042}$	42	$14.430_{-0.044}^{+0.046}$	$1.009_{-0.034}^{+0.038}$	43

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Figure 11 displays the NIR color (including (V − K_s)₀) distribution of the whole sample of clusters (solid-line histogram) and the confirmed GCs (filled histogram) with magnitude uncertainty <0.5 mag. There are some bumps located in the redder wings of the (J − K_s)₀, (H − K_s)₀, and (V − K_s)₀ colors, which are mainly caused by GC candidates. The GC systems of many galaxies reveal bimodal optical color distributions (Forbes et al. 1997; Kundu et al. 1999; Gebhardt & Kissler-Patig 1999; Larsen et al. 2001; Rejkuba 2001). Barmby et al. (2000) have detected the bimodal distribution of the GC colors in M31 using (U − V)₀, (U − R)₀, and (V − K)₀ with the KMM statistics (McLachlan & Basford 1988; Ashman et al. 1994). To check whether the NIR color distributions are bimodal, the KMM algorithm was also performed here. The homoscedastic fitting, with same variances for both groups, was assumed. Table 5 lists the parameters returned by the KMM algorithm. Columns (2) and (3) give the estimated mean value and covariance assuming the whole sample as one group. Columns (4) and (5) give the estimated mean value for each group, assuming the whole sample as two groups, and Column (6) gives their common covariance. Columns (7) and (8) give the number points assigned to each group. Column (9) gives the p value, which is an estimate of the improvement of the two-group fit over a one-group fit, and is interpreted as a rejection of the single Gaussian model at a confidence level of 1 − p. The purely NIR color distributions show bimodality at the ∼100% confidence level; however, the size of the redder subsample is very small, and this subsample is composed mainly of GC candidates, as shown in Figure 11. Although Table 5 presents the mean values and number points for the two groups of (V − K_s)₀ color, these values vary with the initial set, and the p value is unavailable since the sample is not convergent. Some previous studies of purely NIR and optical–NIR GC colors in different galaxies have shown that in some cases the optical color distributions are clearly bimodal; however, the purely NIR and optical–NIR color distributions are not, or they display "differing bimodalities" (Blakeslee et al. 2012) with those of the optical colors alone (see Cantiello et al. 2014, and references therein).

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Table 5. Results from the KMM Homoscedastic Bimodality Tests for the JHK_s Colors of the Sample Clusters in M31

Data set	$\overline{\Delta }$	σΔ	$\overline{\Delta _1}$	$\overline{\Delta _2}$	σΔ	N1	N2	p
	(mag)	(mag)	(mag)	(mag)	(mag)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
(J − H)0	0.600	0.101	0.579	1.573	0.080	840	16	0.000
(H − Ks)0	0.240	0.114	0.193	0.979	0.079	779	38	0.000
(J − Ks)0	0.845	0.155	0.801	1.622	0.120	812	29	0.000
(V − Ks)0	2.423	0.696	2.324	2.560	0.683	718	125	⋅⋅⋅

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Figure 12 shows the relation between metallicity and intrinsic color for the sample clusters with magnitude uncertainty <0.5 mag. Metallicities in the top panels are from Kang et al. (2012), while metallicities in the bottom panels are from Caldwell et al. (2011). The open circles represent the confirmed GCs, while the crosses represent the rest clusters, including candidate GCs, controversial objects, and extended clusters. The (V − K_s)₀ color index, which is often used as a metallicity indicator, shows a clearer correlation with [Fe/H] than other NIR colors, indicating that these NIR intrinsic colors are less sensitive to metallicity (Barmby et al. 2000). Nantais et al. (2006) also reported that stellar spectra in the NIR are much less dependent on metallicity than those in the optical. The relation of (V − K_s)₀ and metallicity shows a notable departure from linearity, with a shallower slope in the redder part. The nonlinear correlation of color with metallicity, which is mainly driven by the horizontal-branch stars, can produce a bimodal color distribution with unimodal metallicity for a group of old clusters (Yoon et al. 2006).

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We also use the KMM algorithm to investigate the bimodality of the metallicity distributions for M31 GCs. Table 6 lists the parameters returned by the KMM algorithm. The average metallicities for Caldwell et al. (2011) and Kang et al. (2012) are nearly consistent, while both groups from Kang et al. (2012) are more metal-rich than the two groups from Caldwell et al. (2011). Metallicities from Kang et al. (2012) show a strong bimodal distribution at the 99.8% confidence level, while the hypothesis of a unimodal distribution is rejected only at a ∼64% confidence level for metallicities from Caldwell et al. (2011). Although many previous studies (Ashman & Bird 1993; Barmby et al. 2000; Perrett et al. 2002; Fan et al. 2008) have reported that the metallicity distribution in M31 is bimodal, Caldwell et al. (2011) suggested that the metallicity distribution in M31 is not generally bimodal, in strong distinction with the bimodal Galactic globular distribution.

Table 6. Results from the KMM Homoscedastic Bimodality Tests for the Metallicities of the Sample Clusters in M31

Data set	$\overline{\rm [Fe/H]}$	σ[Fe/H]	$\overline{\rm [Fe/H]_1}$	$\overline{\rm [Fe/H]_2}$	σ[Fe/H]	N1	N2	p
	(dex)	(dex)	(dex)	(dex)	(dex)
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
Caldwell et al.	−1.010	0.345	−0.934	−1.772	0.287	303	9	0.361
Kang et al.	−1.032	0.350	−0.632	−1.527	0.152	167	127	0.002

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Figure 13 shows the relation between intrinsic color and galactocentric distance R_gc for the sample clusters. Symbols are as in Figure 12. It can be seen that the colors of candidate GCs are on average redder than those of the confirmed GCs. We derived the mean color values in different projected galactocentric distances for the confirmed GCs. For clusters with galactocentric distance R_gc < 30 kpc and R_gc > 30 kpc, the mean values were measured with star clusters located within an annulus at every 3 kpc and 10 kpc radius from the center of M31, respectively, all of which were plotted with squares in Figure 13. Crampton et al. (1985) found that no radial (B − V)₀ color gradient exists for M31 GCs; however, Sharov (1988) reported that the (V − K)₀ color shows a weak correlation with the galactocentric distance. No clear trend is present between the NIR colors and R_gc for the confirmed GCs. It seems that clusters with R_gc around 20 kpc are on average redder than inner clusters in the (V − K_s)₀ color index; however, this may be caused by the crowding blue clusters around the "10 kpc ring" (Gordon et al. 2006), which pull the color index down.

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The color–magnitude diagram (CMD) provides a qualitative model-independent global indication of cluster formation history (Ma 2012a, 2013). Figure 14 displays the CMD of $M_{K_{\rm s}}$ versus (V − K_s)₀ for the sample star clusters with magnitude uncertainty <0.5 mag. Symbols are as in Figure 12. The absolute magnitudes were derived with the distance modulus of (m − M)₀ = 24.47 with a distance of ∼784 kpc. Several models were added to the CMD to obtain a more detailed history of cluster formation. Four fading lines from the simple stellar population (SSP) synthesis model Bruzual & Charlot (2003, hereafter BC03) for a metallicity of Z = 0.004, Y = 0.24, assuming a Salpeter (1955) stellar IMF, and using the Padova-1994 isochrones, are plotted on the CMD of M31 star clusters for four different total initial masses: 10⁶, 10⁵, 10⁴, and 10³ M_☉ (from top to bottom). It seems that many GC candidates in M31 are located out of the evolutionary tracks. The majority of M31 GCs fall between these four fading lines, consistent with previous results (Wang et al. 2010).

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Figure 15 shows the (V − K_s)₀ versus (J − K_s)₀ color–color diagram for M31 star clusters with a magnitude uncertainty <0.5 mag. Symbols are as in Figure 12. The theoretical evolutionary paths from the SSP model BC03 for Z = 0.004, Y = 0.24 (black line) and Z = 0.02, Y = 0.28 (green line) are displayed. The horizontal dashed line represents V − K_s = 3, assuming the reddening value E(B − V) = 0.13. As Galleti et al. (2004) reported, most of the background galaxies have V − K_s ⩾ 3, providing a powerful tool to discriminate between M31 clusters and background galaxies. It can be seen that a large number of GC candidates are located above the dashed line, indicating that some background galaxies were mistaken for GC candidates. However, we should note that the dashed line is just approximate to the criterion V − K_s = 3, since one source with V − K_s > 3 may be located below the dashed line if the reddening value is larger than 0.13 mag. There is a wide distribution of clusters in the color–color diagram, and many GC candidates are located out of the theoretical evolutionary paths, indicating that many GC candidates may not be true GCs.

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