Determination of Fundamental Properties of an M31 Globular Cluster from Main-Sequence Photometry

More than 50 years ago, Sandage (1953) presented the CMD for the Galactic GC M3 and applied an evolutionary theory to the CMD to give a time interval of 5 × 10⁹ yr since the formation of the main sequence. From then on, main-sequence photometry has been considered the most direct method for determining ages of star clusters, because the turnoff of the CMD is mostly affected by age (see Puzia et al. 2002b and references therein). Stellar evolutionary models from the Padova group (Bertelli et al. 1994; Girardi et al. 2000, 2002 and references therein) and the Victoria-Regina (VandenBerg et al. 2000, 2006 and references therein) are widely used. In the Padova stellar evolutionary models, Girardi et al. (2002) provided tables of theoretical isochrones in such photometric systems as ABmag, STmag, Vegamag, and a standard star system, and derived tables of bolometric corrections for the Johnson-Cousins-Glass, HST/WFPC2, HST/NICMOS, Washington, and ESO Imaging Survey systems. The complete database (Girardi et al. 2002) covers a very large range of stellar masses (typically from 0.6 to 120 M_⊙). As a supplement, Girardi et al. (2008) presented several theoretical isochrones including HST/ACS WFC. These models (Girardi et al. 2002, 2008) are computed with updated opacities and equations of state, and moderate amount of convective overshoot. However, the isochrones are presented for only 6 initial chemical compositions: [Fe/H] = -2.2490, -1.6464, -0.6392, -0.3300, +0.0932 (solar metallicity), and +0.5595, which are evidently not dense enough. It is fortunate that Marigo et al. (2008) provide tables for intermediate values of age and metallicity via an interactive interface.⁶ We will discuss this resource in detail in § 4.2. The novel feature of the Victoria-Regina models (VandenBerg et al. 2000, 2006 and references therein) is that they provide a wide range of metallicities; i.e., VandenBerg et al. (2006) presented 72 grids of stellar evolutionary tracks for 32 [Fe/H] values from -2.31 to 0.49, which are dense enough for studying properties of stellar populations with different metallicities. In addition, in these models, convective core overshooting has been treated using a parameterized form of the Roxburgh criterion (Roxburgh 1978, 1989), in which the free parameter, F_over (F_over must be calibrated using observations), is assumed to be a function of both mass and metal abundance.

To determine the main characteristics (age and metallicity) of the population in B379, we fit isochrones to the cluster CMD. We used the Padova theoretical isochrones in the HST/ACS WFC STmag system (Marigo et al. 2008). Via the interactive interface noted in § 4.1,, we can construct a grid of isochrones for different values of age and metallicity, photometric system, and dust properties. We use the default models that involve scaled solar abundance ratios (i.e., [α/Fe] = 0.0). The Salpeter initial mass function (IMF) (Salpeter 1955) is adapted to match the selection of Ma et al. (2007), who used the high-resolution SSP models of BC03, computed using the Salpeter (1955) IMF; circumstellar dust is not included. As we pointed out previously, by comparison to isochrones of VandenBerg et al. (2006), Brown et al. (2004a) derived the age of B379 to be . In addition, the metallicity of B379 is available: Huchra et al. (1991) derived [Fe/H] = -0.7 ± 0.35 using the strengths of six absorption features in the cluster integrated spectra; Holland et al. (1997) used the HST/WFPC2 photometry to construct the deep CMD for B379, and the shape of the red giant branch (RGB) gave an iron abundance of [Fe/H] = -0.53 ± 0.03. These metallicities obtained from different methods are consistent. Based on the age and metallicity of B379 obtained by the previous authors (Brown et al. 2004a; Huchra et al. 1991; Holland et al. 1997), we used the interactive tables noted in § 4.1 to construct a fine grid of isochrones about ages and metallicities, sampling an age range 8.0 ≤ τ ≤ 13.5 Gyr at intervals of 0.5 Gyr, and a metal abundance range 0.00250 ≤ Z ≤ 0.00950 at intervals of 0.00025 dex. The total metallicity [M/H] = log(Z/Z_⊙) where Z_⊙ ≈ 0.019, so this abundance range corresponds to -0.88 ≤ [M/H] ≤ -0.30.

We followed the method of Mackey & Broby Nielsen (2007) of finding the best-fitting isochrone, by locating by eye three fiducial points on the CMD of the cluster: the magnitude and color of the turnoff, the magnitude of the tight clump of red HB stars, and the color of the RGB at a level 3.0 mag brighter than the level of the turnoff. This latter point was selected simply as a point lying on the lower RGB at a level intermediate between that of the red end of the SGB and that of the tight clump of red HB. We then calculated the difference in magnitude between the level of the turnoff and the level of the tight clump of red HB (Δm_F814W), and the difference in color between the turnoff and the RGB fiducial point (Δc_mF606W ). As Mackey & Broby Nielsen (2007) pointed out, the difference in magnitude between the level of the turnoff and the level of the tight clump of red HB is strongly sensitive to cluster age (and weakly sensitive to cluster metallicity), while the difference in color between the turnoff and the RGB fiducial point is sensitive to both cluster age and metallicity. We determined Δm_F814W = 3.77 ± 0.1 and Δc_{mF606W-mF814W}  = 0.28 ± 0.01.

Second, we calculated the same intervals for all isochrones on the grid, and selected only those with values lying within certain tolerances of the cluster measurements. In this article, we adopted ± 0.2 mag for Δm_F814W and for Δc_{mF606W-mF814W} . We fit the selected isochrones to the CMD by eye. At the same time, we calculated the offsets in magnitude and color required to align the turnoff of the isochrone with that of the CMD, and the offsets in magnitude required to align the tight clump of red HB of the isochrone with that of the CMD, and the offsets in color required to align the RGB fiducial point of the isochrone with that of the CMD. We then averaged the offsets in magnitude and in color and applied them to overplot the isochrone on the CMD, and identified the best-fitting isochrone by eye. The resulting offsets δm_F814W and δc_{mF606W-mF814W} provide estimates for the distance modulus to B379 ((m - M)₀) and the reddening value (E(B - V)): δm_F814W = (m - M)₀ + A_F814W, and δc_{mF606W-mF814W}  = A_F606W - A_F814W. The reddening law from Cardelli et al. (1989) is employed in this article. The effective wavelengths of the ACS F606W and F814W filters are λ_eff = 5918 and 8060 Å (Sirianni et al. 2005), so that from Cardelli et al. (1989), A_F606W ≃ 2.8 × E(B - V) and A_F814W ≃ 1.8 × E(B - V) (see Barmby et al. 2007 for details). The reddening value and distance modulus for B379 obtained in this article are: E(B - V) = 0.08 and (m - M)₀ = 24.44 ± 0.10, where the uncertainty is the standard error of the mean.

The best-fitting Padova isochrone can be seen in Figure 1, with the metal abundance 0.009 in Z (or -0.325 in [M/H]) and 11.0 Gyr in age. The age of B379 obtained in this article is 11.0 ± 1.5 Gyr, where the uncertainty is the standard error of the mean.

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The primary purpose of this article is to obtain the age of B379 by comparing its CMD with isochrones of the Padova stellar evolutionary models, and to check whether the age of B379 we obtained is in agreement with the determination of Brown et al. (2004a). From high-resolution stellar spectroscopy, it is shown that GCs in both the halo and the bulge of our Galaxy are α/Fe enhanced with the typical values [α/Fe] ≈ 0.3 ± 0.1 dex (see Thomas et al. 2003 and references therein). For M31 GCs, the estimates of α/Fe ratios by Beasley et al. (2005) and Puzia et al. (2005) showed it may, on average, be ∼0.1–0.2 dex lower than in the Milky Way (see also Colucci et al. 2009). The Padova stellar evolutionary models do not provide isochrones with [α/Fe] = 0.0; however, the luminosities of turnoff, SGB, and the tip of the RGB are nearly unchanged by varying α enhancement except in the intermediate-age regime, where α-enhanced isochrones are slightly fainter than scaled solar ones (see Gallart et al. 2005 and references therein). It is generally known that the turnoff, SGB, and lower RGB are the most age-sensitive features of the CMD, so the age of B379 obtained based on the isochrones with [α/Fe] = 0.0 will not change when using the isochrones with [α/Fe] > 0.0.

The age of B379 (11.0 ± 1.5 Gyr) obtained in this article is consistent with the determination () of Brown et al. (2004a). Brown et al. (2004a) determined the age of B379 by comparing the observed CMD with isochrones of VandenBerg et al. (2006). The result of this article confirmed the conclusion of Brown et al. (2004a) that B379 is 2–3 Gyr younger than the oldest Galactic GCs. The metallicity of B379 obtained in this article is [M/H] = -0.325. Taking into account an enhancement of the α-capture elements by [α/Fe] = 0.3 (Brown et al. 2004a), and using the relation between [M/H], Fe/H, and [α/Fe] from Salaris et al. (1993), we derived [Fe/H] = -0.54, which is in good agreement with the determination of [Fe/H] = -0.53 of Holland et al. (1997) based on the shape of the RGB of the deep CMD observed by the HST/WFPC2.

B379 is located in the M31 halo, so the extinction is mainly from the foreground Galactic reddening in the direction of M31, which has been discussed by many authors (e.g., van den Bergh 1969; McClure & Racine 1969; Frogel et al. 1980; Fusi Pecci et al. 2005), with nearly similar values determined; e.g., E(B - V) = 0.08 by van den Bergh (1969), E(B - V) = 0.11 by McClure & Racine (1969) and Hodge (1992), and E(B - V) = 0.08 by Frogel et al. (1980). In addition, Barmby et al. (2000) determined the reddening for each individual cluster using correlations between optical and infrared colors and metallicity, and by defining various "reddening-free" parameters using their large database of multicolor photometry. Finally, Barmby et al. (2000) determined reddenings for 314 clusters, 221 of which are reliable (see Barmby et al. 2000 for details). For B379, Barmby et al. (2000; also P. Barmby 2002, private communication) obtained its reddening value to be E(B - V) = 0.10 ± 0.05. It is evident that the reddening value of E(B - V) = 0.08 obtained in this article is consistent with these determinations.

Given the importance of M31 as an anchor for the extragalactic distance scale, many studies have presented distance determinations to M31 using different methods. Pritchet & van den Bergh (1987), Holland (1998) and Vilardell et al. (2006) have given a detailed review. Although the stellar populations located in different positions in M31 have different distance moduli, the dispersion can be neglected since the distance of M31 is large enough. For example, Rich et al. (2005) pointed out that the clusters in M31 dispersed over a 20 kpc radius would have up to 0.06 mag random distance uncertainty. So, the distance modulus to B379 obtained in this article should be consistent with the distance of M31 previously determined within 0.06 mag random distance uncertainty.

Here we compare our determination with the most recent and/or important measurements. Freedman & Madore (1990) derived the mean distance modulus to M31 to be (m - M)₀ = 24.44 ± 0.13 based on the Cepheids in Baade's fields I, III, and IV (Baade & Swope 1963, 1965) observed using the Canada-France-Hawaii Telescope (CFHT). Holland (1998) determined the distance moduli to 14 M31 GCs by fitting theoretical isochrones to the observed RGBs, including B379. The distance modulus to B379 obtained by Holland (1998) is (m - M)₀ = 24.45 ± 0.07. Stanek & Garnavich (1998) estimated the distance modulus to M31 as (m - M)₀ = 24.471 ± 0.035 by comparing the red clump stars with parallaxes known to better than 10% in the Hipparcos catalog with the red clump stars in three fields in M31 observed with the HST. A determination of Freedman et al. (2001) based on the Cepheid period-luminosity (PL) relation suggests the distance modulus of (m - M)₀ = 24.38 ± 0.05 to M31 based on the results of the HST Distance Scale Key Project to measure the Hubble constant. Durrell et al. (2001) determined the distance modulus of (m - M)₀ = 24.47 ± 0.12 to M31 from the luminosity of the RGB tip of over 2000 RGB halo stars in a halo field located about 20 kpc from the M31 nucleus along the southeast minor axis. Joshi et al. (2003) obtained R- and I-band observations of a 13^' × 13^' region in the disk of M31 and derived the Cepheid PL distance modulus to be (m - M)₀ = 24.49 ± 0.11. Brown et al. (2004b) determined the distance modulus of (m - M)₀ = 24.5 ± 0.1 to M31 based on brightness of 55 RR Lyrae stars detected on the HST/ACS images of ∼84 hr (250 exposures over 41 days). McConnachie et al. (2005) derived the distance modulus to M31 to be (m - M)₀ = 24.47 ± 0.07 based on the method of the tip of the RGB observed using the Isaac Newton Telescope Wide Field Camera (INT WFC). Ribas et al. (2005) derived the distance modulus of M31 as (m - M)₀ = 24.44 ± 0.12 from an eclipsing binary. Very recently, Sarajedini et al. (2009) presented the HST observations taken with the ACS WFC of two fields near M32 located 4–6 kpc from the center of M31, and identified 752 RR variables with excellent photometric and temporal completeness. Based on this large sample of M31 RR Lyrae variables, and using a relation between RR Lyrae luminosity and metallicity along with a reddening value of E(B - V) = 0.08 ± 0.03, they derived the distance modulus of (m - M)₀ = 24.46 ± 0.11 to M31. For comparison, we list these determinations of M31 distance moduli in Table 1. It is evident that our determination is in good agreement with the previous determinations.