THE PROPAGATION OF UNCERTAINTIES IN STELLAR POPULATION SYNTHESIS MODELING. III. MODEL CALIBRATION, COMPARISON, AND EVALUATION

Data on MW star clusters are taken primarily from the catalog of Harris (1996). This catalog provides UBVRI photometry, [Fe/H] metallicities, and E(B − V) reddening values for 150 star clusters. The photometry is corrected for Galactic extinction using the reddening values provided in the catalog in conjunction with a Cardelli et al. (1989) extinction curve. The Harris (1996) catalog is complemented with JHK photometry from the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) for 106 star clusters as compiled by Cohen et al. (2007). For most of the comparisons between models and data, the MW sample is restricted to star clusters with E(B − V) < 0.2 in order to reduce the uncertainties associated with de-reddening. When presenting V − K and J − K colors, all star clusters with E(B − V) < 0.4 are included since reddening corrections are smaller in the near-IR.

Schiavon et al. (2005) have measured integrated, flux-calibrated, high signal-to-noise spectra of 40 MW star clusters spanning a range of metallicities and HB morphologies. The spectra span the wavelength range 3360 Å < λ < 6430 Å at a FWHM resolution of 3.1 Å and sampling of 1.0 Å. These spectra have been de-reddened using the Cardelli et al. (1989) extinction curve with E(B − V) values tabulated by Schiavon et al. Metallicities are also provided by Schiavon et al. and will be used herein, except for 19 star clusters that have updated [Fe/H] measurements from Carretta et al. (2009).

Photometry of star clusters in M31 is available in the Revised Bologna Catalog (RBC v3.5; Galleti et al. 2004). Available photometry includes UBVRI from a variety of sources, JHK from 2MASS, and new ultraviolet observations from the Galaxy Evolution Explorer (GALEX; Martin et al. 2005) presented in Rey et al. (2007). Spectroscopic metallicities of a subset of M31 star clusters are taken from Barmby et al. (2000) and Perrett et al. (2002). Galactic E(B − V) reddening estimates are derived from Schlegel et al. (1998) via the utilities in the kcorrect v4.1.4 software package (Blanton & Roweis 2007). Photometry is corrected for Galactic extinction via E(B − V) and the extinction curve of Cardelli et al. (1989). In addition to Galactic reddening, the colors of M31 star clusters will also be reddened by dust within M31. Unfortunately it is not straightforward to make such a correction, and therefore no additional correction will be applied. In later sections, attention will be focused on the near-IR colors of M31 star clusters, where dust corrections should be small.

The RBC contains a number of quality flags. Star clusters are selected to be definite globular clusters (classification flag = 1), with a definite confirmation (confirmation flag = 1), and to not be in the young cluster catalog of Fusi Pecci et al. (2005). This sample is further restricted to star clusters with E(B − V) < 0.2 to reduce the uncertainties associated with Galactic reddening.

The final star cluster catalog, which includes the aforementioned quality cuts and metallicity measurements, contains 22 clusters, 20 of which include GALEX NUV photometry, and 12 of which include GALEX FUV photometry.

Data on star clusters in the MCs come principally from two sources. Pessev et al. (2006) presented a carefully constructed catalog of 2MASS JHK photometry for a sample of 75 star clusters. In Pessev et al. (2008), this near-IR photometry is aperture matched to and combined with BV photometry from the literature. These authors construct a "test sample" of star clusters where high-quality data are available to estimate main-sequence turnoff ages from CMDs and metallicities from either the CMDs or individual stellar spectroscopy. These high-quality data are then binned by age to produce average colors and metallicities as a function of star cluster age.

In the youngest age bin, a less direct method is used for determining star cluster ages in the Pessev et al. (2008) sample. The "S-parameter," introduced by Elson & Fall (1985), provides a simple empirical relation between cluster age and a combination of integrated U − B and B − V colors. High-quality age estimates from Hubble Space Telescope observations from Kerber et al. (2007) were used by Pessev et al. (2008) to update the S-parameter–age calibration in order to provide age estimates for the youngest clusters in their sample. We note in passing that the Pessev et al. calibration is substantially different from the commonly used relation of Girardi et al. (1995). It is important to keep in mind that since ages are based on CMD fitting, they will depend on the stellar evolution calculations used to convert main-sequence turnoff points into ages. In particular, the treatment of convective overshooting has a strong impact on the age–main-sequence turnoff point relation for young and intermediate ages (t ≲ 2 × 10⁹ yr). This sensitivity arises because overshooting increases the amount of fuel available for core hydrogen burning and hence prolongs main-sequence lifetimes. We will return to these issues when these data are compared to models.

The second significant source of data on MC star clusters comes from González et al. (2004). These authors analyze 2MASS data to produce not only integrated near-IR colors but also SBFs in the 2MASS JHK filters. In this sample, star clusters were stacked according to the classification system devised by Searle et al. (1980, SWB), which closely tracks the S-parameter described above. The relation between SWB type, age, and metallicity is adopted from Cohen (1982). We have independently verified that these ages agree with the ages inferred from the S-parameter calibration discussed in Pessev et al. (2008). This sample has been augmented with V − I colors (González-Lópezlira et al. 2005), where I magnitudes are from the DENIS survey (Epchtein et al. 1997) and V magnitudes are taken from van den Bergh (1981). We point out that the Pessev et al. and Gonzalez et al. samples are not disjoint. Comparison of these two catalogs will provide a valuable cross-check on the reliability of the derived data products.

The novel feature of these two data sets is that the star cluster data have been stacked in bins of age in order to reduce stochastic effects. A number of authors have discussed the importance of stochastic effects arising from small numbers of very bright stars (Lançon & Mouhcine 2000; Bruzual A. 2002; Cerviño & Luridiana 2004; Bruzual & Charlot 2003). Lançon & Mouhcine (2000) provide estimates of the star cluster mass required to produce cluster-to-cluster fluctuations in luminosity that are less than 10%. The minimum cluster mass required to meet this criterion will of course be band dependent (i.e., the minimum mass will generally be higher in the K band than in the V band).

Pessev et al. (2008) demonstrate that the masses of their stacked clusters satisfy the minimum mass constraint of Lançon & Mouhcine (2000), ensuring that their stacked results can be robustly compared to SPS models. The masses of the stacked clusters in the González et al. (2004) sample are comparable to those in Pessev et al. (2008), and so their results should also be relatively unaffected by stochastic effects.

Finally, we include for completeness the V − K and J − K colors of LMC star clusters from Persson et al. (1983), which have been used in many comparisons between observations and SPS models. These J and K magnitudes are converted from the California Institute of Technology photometric system to the 2MASS system using the transformations provided by Carpenter (2001). Ages for these clusters were assigned based on the S-parameter calibration of Pessev et al. (2008).

Pessev et al. (2006) presented a detailed comparison between their near-IR colors and those of Persson et al. (1983), in some cases noting substantial differences in the sense that colors from Persson et al. were redder than the colors from Pessev et al. They attributed this difference to the fact that Persson et al. centered their photometry on the brightest part of the star cluster in order to maximize the measured flux, while Pessev et al. center their apertures using a Gaussian smoothing technique. In some cases, the Persson et al. centroids are significantly offset from the true dynamical center of the star cluster because of off-center IR-bright sources within the cluster. For this reason, in addition to the fact that the data are stacked to reduce stochastic effects, the results of Pessev et al. should be preferred to the older results of Persson et al. We will return to this issue in Section 4.

For the reasons mentioned in the previous section, star clusters are the ideal observational data to calibrate SPS models. There are however at least two reasons why it is useful to also consider SPS constraints from galaxies: (1) galaxies tend to be of higher metallicity than old star clusters, and so SPS model predictions for old ∼Z_☉ populations can be assessed with galaxies composed of old stars and (2) galaxies contain many more stars than star clusters, and so stochastic effects are never a source of concern when comparing models to galaxy data.

These two benefits of using galaxies to calibrate SPS models are exploited herein by considering two classes of galaxies. The first class consists of bright red sequence, "quiescent" galaxies, which contain old metal-rich stars, and will therefore be useful for calibrating old metal-rich SPS model predictions. The second class are called "K+A," or "post-starburst" galaxies, as their spectra are dominated by a mixture of A and K type stars and show no signs of ongoing star formation (e.g., Dressler & Gunn 1983; Goto et al. 2003). Such galaxies are presumed to have had their star formation abruptly truncated. The presence of A stars in their spectra indicates that the galaxy contains a significant population of ≲1 Gyr old stars (e.g., Leonardi & Rose 1996). These galaxies will provide valuable constraints on the importance of TP-AGB stars, as the prominence of this stellar phase peaks for ages ≈1–2 Gyr (Maraston 2005; Conroy et al. 2009). Since stars spend a small amount of time in this phase, observational constraints from star clusters can be difficult to interpret because of stochastic variations. As mentioned above, a stacking technique is employed when comparing to star clusters so that stochastic effects are minimized. Nonetheless, it will be useful to derive independent constraints from galaxies where stochastic effects are known to be negligible.

As mentioned in Section 1, the use of galaxies to assess the accuracy of SPS models is complicated by at least two effects: the attenuation of starlight by dust and the fact that galaxies contain stars of a range of ages and metallicities. These complications are addressed where relevant in Section 4.

The source of all galaxy positions, optical photometry, and redshifts is the Sloan Digital Sky Survey (SDSS; York et al. 2000), as made available through the hybrid NYU Value Added Galaxy Catalog² (VAGC; Blanton et al. 2005). The SDSS photometry is k-corrected and corrected for Galactic extinction via the kcorrect v4.1.4 software package (Blanton & Roweis 2007). The photometry is k-corrected to z = 0.1—the median redshift of the sample—in order to minimize the magnitude of the k-correction. Spectroscopic indices are provided by the MPA/JHU database.³

Near-IR photometry in YJHK filters is provided by the UKIRT Infrared Deep Sky Survey⁴ (UKIDSS). The UKIDSS project is defined in Lawrence et al. (2007). UKIDSS uses the UKIRT Wide Field Camera (WFCAM; Casali et al. 2007). The photometric system is described in Hewett et al. (2006), and the calibration is described in Hodgkin et al. (2009). The pipeline processing and science archive are described in M. J. Irwin et al. (2010, in preparation) and Hambly et al. (2008). We make use of Data Release 3, which is publically available. UKIDSS photometry is matched to the SDSS with a search radius of 5''. Petrosian magnitudes are used for the UKIDSS photometry. We do not consider colors that are combinations of SDSS and UKIDSS photometry, so it is not essential that the photometric apertures be precisely matched.

The UKIDSS survey goes much deeper than 2MASS, and is therefore more desirable. For example, typical 2MASS photometric errors for galaxies at z ∼ 0.1 are 0.2 mag, compared to ∼0.01 for UKIDSS photometry. Galactic extinction is corrected using E(B − V) reddening estimates (Schlegel et al. 1998) in conjunction with a Cardelli et al. (1989) extinction curve. K-corrections for the UKIDSS photometry are estimated from the kcorrect v4.1.4 software package using the 2MASS JHK filters since those, and not UKIDSS filters, are available in the code. The filter transmission curves for 2MASS and UKIDSS are very similar and so this approximation is justified (and, in any event, the effect of the k-corrections on colors for the samples defined below is always less than 0.01 mag).

As mentioned in Section 2.2.1, we will consider two well-defined classes of galaxies to assess the accuracy of SPS models. The first class are K+A galaxies as identified by Quintero et al. (2004). This sample contains 1194 galaxies. Of these, only 186 have UKIDSS photometry. When considering near-IR colors of the K+A sample we use only these 186 galaxies, while when SDSS colors are considered we use the full sample.

The second sample consists of bright red sequence ("quiescent") galaxies. This sample is defined by M^0.1_r < −20.5, where the superscript indicates that the magnitude has been k-corrected to z = 0.1, and

$$\begin{equation} (g-r)^{0.1}> -0.03\,\big(M_r^{0.1}+23.0\big)+0.94, \end{equation} \tag{ 1 }$$

which divides galaxies by the valley visible in the CMD and effectively isolates red sequence galaxies. Applying these cuts and requiring that galaxies have both SDSS and UKIDSS photometry results in a sample of 5340 galaxies at 0.08 < z < 0.12. A narrow redshift slice was chosen to minimize both evolutionary and k-corrections.

In addition to these two classes, a third sample of galaxies spanning a range in properties is derived from the spectroscopic indices catalog. Galaxies are selected with 0.05 < z < 0.1 and with a median spectroscopic signal to noise of S/N >15. This redshift range was chosen as a trade-off between maximizing the sample size given the S/N cut and minimizing evolutionary corrections. From this catalog, two classes of galaxies are defined: those with velocity dispersions measured from the stellar continuum of 75  km s⁻¹ < σ < 150  km s⁻¹ and those with 200  km s⁻¹ < σ < 300  km s⁻¹. The latter sample contains exclusively red sequence galaxies, while the former contains galaxies spanning a wide range in g − r color. These samples contain 42963 and 9909 galaxies, respectively.

SPS model construction closely follows that of Conroy et al. (2009), to which the reader is referred for details. In brief, this SPS code (referred to below as FSPS, for "flexible" SPS) combines stellar evolution calculations with stellar spectral libraries to produce simple stellar populations (SSPs), which describe the evolution in time of the spectrum of a mono-metallic, coeval set of stars. The initial stellar mass function (IMF) specifies the weights given to stars of various masses. The IMF of Kroupa (2001) is adopted throughout. FSPS also contains a number of phenomenological dust attenuation prescriptions and produces spectra for composite stellar populations. Our SPS code is open source and publicly available.⁵

The novel feature of this SPS model is that the essential aspects of SPS are integrated in a flexible way so that the user may choose, for example, their preferred set of isochrones and stellar spectral libraries. In addition, the user may alter the weights given to and physical parameters of various advanced stages of stellar evolution (see below), and may specify the IMF of their choosing. The details of these various model choices are now described.

We make use of both the Padova⁶ (Girardi et al. 2000; Marigo & Girardi 2007; Marigo et al. 2008) and BaSTI⁷ (Pietrinferni et al. 2004; Cordier et al. 2007) stellar evolution calculations, both in the versions available as of 2009 November. These two groups follow stellar evolution from the main sequence through the TP-AGB phase, although the treatment of the TP-AGB phase is very different between the two groups.

The Padova calculations exist for metallicities in the range 2 × 10⁻⁴ < Z < 0.030, for ages 10^6.6 yr < t < 10^10.2 yr, and for initial masses 0.15 M_☉ ⩽ M ⩽ 67 M_☉. BaSTI outputs cover a somewhat narrow range in ages, 10^7.5 yr < t < 10^10.2 yr, and initial masses 0.5 M_☉ ⩽ M ⩽ 10.0 M_☉ but include higher metallicities: 3 × 10⁻⁴ < Z < 0.040. The BaSTI database provides models for both solar-scaled and α-enhanced chemical compositions, for two assumptions regarding convective core overshooting, and for two stellar mass-loss rates. We use the solar-scaled, no overshooting, and η = 0.4 mass-loss rate parameter model. For both lower and higher masses, the Padova calculations are stitched onto the BaSTI grid. The Padova and BaSTI models are compared in Figure 1. In this figure, isochrones are compared at several ages and metallicities, and in addition the location of the TP-AGB onset is shown.

Zoom In Zoom Out Reset image size

In order to follow stellar evolution from the end of the TP-AGB phase through the post-AGB phase, the above evolutionary tracks have been supplemented with the post-AGB models of Vassiliadis & Wood (1994). The contribution of post-AGB stars to the integrated SED is highly uncertain for a number of reasons, including a lack of calibrating observations and the unknown duration of heavy enshroudment as post-AGB stars leave their M giant phase. Recent observations of M32 have found that post-AGB stars are much less frequent than standard models predict (Brown et al. 2008). Nonetheless, the inclusion of post-AGB tracks may be important for the interpretation of the ultraviolet spectra of old systems (e.g., Greggio & Renzini 1990).

The FSPS code contains a number of parameters that control the weights given to various uncertain phases of stellar evolution. For example, the bolometric luminosity and effective temperature of TP-AGB stars can be arbitrarily modified. These modifications are applied as overall shifts in log(L_bol) and log(T_eff) for the entire TP-AGB phase, and are parameterized by the variables Δ_L and Δ_T (see discussion in Conroy et al. 2009). In addition, an extended blue HB can be included,⁸ as can a population of BS stars. A blue HB population is parameterized as the fraction, f_BHB, of HB stars that are in a blue/extended component. Such a flexible treatment is motivated by the great variety of observed HB morphologies in MW star clusters, even at a given metallicity (Piotto et al. 2002). These stars are uniformly populated in log(T_eff) from the red clump to log(T_eff/K) = 4.2.

BS stars are uniformly populated from one-half magnitude above the main-sequence turnoff to two and a half magnitudes brighter than the turnoff, along the zero-age main sequence (see, e.g., Xin & Deng 2005). The weight given to BS stars is specified as the number of BS stars per unit HB star, S_BS. This parameterization is observationally motivated and has no (known) physical significance. Note that the origin of BS stars remains a mystery. For our purposes, the most important unknown relating to BSs is whether or not they exist in large numbers in the field. If they are the result of binary interactions (e.g., McCrea 1964), then one would expect them to be found in the field, though perhaps in fewer numbers compared to dense star clusters where the production of binary stars is common. While difficult to confidently identify, recent observations suggest that BSs may in fact be quite common in the field (Preston & Sneden 2000).

The weight given to the post-AGB phase can also be arbitrarily modified. The effect of these uncertain aspects of stellar evolution will be investigated in later sections.

Stellar spectral libraries are required to convert the outputs of stellar evolution calculations—surface gravities, g, effective temperatures, T_eff, and metallicities, Z,—into observable SEDs. Two spectral libraries are considered herein. The first is the empirically calibrated theoretical BaSeL3.1 library (Lejeune et al. 1997, 1998; Westera et al. 2002), supplemented with empirical, average TP-AGB spectra from the library of Lançon & Mouhcine (2002). The BaSeL library was constructed from the model atmospheres of R. Kurucz (1995, private communication to R. Buser), Fluks et al. (1994), Bessell et al. (1989, 1991), and Allard & Hauschildt (1995). In the TP-AGB library, spectra are binned both by near-IR color and by whether the star is carbon rich or oxygen rich. Stars hotter than 50,000 K are assumed to be pure blackbodies. This combined library covers the full wavelength range (ultraviolet through infrared) at a sampling of 10 Å in the ultraviolet and 20 Å in the optical.

The second spectral library considered herein is derived from the empirical Miles library (Sánchez-Blázquez et al. 2006). This atlas contains spectra of ≈1000 stars covering the wavelength range 3600 Å < λ < 7500 Å at a sampling of 0.9 Å per pixel (FWHM ≈ 2.3 Å). Derived stellar parameters are presented in Cenarro et al. (2007). Integrating empirical libraries into SPS models is a non-trivial task because the coverage in the stellar parameters g, T_eff, and Z is very inhomogeneous. In fact many regions of this parameter space are not covered by empirical libraries at all, especially at sub-solar and super-solar metallicities and for spectral types earlier than A. An additional problem with the Miles library is that the spectra have been arbitrarily normalized to unity at 5350 Å. This is particularly problematic because the wavelength range is not broad enough to compare the integral of the flux to the expected bolometric luminosity for a star with a given g, T_eff, and Z. Therefore, an additional library must be used to provide the absolute flux scale. The BaSeL library is used for this purpose. In detail, at a given g, T_eff, and Z, the normalization of the flux for stars in the Miles library is chosen such that the flux integrated over the interval 4000 Å <λ < 7000 Å is identical to that predicted by the BaSeL library. This wavelength interval was chosen because it is the range over which the BaSeL library has been most extensively calibrated.

The BaSeL library is also used to provide the relative trends of spectra with stellar parameters for regions of parameter space where the relative trends cannot be self-consistently determined from Miles. The Miles library is binned and, when necessary, linearly interpolated onto the Kurucz/BaSeL log(g)–log(T_eff) grid. This interpolated library is then interfaced with FSPS in a manner identical to the BaSeL library. Five metallicity bins are defined so that roughly similar numbers of stars occupy each bin. The bin boundaries are log(Z/Z_☉) = [ − 2.0, − 1.0, − 0.5, − 0.15, 0.15, 0.75]. The resulting average metallicities within these bins are log(Z/Z_☉) = [ − 1.4, − 0.7, − 0.3, 0.0, 0.2].

At metallicities less than solar, the Miles library stars are α-enhanced (Milone et al. 2009). At [Fe/H] <−1 the library is on average enhanced in Mg by +0.4 dex. This will be relevant in our comparison to the spectra of MW star clusters in later sections.

There are several important caveats to these libraries that the reader should keep in mind. In the empirical TP-AGB library, the metallicities of the stars are unknown. The authors advocate assuming that they are all solar metallicity since a significant fraction are located in the solar neighborhood (although many are also located in the MCs). In order to use the library at non-solar metallicities, the authors suggest the use of the theoretical metallicity-dependent color–temperature relations in Bessell et al. (1991) in order to convert the observed I − K colors into temperatures at other metallicities. We adopt this approach herein, but caution that model predictions of TP-AGB spectra at non-solar metallicities will carry additional uncertainties.

Westera et al. (2002) found that the original theoretical atmosphere calculations used in the BaSeL library could not simultaneously match observed UBVRIJHKL color–temperature relations for individual stars and CMDs of globular clusters. The reason for this was thought to be inadequacies in the stellar evolution calculations used in the CMD comparisons. Because of this problem, Westera et al. (2002) presented two normalizations of the BaSeL library, one which matched the color–temperature relations, and the other which matched the globular cluster data. These two normalizations are referred to as the "WLBC" and "PADOVA2000" libraries, since the Padova evolutionary calculations were used in the globular cluster comparisons. The "WLBC" normalization will be used herein because this calibration is not tied to any particular isochrone set.

In summary, we have in hand a flexible SPS model from which we can choose two sets of isochrones (Padova or BaSTI) and two sets of spectral libraries (BaSeL or Miles). In addition, there are a number of free parameters characterizing various uncertain aspects of advanced stages of stellar evolution. These various options will be exploited in later sections.

For the Padova isochrones, the TP-AGB phase has been modified in order to generate acceptable agreement with the data. Specifically, the TP-AGB phase has been calibrated against both MC star clusters and post-starburst galaxies (Sections 4.2 and 4.6, respectively). These calibrations, presented here for clarity, result in an adjustment of the TP-AGB parameters Δ_L and Δ_T (quantifying overall shifts in log(L_bol) and log(T_eff); see Section 3.1) as follows:

$$\begin{equation} \Delta _L =\left\lbrace \begin{array}{@{}ll@{}} - 1.0+(t-8.0)/1.5, & 8<t<9.1 \\ [1mm] -{\rm max}\lbrace {\rm min}\lbrace 0.4,-{\rm log}(Z/Z_\odot)\rbrace,0.2\rbrace, & t>9.1, \end{array} \right. \end{equation} \tag{ 2 }$$

$$\begin{equation} \Delta _T =\left\lbrace \begin{array}{@{}ll@{}} \left\lbrace \begin{array}{@{}ll@{}} +0.1,& {\rm log}(Z/Z_\odot)<-0.25 \\ 0.0, & {\rm otherwise}\end{array}\right. & 8<t<9.1 \\ [1mm] +0.1-{\rm min}\lbrace (t-9.1)/1.5,0.2\rbrace, & t>9.1, \end{array} \right. \end{equation} \tag{ 3 }$$

where t is the age of the population measured in log(yr). We emphasize that the values of Δ_L and Δ_T chosen to fit the data are not unique, and that, unless explicitly stated in the above definitions, they are independent of metallicity. Throughout, reference to the "modified Padova model" refers to these modifications.

Figure 2 shows the colors of MC star clusters as a function of cluster age. As discussed in Section 2.1.3, individual clusters are binned in age to produce "superclusters" that are relatively immune to stochastic fluctuations (Pessev et al. 2008; González-Lópezlira et al. 2005). The data from Persson et al. (1983) are also included, and have not been binned in age. These data are of inferior quality (see Section 2.1.3) and are only included for comparison because this data is frequently used to constrain SPS models (e.g., Maraston 2005).

Zoom In Zoom Out Reset image size

A variety of SPS models are compared to the MC star cluster data in Figure 2. From our FSPS code, we have generated SSP color evolution for both the Padova and BaSTI isochrones. In this figure, we make use of the modified Padova isochrones (see Section 3.1.3). The models of BC03 and M05 are also included. All model predictions reflect the fact that the average metallicity of the data varies with age. That is, we have interpolated model predictions so that they vary with metallicity in as in the data, where the age–metallicity relation in the MC data is log(t/yr) =(7.0, 7.7, 8.7, 9.0, 9.5, 9.8, 10.1), log (Z/Z_☉) = (−0.28, − 0.28, − 0.28, − 0.28, − 0.38, − 0.68, − 1.5) (Pessev et al. 2008; González-Lópezlira et al. 2005).

The M05 models are significantly redder than the data at intermediate ages, owing to the different TP-AGB treatment in M05 (see also Pessev et al. 2008, who reach similar conclusions). The M05 colors also have an age dependence that does not agree with the data. The MC star cluster ages were estimated via CMD fitting using isochrones that include convective overshooting (Kerber et al. 2007; Pessev et al. 2008), whereas the M05 models do not include overshooting, and so one might expect some mismatch in age between the M05 models and data. However, the difference induced by the treatment of overshooting on the MC ages is ≲0.2 dex at ages 10⁸ yr ≲ t ≲ 10⁹ yr, whereas the discrepancy in the age dependence between M05 and the data is >0.5 dex in age. We therefore regard most of this discrepancy as a shortcoming of the M05 model, with only a fraction being due to different definitions of star cluster age.

The Persson et al. data are substantially redder than the newer data sets from Pessev et al. and Gonzalez–Lopezlira et al. As discussed in Pessev et al. (2006), this is probably due to the fact that the Persson photometry was probably centered on the brightest star in the star cluster, which are often carbon stars, and in any case are very red, and so the photometry will be biased red. The difference between Persson's data and the 2MASS-based data highlights the care that must be taken to construct data sets that can be meaningfully used for SPS model calibration.

The M05 models were calibrated against color–color plots of MC star clusters using the Persson et al. data and very wide age bins. Stochastic effects will tend to smear star clusters along an underling color–color relation because if a cluster is scattered redder in one color, it will be redder in another color as well. The M05 model calibration is therefore sensitive to stochastic effects, and is in addition plagued by the systematics in the Persson data. Moreover, the wide age bins used in M05 explains why the discrepancies in age seen for the M05 model in Figure 2 were not seen in their own calibrations.

The BC03 model and BaSTI isochrones both fare well except for H − K and J − K colors at intermediate ages (log(t/yr) ≈ 9.5), for which models are too blue by ≈0.1 mag compared to the data. This disagreement is probably due to the simplistic treatment of TP-AGB stars in these models. The BaSTI isochrones also produce redder colors at young ages compared to the Padova isochrones, which is probably due a lack of convective core overshooting in the BaSTI isochrones adopted herein (see Figure 1). The data do not definitively favor one model over another at log(t/yr) ≲ 8.0.

The disagreement in V − I between the models and data at intermediate ages is also noteworthy. The models perform better at old ages, although it is unclear if this is because of the old ages or the lower metallicities of the old star clusters. We will return to issues with V − I colors in later sections.

FSPS model predictions using the Padova isochrones are explored further in Figure 3. In this figure, we compare the unmodified Padova isochrones to the modified model, and we also show the effect of a variation in metallicity of ±0.2 dex about the modified model. For ages and colors this variation with metallicity does not bracket the modified model because some metallicity–color relations in the models are not monotonic and in fact some colors are reddest at approximately the average metallicity of the LMC (log(Z/Z_☉) ≈ −0.3). It is clear that use of the unmodified Padova models results in substantial disagreement with observed star clusters. This disagreement is driven to a significant extent by the large population of luminous carbon stars predicted in the Padova calculations. In our modifications to the Padova models, we have substantially lowered the bolometric luminosity along the TP-AGB phase, which has the effect of down-weighting the importance of all TP-AGB stars, including carbon stars.

Zoom In Zoom Out Reset image size

It is worth discussing in more detail the substantial disagreement between the unmodified Padova calculations and the star cluster data. The Padova group provided comparisons between their model and observations in Marigo et al. (2008) for V − K and J − K colors, finding acceptable agreement. The explanation for this difference is twofold.

First, the data used in their comparison are the old Persson data, which suffer from the issues discussed above. Second, and perhaps as important, the conversion from L_bol and T_eff to SEDs was achieved with the theoretical atmosphere models of Loidl et al. (2001) coupled to dust radiative transfer models from Groenewegen (2006). Aside from the fact that the dust models were never demonstrated to fit the V − K colors of AGB stars, the Loidl et al. (2001) carbon star models are substantially bluer (by >1 mag in V − K) than observed carbon stars (R. Gautschy 2009, private communication). We believe that our transformation to SEDs is more robust since we use observed TP-AGB spectra. Our qualitative conclusions are echoed by Gullieuszik et al. (2008) who compared the population of O-rich and carbon stars in the Leo II dwarf galaxy to the Padova models and found that the models predict substantially more, and substantially more luminous O-rich stars than are seen in that galaxy. Finally, the Padova group uses the carbon star luminosity function in the MCs as a constraint on their TP-AGB model, but this observable can only provide valuable insight if the SFHs of the MCs are known to high precision in the age range relevant for carbon star production (≈10^8.5–10^9.15 yr).

Figures 4 and 5 compare near-IR SBFs between several FSPS-based models and MC data. In Figure 4, the evolution of SBF magnitudes and SBF colors are compared to our FSPS model with both the BaSTI and Padova isochrones. We show results both for the unmodified and modified Padova isochrones. The BaSTI isochrones provide an acceptable fit to the K-band SBFs, but produce SBF colors that are somewhat too blue compared to the data. This implies that the AGB and TP-AGB temperatures are too hot in the BaSTI calculations.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The unmodified Padova results show similar discrepancies as seen in Figure 3. SBF magnitudes and colors are too bright and too red, which can be traced back to the overabundance of carbon stars and the overluminous TP-AGB evolutionary phase. As discussed earlier in Section 4.2.1, these problems are rectified by altering the overall luminosity and temperature of stars in the TP-AGB phase. These modifications result in SBF magnitudes and colors that are in better accord with the data, although K-band SBFs tend to be somewhat lower than the observations at intermediate ages.

In Figure 5, we compare models and data in K-band SBFs versus integrated V − I colors. In addition to MC star cluster data, we include data from elliptical galaxies (Liu et al. 2002; Jensen et al. 2003). It is clear that the modified Padova isochrones span the range of data much better than the unmodified isochrones. In addition, notice that there is a subsample of ellipticals that have SBFs fainter than predicted by the models. This has a direct and potentially important impact on the inferred mass-to-light ratios of these systems and consequently on their inferred stellar masses. SBFs in the red/near-IR will correlate inversely with derived stellar masses, and therefore if SBFs are overestimated in the models (as is the case here), then derived stellar masses will be underestimated.

The models shown in Figure 5 assume solar-scaled chemical compositions, while many elliptical galaxies are α-enhanced (e.g., Worthey et al. 1992; Thomas et al. 2005). However, Lee et al. (2009a) have shown that the level of α-enhancement has almost no effect on the model predictions in the V − I versus F160W SBF plane (F160W is similar to the H band), and therefore our conclusions should not be affected by these details.

SPS model predictions for the metallicity-dependent optical and near-IR colors of old populations are compared to observed star clusters in Figure 6. Model predictions are restricted to ages of 12.5 Gyr, although results are not sensitive to this particular age. FSPS models with both the BaSTI and modified Padova isochrones are included, as are the predictions from M05 and BC03. Comparisons to star clusters in both the MW and M31 are included. For all colors except for J − K the MW and M31 data agree (see below), although the scatter in M31 cluster colors is larger. This is probably due to the effects of internal reddening in M31 that has not been accounted for.

Zoom In Zoom Out Reset image size

The models reproduce the trends in the observed colors fairly well, although there are areas of noteworthy disagreement. The U − V and B − V colors are somewhat too blue in the models at log(Z/Z_☉)> − 1.0. The FSPS model with the BaSTI isochrones perform worst of the models considered. We can check whether or not this general disagreement between models and data in UBV filters is due to the adopted spectral library by considering the empirical Miles stellar spectral library. Due to the limited wavelength range of this library, we can only construct the B − V color. The B − V colors from the Miles empirical library are almost indistinguishable from the colors using the BaSeL library, implying that the stellar atmospheres are not the source of the discrepancy. A similar test was performed in BC03, where they compare the BaSeL, STELIB, and Pickles (1998) libraries and find negligible differences in B − V at Z = Z_☉.

A second possible explanation for the differences between models and observations in the U − V and B − V colors is the UBV filter transmission curves. The B filter used for these data actually consists of two slightly different filters (see the discussion in Maraston 2005). It is difficult to determine which filter was used in each observation. The difference between these two filters varies from 0.02 to 0.04 mag. The U − V color is constructed by adding the U − B and B − V colors, so this color will also be subject to this source of uncertainty. The offset between models and data is of order 0.05 mag—the same order as the differences induced by the two B filters—and so we cannot make stronger comparisons between models and data until more reliable optical photometry is available for MW and M31 star clusters.

The morphology of the HB cannot explain the observed discrepancy because in all SPS models shown in Figure 6 adopt a red HB at log(Z/Z_☉)> − 1.0. The HB therefore cannot be modified to produce even redder model colors.

The model V − I colors compare favorably with the data at log(Z/Z_☉)> − 1.0. At lower metallicities, the models diverge from the data, with the FSPS model using BaSTI isochrones performing worst.

All models fail at predicting the observed V − K and J − K colors, both at low and high metallicities for the latter and at higher metallicities for the former. Recall that the near-IR data come from a recent analysis of 2MASS data by Cohen et al. (2007). Our results echo those of Cohen et al. (2007) who compared MW star cluster J − K colors against several SPS models. In addition to the M05 model, they also compare to the models of Worthey (1994) and Buzzoni (1989) and find similar levels of disagreement with these other models.

Cohen et al. (2007) speculate that the cause of the discrepancy could be either (1) inadequate calibration of the stellar libraries or (2) issues with the near-IR filter transmission curves (Cohen et al. used published predictions for Johnson JHK photometry from these models). Since we compute 2MASS magnitudes self-consistently from the model spectra, the second explanation cannot be the source of the discrepancy. The problem is not likely to be due to inadequacies in the theoretical stellar spectral libraries, since the colors of giants compare well with observed stars in the near-IR (Westera et al. 2002). The temperature of the RGB need only be lowered by 200 K in order to achieve better agreement in V − K (although a lower RGB temperature would only exacerbate the disagreement in J − K). As explored in Section 4.3.2, one need only change the RGB at the brightest ∼1 mag to in order to produce substantial changes in V − K colors. Without definitively identifying the source of this problem, we are forced to simply reiterate the conclusions of Cohen et al. that near-IR colors of old populations cannot be used to constrain the metallicities or ages of either GCs or distant galaxies.

It should also be noted that the near-IR photometry presented in Cohen et al., which is based on data from the 2MASS survey, differs substantially from previous near-IR photometry of MW star clusters provided by M. Aaronson et al. (1977, unpublished). The differences in J − K are 0.1–0.2 mag in the sense that the latest results are bluer than the older data. Since previous models were calibrated against the older data (e.g., Maraston 2005), it is not surprising that the models are all redder than the Cohen et al. data.

The most dramatic discrepancy in Figure 6 is between the J − K colors in MW and M31 star clusters. The M31 clusters are approximately 0.2 mag redder in J − K than the MW clusters. Internal reddening in M31 cannot be the source of this discrepancy because reddening effects are quite small in J − K, especially in comparison to the colors considered in Figure 6, which do not show such an offset. Barmby et al. (2000) have also presented JHK photometry of old M31 star clusters. Their results are consistent with the M31 results from the RBC catalog shown in Figure 6, suggesting that the discrepancy in J − K is not due to data reduction issues. We cannot provide a compelling explanation for this discrepancy, but note that without an explanation it is impossible to calibrate the old metal-poor models to better than ∼0.2 mag in the near-IR.

Calibrating data at higher metallicities than shown in Figure 6 is far more desirable from the standpoint of using SPS to model galaxies since galaxies are generally of higher metallicity. Unfortunately, old metal-rich star clusters are rare. Classic metal-rich star clusters include NGC 6791, NGC 188, NGC 6553, and NGC 6528; the latter two belong to the bulge and thus are along sightlines of high extinction.

In Figure 7, we compare multi–color CMDs of NGC 6791 and NGC 188 to our SPS predictions. The consideration of multi-color CMDs provides a powerful constraint on SPS models in part because the reddening value cannot be arbitrarily varied to fit a color without potentially worsening the fit at another color. NGC 188 and 6791 are open clusters in crowded fields and therefore CMD construction is challenging without radial velocities and/or proper motions to isolate true members. This complication should be kept in mind when considering model comparisons below.

Zoom In Zoom Out Reset image size

NGC 6791 is assumed to have an age of 8.5 Gyr, E(B − V) = 0.17, [Fe/H] =+0.4, [α/Fe] =0.0, and m − M = 13.0 (Carney et al. 2005; Origlia et al. 2006; Kalirai et al. 2007). JK and BVI photometry are provided by Carney et al. (2005) and Stetson et al. (2003), respectively. The parameters for NGC 188 are taken to be E(B − V) = 0.09, [Fe/H] =−0.06, [α/Fe] =0.0, m − M = 11.1, and an age of 7 Gyr (Sarajedini et al. 1999; Carretta et al. 2009). UBVRI photometry of this star cluster are provided by Sarajedini et al. (1999); near-IR photometry was extracted directly from the 2MASS point source catalog.¹⁰

We use the BaSTI isochrones for the following comparisons because those isochrones have been computed for log(Z/Z_☉) = +0.3, which is comparable to the metallicity of NGC 6791 and is higher than the highest metallicities available from the Padova isochrones. Our model CMDs are compared to NGC 6791 and NGC 188 in Figure 7 for the same reddening values, ages, and distance moduli as quoted above, but for slightly difference metallicities. Metallicities of log(Z/Z_☉) = +0.3 for NGC 6791 and log(Z/Z_☉) = +0.0 for NGC 188 are adopted based on the available isochrones.

The agreement between data and models is generally good, including the location of the main-sequence turnoff, the sub-giant branch and RGB, and the position of the red HB.

The model fails to reproduce the low-mass end of the main sequence in NGC 6791 in the sense that the model isochrone is on the blue side of the data in B − V and V − K and on the red side in V − I. Of course, this disagreement is of little consequence for integrated properties since low-mass stars are faint. The disagreement at low masses is probably due to uncertainties associated with color–T_eff relations for low-mass stars at high metallicity (Cassisi et al. 2000; Cassisi 2007). It appears that the model RGB V − K color becomes progressively bluer than the RGB of NGC 6791 at higher luminosities, although it is difficult to make firm statements owing to the paucity of data at the bright end. Notice that this discrepancy is in the same sense as was seen for integrated V − K colors of metal-poor MW and M31 star clusters (see Section 4.3.1).

The model CMDs perform well when compared to NGC 188. The only obvious discrepancy is in the U − B color of the RGB. Thankfully, this is of little consequence for integrated colors because the integrated U − B color is determined primarily by the main-sequence turnoff point and sub-giant branch, not the RGB (see below).

For the purpose of modeling entire galaxies, we are interested in the ability of SPS models to accurately predict the integrated colors of star clusters. It is therefore worth considering the extent to which constraints from resolved CMDs can constrain integrated colors of star clusters.

In Figure 8, the integrated colors corresponding to the CMD of NGC 188 are explored. Integrated colors are computed by including all stars fainter than a given apparent V-band magnitude. It is clear from this figure that integrated blue colors such as U − B and B − V are well constrained even at relatively faint magnitudes in V. This occurs because integrated blue colors are determined primarily by the main-sequence turnoff point and sub-giant branch. In contrast, redder and near-IR colors are highly sensitive to the brightest one magnitude (11 < V < 12) for the simple reason that the RGB and AGB dominate the near-IR light. The brightest 1 mag in V contributes ≈20%–40% of the integrated V-band light (depending on whether or not one includes the red clump HB), and the V − I and V − K colors redden dramatically in this brightest magnitude. Thus, unless the brightest, and rarest stars in star clusters can be confidently identified and used for model calibration, model constraints in CMD space will not guarantee accurate model integrated colors.

Zoom In Zoom Out Reset image size

We now turn to an analysis of the spectral indices of old populations over a range of metallicities. Observed indices are measured from the catalog of optical spectra for 40 MW star clusters from Schiavon et al. (2005). The indices are measured in the exact same way for both the observed spectra and the models, ensuring fair comparison. The FSPS model spectra are at slightly higher resolution than the MW star cluster spectra, so the model spectra have been broadened to match the resolution of the data. The spectral resolution of the BC03 model is comparable to the observed spectra, and so the BC03 model spectra were not broadened.

Figure 9 presents the spectral indices D_n4000, Hδ_A, Fe5270, and Mgb as a function of metallicity for the Schiavon et al. sample and the FSPS and BC03 SPS models. The FSPS predictions utilize the Miles empirical stellar library, while BC03 adopt the STELIB empirical library. For these ages, the Padova predictions are very similar to the BaSTI isochrones except for Hδ_A at low metallicity because the BaSTI isochrones have a hotter HB. This figure also demonstrates the importance of BS stars and an extended/blue HB by including models with S_BS = 2 and f_BHB = 0.5 (see Section 3.1).

Zoom In Zoom Out Reset image size

There are several important conclusions to draw from this figure. First, the models are able to reproduce the observed metallicity dependence of the Mgb and Fe5270 indices over the full observed range. At low metallicities, this success is driven largely by the fact that the bulk of the metal-poor stars in the empirical stellar library contains α-enhanced stars (Milone et al. 2009). While the isochrones used in FSPS are solar-scaled,¹¹ recent work has shown that for old populations the effects of α-enhancement are strong on the stellar libraries and weak on the isochrones (Schiavon 2007; Lee et al. 2009c, 2009b).

The default SPS models (i.e., without BS or blue HB stars) predict D_n4000 in excess of observed values at solar metallicity, and the same models predict Hδ_A strengths too weak compared to the observations at both very low and solar metallicity. The disagreement at matching Hδ_A at very low metallicity is a consequence of the well-known fact that most low-metallicity star clusters have a blue HB (Piotto et al. 2002). Indeed, the models do a better job of matching the observed Hδ_A strengths at low metallicity for f_BHB>0.5. Of course, to match the data over the whole range in metallicity, the parameter f_BHB will need to be a complicated function of metallicity.

The failure of the default models for both D_n4000 and Hδ_A at solar metallicity is more striking, and has potentially significant consequences for understanding the spectra of metal-rich galaxies. The two solar metallicity star clusters are NGC 6553 and NGC 6528; both are in the bulge. The CMD of NGC 6553 shows an abundance of BS stars (Zoccali et al. 2001), and so the addition of such stars to the models in Figure 9 is justified. Indeed, adding BS stars results in much better agreement in Hδ_A and moderately better agreement for D_n4000. The addition of blue HB stars also alleviates much of the discrepancy between models and data. Lee et al. (2009b) compare their model to the same data set and find that for solar-scaled models the Hδ_A strengths are too weak compared to the observed Z ≈ Z_☉ clusters, echoing our results regarding our default SPS model.

Main-sequence turnoff stars are the dominant stellar phase controlling the strength of D_n4000. The effective temperature of turnoff stars would have to be increased by ∼1000 K in order to alleviate the discrepancy at solar metallicity. Such an increase would yield a CMD in gross conflict with observations. Modifications to the standard stellar phases thus cannot reconcile the data and model D_n4000 strengths. Either BS, blue HB stars or both are therefore the most likely candidates for explaining the discrepancy.

In the absence of evolved or young stars, the ultraviolet would be sensitive to the temperature of main-sequence turnoff stars. Since the temperature of turnoff stars depends on metallicity at fixed age, ultraviolet colors such as NUV − V and FUV − V will, in the absence of evolved or young stars, redden substantially with increasing metallicity.

In reality, the interpretation of ultraviolet colors of old populations is complicated by the existence of hot, evolved stars, including post-AGB, extended/blue HB, and BS stars (e.g., Burstein et al. 1988; Greggio & Renzini 1990; Dorman et al. 1995; O'Connell 1999; Brown et al. 2000; Han et al. 2007). It is beyond the scope of this paper to consider in detail the constraints on the relative importance of these various populations afforded by ultraviolet colors. Instead, our goal will be to demonstrate that plausible variations in the relative weights assigned to post-AGB and blue HB stars can explain the ultraviolet properties of observed star clusters.

Ultraviolet colors of star clusters in M31 (Rey et al. 2007), the MW (as compiled by Dorman et al. 1995), and M87 (Sohn et al. 2006) are compared to SPS models in Figure 10. The approximate location of elliptical galaxies from the SAURON survey (Jeong et al. 2009) is also included in the figure. We include model predictions from our FSPS code for both the Padova and BaSTI isochrones, along with model predictions from BC03 and M05. Recall that M05 does not include post-AGB stars and so the FUV photometry, which is sensitive to such hot stars, cannot be fruitfully compared to the data. M05 does include blue HB stars for metal-poor models. BC03 includes post-AGB stars but do not include blue HB stars at any metallicity.

Zoom In Zoom Out Reset image size

For the FSPS predictions with the Padova isochrones, we include several model variations. In particular, we consider models (1) without post-AGB and without blue HB stars, (2) with post-AGB but without blue HB stars, (3) without post-AGB stars and with a blue HB fraction of 10%, (4) with post-AGB stars and a blue HB fraction of 100%, and (5) with post-AGB stars and a blue HB fraction that varies with metallicity as f_BHB = −0.75 log(Z/Z_☉) − 0.5 for log(Z/Z_☉) < −0.7 and zero otherwise. In this last model, the trend of f_BHB with Z was chosen to match NUV − V over the full Z range and the FUV fluxes in the low-metallicity regime.

Before considering the insights gained from model comparisons, it is worth drawing attention to the considerable variation in observed ultraviolet colors of star clusters at fixed metallicity. NUV − V colors vary by ≈0.5 mag at fixed Z, while FUV − V varies by >1 mag. This phenomenon is known as the "second-parameter" effect, and has been known for decades (see the review in Catelan 2009). Parameters that might be responsible for the additional variation at fixed Z include age, helium content, mass loss, stellar rotation, and magnetic fields. The particular solution is not as important as the observed variation. It is difficult to imagine constraining an SPS model in the UV given these variations, and it will therefore continue to be very challenging to extract useful information from the UV flux of quiescent galaxies.

The locii of ultraviolet colors of elliptical galaxies do not correspond to star clusters of any metallicity in either M31, the MW, or M87. The NUV − V colors correspond to metal-rich star clusters, while the FUV − NUV colors are similar to metal-poor clusters, and the FUV − V colors of the elliptical sample are significantly redder than any star cluster. This conflict may be due in part to the fact that galaxies are not mono-metallic. The ultraviolet is quite sensitive to metallicity, and so accurate modeling of the metallicity distribution function is essential for these objects. These and related issues regarding stellar population modeling of ellipticals will be discussed in Sections 4.4 and 4.5.

It should also be noted that the star clusters in M87 are significantly bluer in ultraviolet colors at fixed metallicity compared to star clusters from the MW and M31. The source of this difference is not known, although differences in star cluster age or helium abundance are possible explanations (Sohn et al. 2006; Kaviraj et al. 2007). This difference in ultraviolet colors between M87 and MW/M31 star clusters persists over a wide range in V-band magnitude and therefore is not likely to be due to a star cluster mass-dependent process. From the preceding discussion, it is clear that even if models can be constructed that match the MW cluster data, such models may not be applicable to other systems.

Several conclusions can be drawn from the comparisons between models and data. First, a 10% blue HB fraction at all metallicities produces ultraviolet colors that are comparable to post-AGB stars, demonstrating the degeneracy between these two evolutionary phases when considering UV colors. The BaSTI isochrones and the Padova isochrones with variable blue HB fraction are both able to fit a majority of the metallicity-dependent ultraviolet data. At metallicities log(Z/Z_☉)> − 0.7 these models fail to reproduce the observed FUV fluxes, whereas models with 100% blue HB fractions can reproduce this data. Clearly, none of the models presented herein are capable of reproducing all of the metallicity-dependent UV colors. We speculate that modifying the distribution of temperatures along the blue HB may result in more satisfactory agreement.

When modeling the ultraviolet flux of galaxies, a flexible treatment of hot evolved stars is clearly warranted given our inadequate theoretical understanding of the observed ultraviolet colors of star clusters, and given the considerable scatter in observed ultraviolet data at fixed metallicity.

Figure 11 compares galaxies to models in color–color space. We include the FSPS model predictions with the modified Padova isochrones in addition to the BC03 and M05 models.

Zoom In Zoom Out Reset image size

All models predict similar ugr colors. One clearly sees the almost perfect degeneracy between age and metallicity in this color–color space. It is also clear that all models predict ugr colors that are substantially redder than the data, assuming that such galaxies are composed of old, metal-rich stars (e.g., Worthey et al. 1992; Trager et al. 2000; Thomas et al. 2005; Graves & Schiavon 2008). The discrepancy between observed and predicted g − r colors is ≳0.1 mag. Potential remedies to this discrepancy are explored in Section 4.5.

The models perform reasonably well in the near-IR, although there is a notable disagreement in the J − H color for all models. In the r − z versus H − K plane the FSPS and M05 models perform well, while BC03 fits the data only for solar or sub-solar metallicities. The BC03 and M05 models predict strong variations in Y − K with metallicity, in contrast to FSPS. Given the narrow distribution of observed Y − K colors, the BC03 models would predict nearly all red sequence galaxies to be solar metallicity, while M05 would predict that all were super-solar. In contrast, FSPS allows for a range of metallicities from sub- to super-solar, in better agreement with detailed analysis of the spectra of red sequence galaxies (e.g., Thomas et al. 2005).

Perhaps the most important conclusion to be reached from these color–color comparisons is that no model is able to reproduce the observed locus of red galaxies for all near-IR colors, and in addition no model can match the locus of galaxies in the u − g versus g − r plane.

It should also be noted that within the context of one model, optical and near-IR color combinations, in particular g − r versus Y − K, are in principle capable of breaking the age–metallicity degeneracy. In practice it is not currently possible to use such colors to break this degeneracy for the twofold reason that the models cannot predict the observed locus of galaxies in that space and the models do not compare well with each other (see also Eminian et al. 2008; Carter et al. 2009).

The red and near-IR colors of these galaxies are determined primarily by the RGB. As noted in Section 4.3.2, resolved CMDs do not help constrain model predictions for integrated red and near-IR colors because these colors are determined by the brightest and rarest RGB stars. Homogeneous, integrated near-IR photometry for a sample of old metal-rich star clusters would be invaluable for constraining models in this regime.

Figure 12 compares spectral indices of SDSS galaxies to the FSPS model constructed with the Padova isochrones and the Miles empirical stellar library. The data are split according to their velocity dispersions. All models in this section are broadened to σ = 200 km s⁻¹ in order to compare to the more massive data bin. Model predictions are shown for dust-free SSPs at Z = Z_☉ and Z = 1.5 Z_☉ for ages 2 Gyr < t < 13 Gyr.

Zoom In Zoom Out Reset image size

The index D_n4000, which measures the strength of the Ca ii lines and the Balmer decrement, is often taken to be sensitive primarily to age, and this is borne out in the figure. The majority of the other indices are sensitive to a combination of age and metallicity. The index C4668, which is a blend of many metallic lines including Fe, Ti, Cr, Mg, Ni, and C₂ (Worthey et al. 1994), is the most sensitive to metallicity of the indices shown, at fixed D_n4000. The index [MgFe]' was defined by Thomas et al. (2003) in order to be insensitive to α-enhancement. The G4300 index is sensitive to both CH and Fe i (Worthey et al. 1994). Hδ_A is often used to constrain the fraction of recent star formation because of its sensitivity to A stars (e.g., Kauffmann et al. 2003), and so the reliability of SPS models in the Hδ_A–D_n4000 plane is particularly important. For old populations, the interpretation of Hδ_A is complicated by the presence of nearby absorption due to the CN molecule. This implies that the Hδ_A index can be sensitive to non-solar abundance ratios, depending on the precise definition of the index (Prochaska et al. 2007). The trend is such that increasing α-enhancement will result in a weaker measured Hδ_A index because the flux redward of the line will be depressed due to increased CN absorption.

Figure 13 compares several SPS models to the 200  km s⁻¹ < σ < 300 km s⁻¹ SDSS sample. The BC03 model predictions are compared to the FSPS models using both the modified Padova and BaSTI isochrones with the Miles empirical stellar library. The M05 model is not included because the spectral library in that model (the Kurucz library) is not of sufficient resolution to measure these indices.

Zoom In Zoom Out Reset image size

The BaSTI and Padova isochrones have very similar main-sequence turnoff points at old ages (see Figure 1). The primary difference between these isochrones is that the BaSTI RGB is approximately 100 K hotter than the Padova RGB. Blueward of λ = 5000 Å the main-sequence and sub-giant branch dominate the flux, and so one would expect very little differences in spectral indices between these two isochrone sets in the blue, for a fixed set of empirical stellar spectra. Indeed, in Figure 13 there are almost no differences between the two isochrone sets for those indices that are measured at λ ≲ 5000 Å. The only index that shows appreciable differences is TiO₂, which is measured at λ ≈ 6230 Å, where the RGB contributes more to the integrated flux than the main-sequence turnoff point.

The BC03 and FSPS models compare favorably for the D_n4000, [MgFe]', G4300, and TiO₂ indices (except for the Z = 1.5 Z_☉ models for the latter index). The Z = 1.5 Z_☉ FSPS model predictions for C4668 compare more favorably with the data than the BC03 model of comparable metallicity, while the converse is true for the Fe4531 index. The differences between the BC03 and FSPS models are most dramatic in the Hδ_A–D_n4000 plane, where the BC03 Hδ_A EWs are larger than the FSPS predictions by ≈1.3 Å, and are in significant discord with the data. This discrepancy has been noted previously by Wild et al. (2007), and is most likely due to deficiencies in the STELIB empirical stellar library used by BC03.

One of the most striking results of these comparisons is that all models tend to overpredict the D_n4000 strength and underpredict the Hδ_A strength, especially for Z = 1.5 Z_☉. As can be gleaned from the symbols in the upper left panel of Figure 12, at Z = 1.5 Z_☉ the models reproduce the observed D_n4000 locus of massive galaxies for ages 3–6 Gyr, while for Z = Z_☉ the observations are reproduced for somewhat older ages of 6–10 Gyr. From a detailed analysis utilizing multiple spectral indices, Thomas et al. (2005) find most massive ellipticals to have solar to super-solar metallicities and ages >10 Gyr (see also, e.g., Bower et al. 1992; Bender et al. 1996; Ziegler et al. 1999; Graves & Schiavon 2008). In light of these results, an explanation is required for the discrepancy between observed and predicted D_n4000 and Hδ_A values. Possible explanations are discussed in the following section.

Before considering possible explanations, we point out that the proper inclusion of α-enhancement will only exacerbate the discrepancy between models and data in regards to the Hδ_A index. This occurs because α-enhancement will result in an even weaker Hδ_A index (Prochaska et al. 2007).

We find that the discrepancies are unlikely to be due to the stellar spectral libraries. We have considered both the theoretical BaSeL library and the empirical Miles library and find similar disagreements in the colors. The empirical STELIB library used in BC03 yields similar disagreements between both the g − r color and the D_n4000 strengths, further suggesting that the libraries are not at fault.

These results are at odds with Maraston et al. (2009) who showed that the optical spectra and g − r colors of SSPs at Z_☉ are markedly different when comparing models constructed using the BaSeL spectral library and the empirical Pickles (1998) library. The purported discrepancies caused by the spectral libraries cannot be reproduced with our FSPS code when using the empirical Miles library. To test that this was not due to differences between the Miles and Pickles library, we also compared spectra generated from the BC03 model using both the BaSeL and Pickles library and again find no substantial difference in g − r colors. We have also compared the spectra of individual stars from the Pickles, Miles, and BaSeL libraries for the same stellar parameters discussed by Maraston et al. and again cannot reproduce their results (i.e., their Figure 4).

We also point out that in the context of the explanation proposed by Maraston et al., the BaSeL library should provide a better fit to the data when the colors are k-corrected to z = 0.0 (as opposed to z = 0.1), because their discrepancy arises in a region of the spectrum sampled by the r band redshifted to z = 0.1. However, when we compare the colors of z ≲ 0.03 red sequence galaxies we find that the model discrepancies remain.

The discrepancy is also unlikely to be due to unresolved issues with canonical phases of stellar evolution such as the main sequence and RGB, because the aspects of stellar evolution relevant for ugr colors are adequately described by the models for old, metal-rich star clusters. This claim is based on two observations. First, from Figure 7 it is clear that FSPS can reproduce the main-sequence turnoff point and base of the RGB of old, metal-rich star clusters. This is especially the case for UBV colors of the clusters NGC 6791 and NGC 188. To the extent that there are residual discrepancies, the models tend to be somewhat too blue compared to the data (except for V − I for NGC 6791), which is in the opposite sense of the discrepancies noted for galaxies. Second, in Figure 8 it was shown that for blue colors such as B − V, one need not probe the brightest portions of the RGB in order to achieve convergence in the integrated color (the results for g − r are very similar to B − V). Therefore, integrated g − r colors of old metal-rich clusters should be accurate because the main-sequence turnoff and the base of the RGB are relatively well fit by the model in the CMD.

In sum, we believe that the discrepancies between model and data ugr colors and spectral indices are not a reflection of the limitations of either stellar atmospheres or canonical aspects of stellar evolution.

There are two remaining possible solutions to the disagreement between models and data for massive red sequence galaxies, which we lump into one class because they share the common property of being minority additions to the dominant, old metal-rich population. The first solution consists of modifications to the SSPs. This solution includes the addition of BS or blue HB stars.

The second solution to these disagreements lies not in trying to fix the SSPs but rather in relaxing the implicit assumption that these massive red galaxies are composed of mono-metallic, coeval stars. Indeed, estimates of the metallicities of stars in our own and nearby galaxies, either from direct spectroscopy or via the colors of giant branch stars, always yield a metallicity distribution function that is approximately lognormal with a full width of ≈1 dex (Rich 1990; Grillmair et al. 1996; Harris et al. 1999; Zoccali et al. 2003; Worthey et al. 2005; Sarajedini & Jablonka 2005; Koch et al. 2006). Models for the chemical evolution of galaxies also generically predict that galaxies should contain stars of a range of metallicities, including a non-trivial fraction of metal-poor stars (Searle & Sargent 1972; Pagel 1997).

Extreme/blue HBs exist in the solar and super-solar metallicity star clusters NGC 188 (Landsman et al. 1998) and NGC 6791 (Landsman et al. 1998; Kalirai et al. 2007), in addition to moderately sub-solar clusters (Rich et al. 1997). Resolved UV photometry in the nucleus of M32 has shown that ∼5% of HB stars are hot (Brown et al. 2000). Moreover, many metal-rich star clusters contain copious BS populations (e.g., Landsman et al. 1998; Zoccali et al. 2001; Kalirai et al. 2007). These observations suggest that it would be prudent to consider models that allow for blue/extended HBs and BS stars to be associated with metal-rich populations. In our treatment, we are agnostic as to which population (metal rich or metal poor) these hot stars are associated with.

These various effects are explored in Figures 14 and 15. In these figures, we explore the impact of various modifications to the base dust-free SSP models. Modifications include: (1) a small fraction (1%–3%) of young (t = 1.0 Gyr) stars; (2) a population of blue HB stars (f_BHB = 0.2); (3) a population of BS stars (S_BS = 2); and (4) a minority population (5%) of metal-poor stars.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

All of these modifications move the base SSP models in the correct qualitative sense; that is they result in lower values of D_n4000 and larger Hδ_A strengths, and bluer ugr colors. These trends result from the fact that each modification is adding a population of relatively hot stars, whether in the form of a hotter main-sequence turnoff (as happens when metal-poor stars are added), hot evolved stars including BS and blue HB stars, or hot young stars. Hotter stars have lower values of D_n4000, larger Balmer absorption, and bluer continua.

The discrepancies discussed above between SSP predictions and massive elliptical galaxies have been known, in one form or another, for decades (e.g., Faber 1977; Oconnell 1980; Gunn et al. 1981; Rose 1985; Schweizer et al. 1990; Schweizer & Seitzer 1992; Worthey et al. 1996; Maraston & Thomas 2000; Caldwell et al. 2003; Trager et al. 2005; Schiavon 2007). Our contribution to this issue has been twofold: (1) to highlight the connection between the indices and broadband ugr colors, in the sense that whatever hot star population is adopted, it should alleviate the discrepancies seen in both indices and colors; and (2) to remind the user of SPS models that reliable results concerning the ages and metallicities of massive ellipticals cannot be attained unless these minority hot star populations are carefully considered.

Many previous studies have considered one or more of the minority populations considered herein as a remedy to the model discrepancies. Schiavon (2007) disfavors appeals to blue HB stars based on the consideration of higher-order Balmer lines in ellipticals. However, this conclusion rests on the assumption that the temperature distribution along the HB in other galaxies is similar to that seen in the MW. While this may seem to be a relatively benign assumption, recent UV observations of star clusters in M87 cast doubt on this (Sohn et al. 2006). If the M87 result holds up to further scrutiny, it will mean that insights from the MW star cluster population cannot be readily applied to other galaxies. This would be a dramatic setback to our attempts at understanding the stellar populations in other galaxies, and would have potentially profound implications for our understanding of stellar evolution.

The addition of metal-poor stars is rarely considered without also considering the addition of blue HB stars. In this work, we have separated these two populations, since the observed relation between metallicity and HB morphology in the MW is complex and displays considerable scatter (Piotto et al. 2002). It should be remembered that in every galaxy where metallicity distribution functions have been measured, there exist metal-poor stars (log(Z/Z_☉) ≲ −1) in sufficient numbers (≈1%–5%; e.g., Harris et al. 1999; Zoccali et al. 2003; Sarajedini & Jablonka 2005) to effect the integrated blue/UV flux. Given this observational fact, metal-poor stars must be included in the models; the only question that remains is in what proportion.

It is something of a disappointment that the identification of the hot star population in ellipticals remains unresolved more than three decades after it was first discussed. Despite enormous advances in both the SPS models and the quality of the data, the fundamental problem remains that these various hot star populations occupy essentially the same (or similar) ranges in T_eff. We briefly comment on several possible future directions that might help separate the relative influence of these various populations.

If young stars or metal-poor stars are the dominant contributor, then because their metallicities will differ from the bulk population, metallicities measured in the blue/UV will differ from metallicities measured in the red. Models will have to be constructed to determine whether or not this effect is measurable. If it is then this could be a promising means of discriminating between the various hot star candidates.

If young stars (t ∼ 1 Gyr) are the culprit, then lookback studies of ellipticals will uncover ellipticals with even younger minority populations. Such a study would suffer from the typical difficulties in lookback studies, namely, that precise descendant–progenitor relations are difficult to construct. However, with new and upcoming wide-field spectroscopic surveys we believe that the difficulties associated with lookback studies can be mitigated. Lookback studies might also be able isolate the importance of blue HB stars (Atlee et al. 2009).

A final, somewhat more speculative possibility, is the use of SBFs in the blue to discriminate between these various populations. While these populations occupy approximately the same range of T_eff, they all occupy rather different ranges in L_bol, and they will thus produce different SBF magnitudes in the blue. We are currently constructing models aimed at determining whether or not such an effect is observable with current facilities.