Since the absolute SBF magnitude
in a given wavelength range
depends on the underlying stellar population properties, SBF
luminosities in different bandpasses probe different stages of the stellar
evolution within the unresolved stellar host system.
UV and blue SBF magnitudes are sensitive to the evolution of stars
within the hot Horizontal Branch (HB) and post-Asymptotic Giant Branch
(post-AGB) phase; optical SBF magnitudes serve as a measure of stars
within the RGB and the HB range, whereas NIR SBF luminosities represent
a stellar evolution stage within the AGB and the TP-AGB phase. Since the
SBF luminosity is weighted by the square of the stellar luminosity (see
equation 2), it is extremely sensitive to the most luminous giant stars.
Figure 7 illustrates the higher sensitivity of
the SBF signal to luminous cool giant stars in their late
stellar evolutionary stages compared to the total integrated stellar flux
signal. The stellar evolutionary phases are ordered according to their
relative contribution to the total integrated light and range from the
Zero-Age Main-Sequence (ZAMS) to the late AGB phase. For the computation
a 12-Gyr old, solar-metallicity, single-burst stellar population model
using the Padova evolutionary tracks with semi-empirical spectral energy
distributions (SEDs) was adopted
(Marigo et al. 2008).
It can be seen that the integrated flux arises from stars throughout all
stellar evolutionary phases. In contrast, the IR SBF signal originates
entirely from the RGB and AGB phase (∼ 88% of the flux is within
the two brightest IR magnitudes), whereas for optical SBFs some
additional contribution comes from stars along the HB (∼ 60% of
the signal is concentrated within the 3-4 brightest optical magnitudes).
 |
Figure 7. Origin of the SBF signal. For an old, solar-metallicity, single-burst stellar population model (see text) the fraction of the total light is given as a function of stellar evolutionary phase. Evolutionary phases range from the ZAMS (left) to the late AGB phase (right) in arbitrary units. The integrated flux (dotted line) arises from all stars of all stellar evolutionary phases. In contrast, the IR SBF signal (solid line, top panel) originates entirely from the RGB and AGB phase, with some additional contribution from the HB for optical SBFs (solid line, lower panel). |
From Figure 7 it becomes evident that the SBF luminosities and colours should closely trace the peak luminosity and colours of late cool stars along the giant branch. Therefore, in the following section we give a brief summary of the main evolutionary phases of cool giant stars, the RGB phase (H-shell burning) and the AGB phase (He-shell burning).
5.2. The Late Stages of Cool Giant Stars
Low- and intermediate-mass stars of the main-sequence (1 ≤ M / M⊙ ≤ 8) experience high luminosities as red giants in their late evolutionary phase along the AGB (up to a few 104 L / L⊙). However, due to their low effective temperatures (Teff ≤ 3500 K), these AGB stars are highly extended objects with radii and cool atmospheres of several 100 R / R⊙. Depending on the initial ZAMS mass of the star, the hydrogen (H) in the stellar core will be exhausted, and subsequently the Helium (He) core will rapidly contract and heaten up, while the H will continue to burn in a shell around the nucleus. This stage marks the start of the RGB phase (H-shell burning). The beginning of the AGB phase is characterised by the end of He-burning in the stellar core. In the first phase, the He-burning continues in a shell around the core (early AGB). On the upper part of the AGB, the star becomes unstable because of strong radial pulsations with typical time scales of a few tens to hundreds of days (e.g., Iben & Renzini 1983; Vassiliadis & Wood 1993; Dorfi & Hoefner 1998). The star enters the TP-AGB phase of alternate H- and He-shell burning. In this phase, the AGB star shows thermal pulses: The H-shell burning gets interrupted at regular times by the ignition of the He that has accumulated under the H-shell beforehand as a byproduct of H-burning. Further, the variable convection zones mix material processed in the nucleus to the stellar surface, thereby changing the chemical composition of the convective layers, i.e. dredge-up phases (Iben & Renzini 1983). Heavier elements (mainly carbon, nitrogen, oxygen, and s-process elements) are ultimately transported outwards into the stellar photosphere and modify the stellar surface composition. The relative fraction of carbon-to-oxygen ([C/O]-ratio) is defined by the core mass and this mixing ratio regulates the generation of different molecules in the cool stellar atmosphere (e.g., TiO, C2, CN). Most frequent are oxygen-rich AGB stars (M-star: [C/O] < 0.95), but sometimes a carbon-rich star ([C/O] > 1.0) is formed, which eventually develops into a central white dwarf star within a planetary nebula.
Since the SBF signal is more sensitive to the most luminous stars in a stellar system, the SBFs provide tighter constraints on the evolution of evolved cool giant stars in elliptical galaxies than integrated luminosities alone. However, several aspects of the physics of cool giant stars (e.g., interior structure, mass-loss, mixing length for internal convection, pulsation modes), as well as their exact spectral energy distributions, remain poorly understood. Ideal observational targets would be nearby GCs, which host these stellar populations of homogenous composition and age. However, because cool giants cross their late evolutionary phase rapidly, only a handful of these stars exist at the present time for a given cluster. To overcome uncertainties in GC analyses such as small number statistics or cosmic variance, SBF measurements of individual galaxies can represent an alternative approach, as the SBF signal is dominated by much higher luminosities and the stellar systems are expected to comprise metal-rich populations.
Since a few years, more attention has been dedicated in observing young intermediate-age stellar populations (0.5 ≲ Gyr ≲3) in GC and stellar clusters in the MCs (see e.g. González-Lópezlira et al. 2005; Raimondo et al. 2005; Mouhcine et al. 2005; Lee et al. 2010). The brightest cluster stars are carbon-type AGB stars with bolometric luminosities that exceed the brightness of the tip of the RGB. In contrast to M-type AGB stars, carbon AGB stars are more luminous and also redder.
Therefore, IR SBFs represent a promising tool to disentangle the effects and contributions of ages and metallicities in unresolved stellar systems. In particular, mid-infrared (MIR) SBFs in the L (3.5 µm) and M-band (4.8 µm) are expected to be highly sensitive tracers of the age of a stellar population. Because of the extremely high thermal IR background of the earth’s atmosphere, however, observations from the ground are only possible for the very nearest and bright galaxies (e.g., M31). In the foreseeable future, a new window on MIR SBFs will be opened with the James Webb Space Telescope (JWST), which will access both the L- and M-bands.
5.3. Improved Stellar Population Models
Today the whole extragalactic community is aware that stars in their TP-AGB phase are responsible for more than half of the NIR flux that originates from a stellar population at intermediate ages (several 100 Myr to 1-2 Gyr). In some cases, the TP-AGB contribution may even increase to 80%, depending on the range of ages and metallicities adopted in the modelling (Maraston 2005). As a consequence, AGB stars are the dominant contributors to the NIR mass-to-light ratio of intermediate-age stellar populations (see Section 5.2), which are particularly sensitive for stellar mass estimates of high-redshift galaxies at z ∼ 2 (Maraston et al. 2006).
Over the past few years, significant progress has been made in our theoretical understanding of stellar populations (Biscardi et al. 2008; Raimondo 2009; Conroy et al. 2009; Lee et al. 2010; González-Lópezlira et al. 2010; Conroy & Gunn 2010). Substantial improvements were obtained regarding the uncertain and unknown late stages in the evolution of massive, evolved stars, in particular the (post)-AGB phase and the horizontal branch phase. A full discussion is beyond the scope of this review. However, one of the most promising techniques using a flexible synthesis of stellar populations, which has been verified through detailed comparisons with high-quality observations, is presented in the following.
Flexible Stellar Population Synthesis (FSPS) descriptions take a novel approach in the construction of synthetic SSP spectra (Conroy et al. 2009; Conroy & Gunn 2010). Through a parameterisation of uncertain stages in the stellar evolution and by allowing these variables to alter freely, both the theoretically motivated galaxy properties as well as different sets of observational constraints can be modelled in combination. In particular, FSPS allows researchers to compute SSPs for a range of Initial Mass Functions (IMFs) and metallicities and for a variety of assumptions regarding the morphology of the horizontal branch, the blue straggler population, the post-AGB phase, and the location in the Hertzsprung-Russell diagram of the TP-AGB phase. From these SSPs, composite stellar populations (CSPs) for a variety of star formation histories (SFHs) and dust attenuation prescriptions can be generated. Furthermore, the descriptions offer different choices of isochrones and stellar libraries and provide weights to various stages of the stellar evolution.
Conroy et al. point out that the unmodified Padova models result in a substantial disagreement with the observed properties of stellar clusters. This disagreement is amplified to a significant extent by a large population of luminous carbon stars as predicted by the models. Therefore, the bolometric luminosity (Lbol) along the TP-AGB phase was substantially lowered by accounting for shifts in the effective temperature Teff (ΔT) and Lbol (ΔL) along the entire TP-AGB phase (Conroy et al. 2009). These applied changes are independent of metallicity and reduce the importance and dominance (both the overall number and higher number of more luminous O-rich stars) for all TP-AGB stars, including carbon stars. The resulting modified Padova isochrones are in much better agreement with both MC star clusters and post-starburst galaxies (see Figure 8). The substantial disagreement between the unmodified Padova calculations and the star cluster data arises from three differences: the usage of old inferior-quality data, the application of theoretical atmosphere models with dust radiative transfer descriptions (Loidl et al. 2001), and the application of the LF of carbon stars found in the MCs. In particular, the carbon star models are significantly bluer (> 1 mag in V − K) than observed carbon stars (R. Gautschy 2009, private communication). Further, the new models by Conroy & Gunn (2010) adopt observed stellar spectra including the TP-AGB phase, which provide a better description of the observed properties of O-rich and carbon stars, rather than synthetic modelled spectra.
 |
Figure 8. K-band SBF fluctuation magnitude measurements and FSPS models as a function of integrated (V − I)0 colour. Different FSPS model predictions (Conroy & Gunn 2010, lines) are compared to observations of elliptical galaxies in the Fornax cluster (Liu et al. 2002, green squares), ellipticals in a variety of environments (Jensen et al. 2003, blue circles), and star clusters in the MCs (González-Lópezlira et al. 2005, solid black circles). For clarity reasons, uncertainties are displayed only for the Fornax galaxies and MC clusters. Model predictions are shown as lines of constant age (solid lines; in units of log(t/yrs)) and lines of constant metallicity (dashed lines; in units of log(Z / Z⊙)). Arrows denote the direction of 0.02 and 0.05 dex variations in age and metallicity, respectively. FSPS models are constructed with the modified Padova isochrones (left panel) and the BaSTI isochrones (right panel). The unmodified Padova isochrones used in the past give only a poor approximation of half of the data. Large metallicity variations of Δlog(Z / Z⊙) ∼ 0.60 dex can be seen, depending on the treatment of the convection (convective core overshooting) and the TP-AGB phase in the stellar evolution. |
In Figure 8, a comparison between FSPS models with both the Bag of Stellar Tracks and Isochrones (BaSTI) and modified Padova isochrones (Conroy & Gunn 2010) and NIR SBF measurements within the K-band versus integrated (V − I)0 colour plane is given. Further, data for star clusters in the MCs (González-Lópezlira et al. 2005) and SBF data for elliptical galaxies (Liu et al. 2002; Jensen et al. 2003) are included. The left-hand panel shows the results for the modified Padova isochrones, whereas the right-hand panel displays the findings for the modified FSPS+BaSTI models. It is clearly evident that the new modified FSPS+BaSTI prescriptions define a much broader range in the magnitude-colour plane than the unmodified predictions. In particular, the range of constant ages is more pronounced in the FSPS predictions and extends to the far blue colours of stellar MC clusters. Nevertheless, a subsample of galaxies comprises fainter SBF magnitudes and therefore is still poorly represented in both the BaSTI and Padova calculations. Because of the inverse relationship between NIR SBF luminosities and (stellar) masses, if SBF luminosities are overestimated as in the current model predictions then the derived stellar masses and mass-to-light ratios will be underestimated.
Many observations of elliptical galaxies provide strong evidence for an α-enhancement of heavy elements in their stellar populations (e.g., Worthey et al. 1992; Ziegler et al. 2005). For the theoretical prescriptions in Figure 8, solar-scaled chemical compositions were assumed. However, recent models including an accurate description of the (TP)-AGB-phase suggest that the level of α-enhancement has a minor effect on the model predictions in the (V − I) versus F160W SBF coordinate space (Lee et al. 2010). Nevertheless, the impact of other enrichments on the SBF luminosities and galaxy colours has not yet been explored in detail.
Overall, the new modifications and prescriptions such as FSPS offer a much better representation of the K-band SBF luminosities and colours, thereby addressing several limitations in the unmodified models (Conroy & Gunn 2010): BaSTI isochrones predict too-blue SBF colours compared to the observations (implying too-hot AGB and TP-AGB temperatures), whereas the Padova isochrones provide too-bright and too-red SBF magnitudes and colours (which originate from an overabundance of carbon stars and an overluminous TP-AGB evolutionary phase).
So far, the current status of the theoretical modelling of SBF quantities provides some constraints on the average metallicities of unresolved stellar populations. For the future, a better understanding of the whole range of stellar evolution phases is crucial to gain detailed insights into the stellar population properties of globular clusters and galaxies.