2 Model ingredients

The basic model ingredients are identical to those described in Schaerer & Vacca (1998, hereafter SV98) and the Pop III models of Schaerer (2002a, henceforth S02). A brief summary is provided subsequently including the new features introduced in the present work.

2.1 Atmosphere models

Depending on the metallicity different sets of atmosphere models are used.

For metallicities $Z \le 10^{-5}$ we follow S02 in using the grid of plane parallel non-LTE atmospheres computed with the TLUSTY code of Hubeny & Lanz (1995) for $T_{{\rm eff}}$  $\ge$  20 000 K and the plane parallel line blanketed LTE models of Kurucz (1991) with a very metal-poor composition ([Fe/H] = -5.) otherwise. Possible uncertainties in the predicted ionising spectra of very metal-poor stars are discussed in S02. In comparison with the computations of S02, the use of an extended grid of TLUSTY models in the present paper leads to some small differences related to the highest energy range considered here.

For higher metallicities we follow SV98 and adopt for O stars the CoStar non-LTE models including stellar winds (Schaerer & de Koter 1997), for Wolf-Rayet (WR) stars the pure He spherically expanding non-LTE models of Schmutz et al. (1992), and Kurucz (1991) models of appropriate metallicity otherwise.

As discussed in earlier publications (e.g. Mihalas 1978; Schaerer & de Koter 1997; Schmutz et al. 1992; Tumlinson & Shull 2000; Kudritzki 2002) the inclusion of non-LTE models is the most important ingredient to obtain accurate predictions for the ionising spectra of massive stars.

   
2.2 Stellar tracks

To explore a wide range of metallicities covering populations from zero metallicity (Pop III), over low metallicities ( $Z \sim 4 \times 10^{-4}$) such as observed in H  II regions in the local Universe, up to solar metallicity (Z=  $Z_{\odot }$  =  0.02), we use the following stellar evolution tracks: 1) the Pop III tracks covering masses from 1 to 500 $M_{\odot }$ with no/negligible mass loss compiled in S02 (Marigo et al. 2001; Feijóo 1999; Desjacques 2000), 2) new main sequence stellar evolution tracks from 1 to 500 $M_{\odot }$ computed with the Geneva stellar evolution code for Z=10^-7, 3) non-rotating stellar models from 2 to 60 $M_{\odot }$ from Meynet & Maeder (2002) complemented with new calculations for 1 $M_{\odot }$ and 85-500 $M_{\odot }$ for Z=10^-5, 4) non-rotating Geneva stellar evolution tracks for masses 0.8-120 $M_{\odot }$ (up to 150 $M_{\odot }$ for Z=0.0004) from the compilation of Lejeune & Schaerer (2001) for the remaining metallicities Z=0.0004, 0.001, 0.004, 0.008, 0.02 (=$Z_{\odot }$), and 0.04. For massive stars we have adopted the high mass loss tracks in all cases, as these models reproduce best various observational constraints from the local Universe (cf. Maeder & Meynet 1994).

Our calculations at $Z \le 10^{-5}$ include only the H-burning phase. As He-burning is typically less than 10% of the main sequence lifetime, and is generally spent at cooler temperature, neglecting this phase should have little or no consequences on our predictions. A possible exception may be if stars at very low Z are very rapid rotators (cf. Maeder et al. 1999) which could suffer from non-negligible rotationally enhanced mass loss and could therefore become hot WR-like stars (cf. Sect. 6.2).

To verify our new computations we have compared several Z=10^-5 tracks with the independent calculations of Meynet & Maeder (2002) done with a strongly modified version of the Geneva stellar evolution code. Good agreement is found regarding the zero age main sequences (ZAMS), H-burning lifetimes, and the overall appearance of the tracks.

The HR-diagram showing the main sequence tracks at very low metallicity is given in Fig. 1. As expected one finds an important shift of the ZAMS and main sequence toward hotter $T_{{\rm eff}}$ with decreasing Z. For massive stars the trend shown by the low Z tracks is expected to continue down to a limiting metallicity $Z_{\rm lim}$ of the order of 10^-9, below which the stellar properties essentially converge to those of metal-free (Pop III) objects. This is the case, as massive stars ($M \ga 50$ $M_{\odot }$) with $0 \le Z < Z_{\rm lim}$ will rapidly build up a mass fraction $X_{\rm CNO} \sim 5 \times 10^{-10}$ to $2 \times 10^{-9}$ of CNO during pre main sequence contraction or the early main sequence phase (e.g. El Eid et al. 1983; Marigo et al. 2001). A typical value of $Z_{\rm lim} = 10^{-9}$ will be adopted subsequently for various simplified models fits.

Figure 1: HR-diagram showing the main sequence tracks of stars with masses $\protect\ga$3 to 500 (150 for Z=0.0004) $M_{\odot }$ of metallicities Z=0 (Pop III, dotted line), Z=10-7 (solid), Z=10-5 (long-dashed), and $Z=4 \times 10^{-4}$ (short-dashed) and their ZAMS. The source of the tracks is given in Sect. 2. Note the important shift of the ZAMS to high $T_{{\rm eff}}$ from low metallicity to Z=0.

2.3 Evolutionary synthesis models

As described in S02 the above stellar atmosphere models and evolutionary tracks have been included in the evolutionary synthesis code of Schaerer & Vacca (1998). Using the prescriptions summarised below we compute the predicted properties of integrated stellar populations at different metallicities for instantaneous bursts and constant star formation, the two limiting cases of star formation histories.

 

 

Table 1: Line emission coefficients $c_{\rm l}$ in [erg] for Case B, $n_{\rm e} = 100$ cm^-3 and the different adopted $T_{\rm e}$. See S98 and S02 for sources of the atomic data.

Line	$c_{\rm l}$	$c_{\rm l}$	appropriate $Q_{\rm i}$
	$T_{\rm e}=30$ kK	$T_{\rm e}=10$ kK
Ly$\alpha$	$1.04 \times 10^{-11}$	$1.04 \times 10^{-11}$	$Q({\rm H})$
He  II $\lambda$1640	$5.67 \times 10^{-12}$	$6.40 \times 10^{-12}$	$Q({\rm He^+})$
H$\alpha$	$1.21 \times 10^{-12}$	$1.37 \times 10^{-12}$	$Q({\rm H})$

2.3.1 Nebular emission

Among other predictions the code in particular computes the recombination line spectrum including Ly$\alpha$, He  II $\lambda$1640, He  II $\lambda$3203, He  I $\lambda$4026, He  I $\lambda$4471, He  II $\lambda$4686, H$\beta$, He  I $\lambda$5016, He  I $\lambda$5876, and H$\alpha$. Case B recombination is assumed for an electron temperature of $T_{\rm e}=30~000$ K at $Z \le 10^{-5}$ and $T_{\rm e}=10~000$ K otherwise, and a low electron density ( $n_{\rm e} = 100$ cm^-3). Ly$\alpha$ emission is computed assuming a fraction of 0.68 of photons converted to Ly$\alpha$ (Spitzer 1978). The line emission coefficients $c_{\rm l}$ (defined by Eq. (7)) of interest here are listed in Table 1.

Some differences, e.g. due to a more realistic temperature structure or due to collisional effects on H lines, can be expected between the adopted prescriptions and predictions from detailed photoionisation models (see e.g. Stasinska & Tylenda 1986; Stasinska & Schaerer 1999). However, for the scope of the present investigation these effects can quite safely be neglected.

As shown by S02 the inclusion of nebular continuous emission processes is crucial for very metal-poor objects with intense ongoing star formation. Following S02 we include free-free, free-bound, and H two-photon continuum emission assuming $T_{\rm e}=20~000$ K or 10 000 K and the above value of $n_{\rm e}$.

 

 

Table 2: Summary of IMF model parameters. All models assume a Salpeter slope for the IMF.

Model ID	$M_{\rm low}$	$M_{\rm up}$	$c_{\rm M}$
A	1	100	2.55
B	1	500	2.30
C	50	500	14.5

Figure 2: Predicted SEDs including Ly$\alpha$ and He  II $\lambda$1640 emission lines for zero age main sequence (ZAMS) models at different metallicities. Left panel: dependence of the SED on metallicity: the metallicities Z= 0. (Pop III), 10-7, 10-5, 0.0004, and 0.02 (solar) are from top to bottom in the EUV ( $\lambda < 912$ Å), and reversed at longer wavelengths. The dashed lines are the pure stellar emission, the solid lines show the total (stellar + nebular) emission. IMF A in all cases. Right panel: dependence of the SED on $M_{\rm up}$ for metallicities Z= 0. (Pop III), 10-7, and 10-5. Solid lines show the IMF A, dashed lines the IMF B. In all cases the total (stellar + nebular) emission is shown. Discussion in text.

2.3.2 Initial mass function

In view of our ignorance on the massive star IMF at very low metallicities ( $Z < 4 \times 10^{-4}$) we adopt as in S02 a powerlaw IMF and different upper and lower mass limits with the aim of assessing their impact on the properties of integrated stellar populations. The main cases modeled here are summarised in Table 2. For all models the IMF slope is taken as the Salpeter value ( $\alpha = 2.35$) between the lower and upper mass cut-off values $M_{\rm low}$ and $M_{\rm up}$ respectively.

The model A IMF is a good description of the IMF in observed starbursts (e.g. Leitherer 1998; Schaerer 2002b) down to $\sim$1/50 $Z_{\odot }$ (= $4 \times 10^{-4}$), the metallicity of I Zw 18 representing the most metal-poor galaxy known to date. It is adopted in all calculations for $Z \ge 4 \times 10^{-4}$. IMFs B and C, favouring the formation of very massive stars, could be representative of stellar populations at metallicities $Z \la Z_{\rm crit} \approx 10^{-5}$ (e.g. Bromm et al. 2001a), where altered fragmentation properties may form preferentially more massive stars (cf. Abel et al. 1998; Bromm et al. 1999; Nakamura & Umemura 2001). The computations at $Z \le 10^{-5}$ consider all IMF cases (A, B, and C). Note that our calculations obviously do not apply to cases where only one or few massive stars form within a pre-galactic halo, as suggested by the simulations of Abel et al. (2002).