6 The Q(He⁺)/Q(H) hardness of starbursts at various metallicities

6.1 Predictions at metallicities Z = 0 to Z$_\odot$

6.1.1 Burst models

The time evolution of the predicted hardness $Q({\rm He^+})/Q({\rm H})$ is shown in the lower panel of Fig. 3. The corresponding hardness reached at equilibrium in the case of constant star formation (cf. below), plotted on the right, illustrates the non negligible difference with the hardness derived from a simple ZAMS population neglecting stellar evolution effects (cf. S02).

Note that predicted quantities such as $Q({\rm He^+})/Q({\rm H})$ rely obviously strongly on the adopted value of $M_{\rm up}$. For very low Z this quantitative dependence can be estimated from the tabulated ZAMS properties (Table 3).

As expected from the strong decrease of the stellar temperatures with increasing metallicity (cf. Fig. 1) both the maximum hardness (at age  =  0) and the overall $Q({\rm He^+})/Q({\rm H})$ decreases for Z between 0 and 10^-5. The typical timescale for a decrease of $Q({\rm He^+})/Q({\rm H})$ by $\sim$2 dex in a burst is driven by the redward stellar evolution, and is short ($\sim$2-3 Myr), with obvious potential implications for the detection of sources with very hard spectra (cf. Sect. 7). Possible additional sources of He⁺ ionising photons not included here are discussed in Sect. 6.2.

At higher metallicities ( $Z \ga 4 \times 10^{-4}$) the present models predict a re-increase of $Q({\rm He^+})/Q({\rm H})$ after $\ga$3-4 Myr, due to presence of WR stars, among which a fraction is found at high temperatures (cf. Schmutz et al. 1992; SV98). Albeit with minor quantitative differences, a qualitatively similar behaviour is predicted by the Starburst99 models based on very similar input physics (Leitherer et al. 1999). However, these predictions depend especially on the procedure adopted to link stellar tracks with atmospheres in WR phases with strong winds, and on the neglect of line blanketing in the adopted WR model atmospheres. The reality and the extent of such a trend remains therefore questionable, especially at the largest metallicities (cf. review of Schaerer 2000). Indeed using recent line blanketed O and WR atmospheres and different prescriptions to connect the interior and atmosphere models Smith et al. (2002) find a considerably softer spectrum - i.e. reduced $Q({\rm He^+})/Q({\rm H})$ - before and during the WR phase. To circumvent this theoretical uncertainty we will subsequently derive an empirical estimate of the hardness at metallicities $Z \ga 4 \times 10^{-4}$ (Sect. 6.3).

Figure 5: Hardness $Q({\rm He^+})/Q({\rm H})$ of the He+ ionising flux for constant star formation as a function of metallicity (in mass fraction) for all models given in Table 4. At metallicities above $Z \ge 4 \times 10^{-4}$ the predictions from our models (crosses), as well as those of Leitherer et al. (1999, open circles), and Smith et al. (2002, open triangles) are plotted. The shaded area and the upper limit (at higher Z) indicates the range of the empirical hardness estimated from H  II region observations (see Sect. 6.3). At very low Z solid lines show the fits to the data given by Eq. (2). Discussion in text.

Figure 6: Additional contribution $Q({\rm He^+})$$^\star /$ $Q({\rm H})$ of putative hot WR-like stars to the total He+ hardness of the ionising flux $Q({\rm He^+})$/ $Q({\rm H})$ (cf. Eq. (6)) for SFR  =  const. as a function of the minimum mass $M_{\rm WR}$. Shown are the cases for $t_{\rm WR}/t_\star = 10$ and 50% (solid and dashed curves), temperatures T= 100 and 160 kK (lower and upper curve respectively), and $M_{\rm up}$  =  500 (100) $M_{\odot }$ extending over the full plot (lower left only).

6.1.2 Constant star formation

For the case of constant star formation at equilibrium the metallicity dependence of the hardness $Q({\rm He^+})/Q({\rm H})$ of the ionising flux is shown in Fig. 5 for all the IMFs considered. As apparent, for the very metal-poor cases ( $Z \la 10^{-4}$) $Q({\rm He^+})/Q({\rm H})$ can be well fitted by

$$\log(Q({\rm He^+})/Q({\rm H})) = a \times \log(Z) + b$$ (2)

with the numerical coefficients listed in Table 5, again taking into account that all metallicities $Z \le Z_{\rm lim} \approx 10^{-9}$ are equivalent.

At higher metallicities - in the Z range of known objects - the theoretical predictions for $Q({\rm He^+})/Q({\rm H})$ are probably less clear, due to possible presence of hot WR stars, difficulties in their modeling (cf. above), and the neglect of non-stellar emission processes which could contribute especially to $Q({\rm He^+})$. Indeed as shown in Fig. 5 rather important differences are obtained between various evolutionary synthesis models (SV98, Starburst99 of Leitherer et al. 1999, and the latest computations of Smith et al. 2002). In the metallicity range $Z \sim 8 \times 10^{-4}$ to $5 \times 10^{-3}$ our empirical estimate of $Q({\rm He^+})/Q({\rm H})$ (Sect. 6.3), also shown in this figure, is likely more reliable than the models. At still larger Z the average ionising spectrum of starbursts should be softer, as indicated by the tentative empirical upper limit and predicted by the Smith et al. (2002) models.

   
6.2 Uncertainties on the He⁺ ionising flux at very low metallicities ( $\mathsfsl{Z \protect\la 10^{-4}}$)

In contrast to the Lyman continuum flux $Q({\rm H})$, the He⁺ ionising flux (and in general spectral features at high energy) show a very strong dependence on the stellar temperature in the $T_{{\rm eff}}$ range $\sim$50-100 kK typical of very low metallicity stars (Fig. 2 in S02). Therefore their prediction is naturally more sensitive to even small modifications of the exact stellar $T_{{\rm eff}}$ or evolutionary scenario.

For example, one may wonder how reliable the above predictions of the metallicity dependence of $Q({\rm He^+})/Q({\rm H})$ (Fig. 5) are at $Z \la 10^{-4}$, where presently no observational constraints are available. In fact, studies of massive stars in the Local Group suggest that their average rotation rates increase towards low Z(Maeder et al. 1999), which - when combined with their increased compactness - can lead to non-negligible mass loss despite the low metallicity (Maeder & Meynet 2000; Meynet & Maeder 2002). If this effect is large enough, one could imagine that fast rotators could loose sufficient mass to follow a WR star like evolution leading possibly to a He/C/O core at temperatures $T \ga 100$ kK, a scenario known for metal-rich massive WR (e.g. Maeder & Meynet 1988). Despite high rotational velocities, the detailed calculations of Meynet & Maeder (2002) for Z=10^-5 do not show important alterations of the evolution for stars with $M \le 60$ $M_{\odot }$. Exploratory calculations of Marigo et al. (2002) for Pop III stars treating in an simplified manner the effects of rotation on stellar mass loss find such a scenario for stars with initial mass $\ga$750 $M_{\odot }$.

Quantitatively the effect of such a putative hot "WR-like'' population on the hardness of the ionising flux can be estimated for the case of constant star formation only in the following way. Suppose that stars of given initial mass $M \ge M_{\rm WR}$ spend this phase at constant luminosity L and (hot) temperature Tduring a constant fraction $f_{\rm WR}=(t_{\rm WR}/t_\star)$ of their lifetime $t_\star$. Assuming their winds are optically thin at $\ge$54 eV, the He⁺ flux in this phase is then

$$Q({\rm He^+})^\star(M) = \frac{L(M)}{\sigma T^4} q({\rm He^+})^\star(T),$$ (3)

where $q({\rm He^+})$$^\star$ is the photon flux per unit surface area (cf. Fig. 2 in S02). The hardness ( $Q({\rm He^+})$/ $Q({\rm H})$$)^\prime$ of a population including both the "normal'' (main sequence) population and the additional hot population can be approximated by (for SFR  =  const.)

$$\left(\frac{Q({\rm He^+})}{Q({\rm H})}\right)^\prime \approx ... ...R}\right) + \frac{Q({\rm He^+})^\star}{Q({\rm H})} f_{\rm WR},$$ (4)

assuming small changes in the total Lyman continuum photon output $Q({\rm H})$. Here the first term stands for the unperturbed "normal'' hardness shown in Fig. 5. The second term describes the additional contribution due to hot WR-like objects, which can be expressed as

$$\frac{Q({\rm He^+})^\star}{Q({\rm H})} = \frac{\int_{M_{\rm... ...rm up}} t_\star(M) \overline{Q}({\rm H})(M) \Phi(M) {\rm d}M},$$ (5)

where Q have(M) is the lifetime average of the Lyman continuum production, and $\Phi(M)$ is the IMF. The quantities $t_\star(M)$ and $\overline{Q}({\rm H})$(M) are taken from S02, and we assume as first approximation a luminosity L(M) corresponding to the ZAMS. The hardness contribution $Q({\rm He^+})$$^\star /$ $Q({\rm H})$ is then computed assuming temperatures T= 100-160 kK as expected from stellar evolution models, and durations $f_{\rm WR}=(t_{\rm WR}/t_\star)$ of 0.1 or 0.5. The result is plotted for the IMFs A and B ( $M_{\rm up}$  =  100 and 500 $M_{\odot }$) in Fig. 6 as a function of the minimum mass $M_{\rm WR}$, above which all stars are assumed to reach this hot "WR'' phase.

The case of $(t_{\rm WR}/t_\star)= 50$%, which would require very strong mass loss already during the main sequence or a nearly homogeneous evolution leading early to a blueward evolution, appears extremely unlikely and is shown here to mimic the "strong mass loss'' models adopted in the Pop III models of S02. For a hot phase of a duration typical of the post main sequence evolution ($\sim$10% of total lifetime) Fig. 6 shows that such putative "hot WR'' could in the "best'' case contribute an additional hardness $Q({\rm He^+})$$^\star /$ $Q({\rm H})$ of the order of $10^{-2 \ldots -3}$, comparable to the hardness of normal stellar populations with metallicities $Z \la 10^{-4}$ (cf. Fig. 5). To examine how realistic such cases may be, will require a detailed understanding of the coupled processes of stellar mass loss, rotation, and internal mixing. At present the available limits are $M_{\rm WR} > 60$ $M_{\odot }$ at Z= 10^-5 and $(t_{\rm WR}/t_\star) \la 10$% from the rotating stellar models of Meynet & Maeder (2002), and $M_{\rm WR} \ga 750$ $M_{\odot }$ at Z=0 from the simplified models of Marigo et al. (2002).

Although the above exercise shows that at very low metallicity ( $Z \la 10^{-5}$) the hardness $Q({\rm He^+})/Q({\rm H})$ due to stellar sources could be higher than shown in Fig. 5, it seems that such scenarios are quite unlikely. If star formation takes place on much longer time scales, and massive stars would not form (or in much smaller quantities), hot planetary nebulae could also be a source of hard ionising photons, as illustrated by the scenario of Shioya et al. (2002). In any case, a major uncertainty stems from our limited knowledge of the IMF at very low metallicities.

 

 

Table 5: Fit coefficients for Eq. (2).

IMF	a	b
A	-0.66 $\pm$ 0.071	-8.22 $\pm$ 0.51
B	-0.37 $\pm$ 0.028	-5.04 $\pm$ 0.20
C	-0.39 $\pm$ 0.028	-4.98 $\pm$ 0.20

   
6.3 Empirical constraints on the He⁺ ionising flux of starbursts at Z $\protect\ga$ 4 $\times$ 10 $^\mathsfsl{-4}$

Spectroscopic observations of extra-galactic giant H  II regions probing He  II recombinations lines can yield empirical information on the "average'' hardness $Q({\rm He^+})/Q({\rm H})$ of starbursts. Indeed it is well known that a fairly large fraction of metal-poor H  II regions show the presence of nebular He  II $\lambda$4686 emission indicative of a hard ionising spectrum (see e.g. Guseva et al. 2000; compilation of Schaerer et al. 1999). A complete explanation of the origin of the required high energy photons (shocks, X-rays, WR stars) remains to be found (e.g. Garnett et al. 1991; Schaerer 1996, 1998; Guseva et al. 2000; Izotov et al. 2001; Stasinska & Izotov 2002).

The largest sample of high quality data is that of Izotov and collaborators (cf. Guseva et al. 2000 and references therein), which shows He  II $\lambda$4686 detections with typical relative intensities of I(4686)/I(H$\beta$$) \sim 1$-2%.

From such a sample we may estimate an average hardness from

$$\frac{Q({\rm He^+})}{Q({\rm H})} = \frac{I(4686)}{I({\rm H}\b... ... f_{\rm detect} \times \frac{t_{\rm detect}}{t_{\rm total}},$$ (6)

where a (=0.47 (0.63) for $T_{\rm e}=10~000$ (30 000) K) translates the relative line emissivities, $f_{\rm detect}$ is the fraction of objects showing nebular He  II among the spectra of sufficient quality to allow detections down to this intensity level, $t_{\rm detect}$ is the age range covered by the observed H  II regions, and $t_{\rm total}$ is the total lifetime of H  II regions. Inspection of the data of Izotov yields $f_{\rm detect} \sim 1/3$-1/2 for metallicities $Z \in [8 \times 10^{-4}, 4.8 \times 10^{-3}]$ and possibly even a larger fraction at the lowest observed metallicities. (Stasinska & Izotov 2002). One has $t_{\rm total} \sim 10$ Myr, the typical H  II region lifetime or the time over which bursts show detectable line emission. However, it is now well established that H  II region samples suffer from selection biases leading to an absence of advanced bursts, or in other words a preferential selection of bursts with ages younger than $\la$4-5 Myr (e.g. Bresolin et al. 1999; Raimann et al. 2000; Moy et al. 2001; Stasinska et al. 2001). For this range of values and a=0.47 one obtains an estimate of $\log ($ $Q({\rm He^+})/Q({\rm H})$ $) \sim -3.2$ to -2.6. Postulating the extreme case that additionally the first 2 Myr of the H  II phase are also not detected in the optical one has a minimum value of $t_{\rm detect} \sim 1$ Myr, yielding a lower limit of $\log ($ $Q({\rm He^+})/Q({\rm H})$ $) \ga -3.5$. In principle this average value could be even larger as the (conservative) assumption of $t_{\rm detect}/t_{\rm total} < 1$implies that older (non-detected) H  II regions have no He⁺ ionising flux.

This estimate is obviously independent of the nature of the hard (He⁺) ionising radiation. By construction Eq. (6) provides an estimate of the average $Q({\rm He^+})/Q({\rm H})$ expected in objects with constant ongoing star formation for metallicities $Z \sim 8 \times 10^{-4}$ to $5 \times 10^{-3}$. The near absence of nebular He  II detections in H  II regions at higher metallicity (cf. Schaerer 1998; Guseva et al. 2000) indicates softer spectra. However, it is difficult to establish a firm upper limit on $Q({\rm He^+})/Q({\rm H})$ for $Z \ga 5 \times 10^{-3}$. We here retain $\log ($ $Q({\rm He^+})/Q({\rm H})$ $) \ll -3.3$ as a tentative limit.

   
6.4 Overall behaviour of Q(He⁺)/Q(H) with metallicity

In short, from the considerations above, we find the following two cases for the most plausible metallicity dependence of the average $Q({\rm He^+})/Q({\rm H})$ hardness ratio of starbursts with metallicity (see Fig. 5).

1) If a universal Salpeter like IMF with a "normal'' upper mass limit of $\sim$100 $M_{\odot }$ prevails for all metallicities the hardness decreases by more than 2 orders of magnitude from Z=0 to $\sim$10^-4, re-increases thereafter (up to $Z \la 5\times 10^{-3}$, in metal-poor starbursts) to a level $\sim$2 to 10 times smaller than that of Pop III objects, and decreases again to low levels for higher metallicities.

2) If very massive stars are favoured at metallicities $Z \la 10^{-4}$, the hardness $Q({\rm He^+})/Q({\rm H})$ of Pop III objects is considerably enhanced (corresponding to a powerlaw spectrum with spectral index $\alpha \sim 2.3$-2.8 in $F_\nu$), then decreases down to levels somewhat smaller or comparable to that of metal-poor starbursts, before decreasing further to levels at least two orders of magnitude softer than at zero metallicity.

Figure 7: Temporal evolution of the Ly$\alpha$ equivalent width (left panel) and He  II $\lambda$1640 equivalent width (right) for instantaneous bursts at all metallicities. The very metal-poor models (Z= 0, 10-7, and 10-5) with the IMFs C (50-500 $M_{\odot }$), B (1-500 $M_{\odot }$) and A (1-100 $M_{\odot }$) are shown as short-dashed, solid, and dotted lines respectively from top to bottom. The remaining metallicities (for IMF A) are shown with dashed lines. The equilibrium values for SFR  =  const. at metallicities $Z \le 10^{-5}$ are plotted on the right (at arbitrary ages) using open squares for the IMF C, filled triangles for IMF B, open circles for IMF A, and using short lines for higher metallicities (with IMF A). Note the very large maximum W(Ly$\alpha$) predicted at young ages. W(He  II $\lambda$1640 $) \protect\ga 5$ Å are only expected at the lowest metallicities ( $Z \protect\la 10^{-7}$), except if hot WR-like stars not included in the tracks were formed e.g. through important stellar mass loss (Sect. 6.2). See text for further discussion.