4 Ionising properties of starbursts at various metallicities

Figure 3: Temporal evolution of the H ionising photon flux $Q({\rm H})$ (upper panel) and the hardness $Q({\rm He^+})/Q({\rm H})$ (lower panel) for instantaneous bursts at all metallicities Z between 0.04 (=2 $Z_{\odot }$) and 0. (Pop III). The very metal-poor models ( Z= 10-5, 10-7 and 0) with the IMF B (1-500 $M_{\odot }$) are shown as solid lines from top to bottom. The remaining metallicities computed for the IMF A (1-100 $M_{\odot }$) are shown with dashed lines. Their hardness $Q({\rm He^+})/Q({\rm H})$ reached for constant star formation at equilibrium is shown by the solid triangles ( $Z \le 10^{-5}$), and the short lines for larger Z. Predictions for IMF cases not shown here are summarised in Tables 3 and 4. Discussion in text.

   

Table 3: ZAMS properties of integrated stellar populations at various metallicities. All quantities are normalised to a total burst mass of 1 $M_{\odot }$. The logarithm of the ionising photon fluxes $Q_{\rm i}$ in units of [photon s^-1 $M_{\odot }$^-1] is given.

Z	IMF	$Q({\rm H})$	$Q({\rm He^0})$	$Q({\rm He^+})$	$Q({\rm LW})$
		[log((photon s-1)/($M_{\odot }$ yr-1))]
0.	A	46.98	46.75	45.54	46.22
0.	B	47.29	47.10	46.26	46.40
0.	C	47.98	47.80	47.05	46.96
10-7	A	46.94	46.65	43.45	46.36
10-7	B	47.30	47.06	45.61	46.55
10-7	C	48.01	47.78	46.39	47.14
10-5	A	46.90	46.55	42.39	46.44
10-5	B	47.30	46.99	44.56	46.64
10-5	C	48.02	47.73	45.35	47.24
0.0004	A	46.88	46.43	43.4	46.45
0.0004	A$^\star$	47.01	46.58	43.62	46.53
0.001	A	46.86	46.41	43.44	46.47
0.004	A	46.85	46.38	43.41	46.50
0.008	A	46.84	46.35	43.67	46.52
0.020	A	46.85	46.34	43.72	46.54
0.040	A	46.87	46.33	43.71	46.59

$^\star$ Salpeter IMF with $M_{\rm up}$  =  150 $M_{\odot }$ and $M_{\rm low}$  =  1 $M_{\odot }$.

   
4.1 Burst models

The basic quantities describing the ionising spectrum are the emitted number of H, He, and He⁺ ionising photons, denoted by $Q({\rm H})$, $Q({\rm He^0})$ and $Q({\rm He^+})$ respectively, and the hardness $Q({\rm He^+})/Q({\rm H})$ ( $Q({\rm He^0})/Q({\rm H})$) tracing the energy range between 54 (24.6) and 13.6 eV. The predicted temporal evolution of $Q({\rm H})$ is shown in Fig. 3 (upper panel) for all metallicities between Z=0. and $2 \;\times ~~$$Z_{\odot }$. For the very low metallicities ( $Z \le 10^{-5}$) only the models with an IMF extending to 500 $M_{\odot }$ (model B) are shown for clarity sake. Adopting a larger value of $M_{\rm up}$ affects only the predictions at very young ages (ages $\la$2.5 Myr) due to the very short lifetime of these stars.

The predicted $Q_{\rm i}$ of ZAMS populations (age  =  0) for all IMF cases are listed in Table 3. For completeness with S02 the photon flux in the Lyman-Werner band (11.2-13.6 eV) capable to dissociate H₂ is also listed.

The main difference in the Lyman continuum photon output at different Z is a slower decline of the ionising photon production at low metallicities, due to the blueward shift of the main sequence. The temporal evolution of $Q({\rm H})$ at Z=10^-7 is essentially undistinguishable from the Pop III case. The larger $Q({\rm H})$ apparent for $Z \le 10^{-5}$ at ages $\la$2.5 Myr are essentially due to the larger value of $M_{\rm up}$ adopted at very low Z. The difference at older ages (when stars with masses >100 $M_{\odot }$ have disappeared) represents the pure metallicity difference.

As can be seen from Table 3 the $Q({\rm H})$ production of ZAMS populations at different metallicities increases somewhat with decreasing Z; the changes remain fairly small ($\la$40%) in reasonable agreement with other estimates (e.g. Tumlinson & Shull 2000). However, in cases such as constant star formation (equivalent to a temporal average) the Z-dependence is more pronounced (cf. Sect. 4.2).

   
4.2 Constant star formation

The main predictions for models with constant star formation at all metallicities and for all the IMF cases are listed in Table 4. In this case most quantities of interest here reach rapidly (over timescales $\la$6-10 Myr; except for the Lyman-break and $Q({\rm LW})$ requiring $\ga$200 Myr) an equilibrium value given in the table, normalised to a star formation rate (SFR) of 1 $M_{\odot }$ yr^-1. In addition to the ionising photon production $Q_{\rm i}$ (Cols. 3-5), and the H₂photodissociating photon flux ( $Q({\rm LW})$, Col. 6), we list the average energies $\overline{E}($ $Q({\rm H})$) and $\overline{E}($ $Q({\rm He^+})$) of the Lyman continuum photons and the photons with energies above 54 eV (Cols. 7 and 8). These quantities, not further discussed here, are e.g. of interest to estimate the thermal evolution of the ISM. Most of the data for Z=0, 0.0004, an 0.02 were already given in Table 3 of S02. Due to the use of a finer grid of atmosphere models at Z=0 some small changes are found for these models. The values in Table 4 supersede those of S02.

As expected from the earlier discussion (see Fig. 3), the Lyman continuum flux $Q({\rm H})$ shows an increase with decreasing metallicity, which can be fitted to an accuracy better than 10% by

$$\log(Q({\rm H})) = -0.0029 \times (\log(Z) + 9.)^{2.5} + 53.81$$ (1)

valid for the Salpeter IMF from 1-100 $M_{\odot }$ (models A) and $Z \in [10^{-9},0.04]$. The limitation to $Z \ge Z_{\rm lim} \approx 10^{-9}$ is due to the fact that all metallicities below $Z_{\rm lim}$ are equivalent to the Pop III (Z=0) case, as discussed in Sect. 2.2. Fits for alternate IMFs may be derived from the data in Table 4.

Overall, while ZAMS Lyman continuum fluxes vary by less than $\la$40% over the entire metallicity range for IMF A (Table 3), the ionising output at SFR  =  const. shows an increase of an factor $\sim$1.9 (2.8) between solar and 1/50 $Z_{\odot }$ (zero metallicity). Even larger increases are of course predicted in the case of IMFs extending to higher masses (models B and C).

Figure 4: Temporal evolution of the Lyman-break for instantaneous bursts and constant star formation at all metallicities $0 \le Z \le ~~$$Z_{\odot }$. Dashed lines show the predictions for $Z \ge 4 \times 10^{-4}$, solid lines are for Z=0 (Pop III), Z=10-7, and 10-5(from bottom to top). The behaviour of the very low Z burst models at ages $\log t \sim 7.2$, 7.5, 7.7 are artifacts due to the limited numerical resolution in tracks. Note the small sensitivity of the Lyman-break for SFR  =  const. at metallicities $Z \ge 4 \times 10^{-4}$ and the important decrease at lower metallicities.

   

Table 4: Predictions for the case of constant star formation models (equilibrium values) with different IMFs (Col. 2) at all metallicities (Col. 1). Given are the ionising photon fluxes $Q_{\rm i}$ (Cols. 3-6), their average energies $\overline {E}$ (Cols. 7 and 8), and the proportionality constants $f_{\rm l}$ between recombination line luminosities and the SFR (cf. Eq. (7), Cols. 9-11). The logarithm of the ionising photon fluxes $Q_{\rm i}$ is given in units of [photon s^-1 $M_{\odot }$^-1].

Z	IMF	$Q({\rm H})$	$Q({\rm He^0})$	$Q({\rm He^+})$	$Q({\rm LW})$	$\overline{E}($ $Q({\rm H})$)	$\overline{E}($ $Q({\rm He^+})$)	$f_{\rm Ly-\alpha}$	$f_{\rm H\alpha}$	f1640
		[log((photon s-1)/($M_{\odot }$ yr-1))]				[eV]		[erg s-1]
0.	A	53.81	53.50	51.49	53.57	26.61	66.08	6.80e+42	7.91e+41	1.74e+40
0.	B	53.93	53.64	52.23	53.57	27.78	68.15	8.86e+42	1.03e+42	9.66e+40
0.	C	54.44	54.19	53.03	53.65	29.60	68.22	2.85e+43	3.32e+42	6.01e+41
10-7	A	53.80	53.43	50.39	53.67	25.18	61.68	6.59e+42	7.67e+41	1.40e+39
10-7	B	53.95	53.61	51.42	53.69	25.99	64.86	9.26e+42	1.08e+42	1.49e+40
10-7	C	54.51	54.21	52.21	53.84	27.16	64.80	3.34e+43	3.89e+42	9.29e+40
10-5	A	53.70	53.26	48.71	53.65	23.95	59.62	5.15e+42	5.99e+41	2.91e+37
10-5	B	53.88	53.48	50.71	53.69	24.74	61.77	7.82e+42	9.10e+41	2.88e+39
10-5	C	54.49	54.13	51.50	53.97	25.61	61.77	3.20e+43	3.73e+42	1.82e+40
0.0004	A	53.63	53.10	50.08$^\dagger$	53.74	21.62	52.60$^\dagger$	4.38e+42	5.10e+41	6.79e+38$^\dagger$
0.0004	A$^\star$	53.70	53.20	50.42$^\dagger$	53.75	21.96	61.54$^\dagger$	5.22e+42	6.08e+41	1.50e+39$^\dagger$
0.001	A	53.59	53.04	50.17$^\dagger$	53.72	21.47	60.38$^\dagger$	4.01e+42	4.67e+41	8.39e+38$^\dagger$
0.004	A	53.50	52.93	$^\P$	53.67	21.27	$^\P$	3.37e+42	4.36e+41	$^\P$
0.008	A	53.44	52.83	$^\P$	53.63	20.90	$^\P$	2.89e+42	3.73e+41	$^\P$
0.020	A	53.36	52.75	$^\P$	53.56	20.84	$^\P$	2.44e+42	3.16e+41	$^\P$
0.040	A	53.28	52.65	$^\P$	53.49	20.88	$^\P$	2.00e+42	2.59e+41	$^\P$

$^\star$ Salpeter IMF with $M_{\rm up}$  =  150 $M_{\odot }$ and $M_{\rm low}$  =  1 $M_{\odot }$.
$^\dagger$ uncertain predictions for reasons discussed in Sect. 6.
$^\P$ no data provided, as predictions uncertain and strongly overestimated (cf. Sect. 6).