3. Stellar contribution to the radiation field at the hydrogen Lyman edge
We now use equation (2) to calculate the radiation background at the Lyman limit . The luminosity density is assumed to have a proper evolution parameterized as . Given the severe uncertainties about the spectral energy distribution of photons from star formation, the normalized spectral shape is modelled as while we add a factor for the absorption of stellar photons by the neutral hydrogen of each galaxy. In order to account for the frequency dependence of the absorption of photons by neutral hydrogen, we have written the absorption factor as . The equivalent HI column density term () is introduced for calculation purpose (we make no assumption as to the geometry and distribution of the gas) and the resulting effective escape fraction will be defined as . In these conditions the background radiation J912 (in erg cm-2 s-1 Hz-1 sr-1) at the Lyman limit writes as
3.1. Adopted parameters
By comparison with existing models for the spectral energy distribution of star-forming population (e.g. Bruzual & Charlot 1993), a value of in the range -1 to -3 is a realistic approximation. An upper bound to is also realistic since it cuts any stellar flux contribution below 304 Å . The local luminosity density of galaxies at the Lyman edge can be derived from the total H luminosity per unit volume of 1.26 1039 ergs s-1 Mpc-3 evaluated for star-forming galaxies in the local universe by Gallego et al. (1995). Under the current conditions valid in the ionized gas of galaxies (T=104 K and case B) we get a local density of photons of 9.2 1050 s-1 Mpc-3 (Osterbrock 1989). This number can be considered as a lower limit since photons in optically thin gas produce fewer H photons than those in optically thick gas. The relation between the photons density and the luminosity density at 912 Å depends on the value of . For our average case , we find a luminosity density at 912 Å of erg s-1 Å-1 Mpc-3 or erg s-1 Hz-1 Mpc-3. Incidentally, the relation (photons Å erg-1) established by Leitherer et al. (1995) would give the same value. This relation was established for starbursts with different star formation histories and initial mass functions while the simplifying assumption of a continuous star formation rate is probably valid at the scale of the local universe.
Significant evolution of galaxies is now well established (e.g. Ellis et al. 1996, Lilly et al. 1996, Fall et al. 1996) and we adopt from to as found by Lilly et al. (1996) for the evolution of the luminosity density of the universe at 2800 Å_Insofar as the light at this latter UV wavelength is essentially tracing on-going star formation, we think that the same exponent should be valid at our shorter wavelengths. Beyond the evolution is known to slow down but the situation is less certain. We have adopted , bearing in mind that this choice is not critical since the contribution to the background at from objects at high redshifts is small as soon as the evolution is not strong. Last, the calculation is independent of the value of since luminosity densities scale as .
3.2. The intergalactic opacity term
At the Lyman edge and ignoring absorption due to HeII for a line of sight limited to (HeI absorption is negligible), the effective optical depth in equation (4) writes as
Assuming a power-law of exponent -1.5 for the column density distribution (Petitjean et al. 1993, Songaila et al. 1995), and adopting the line densities per unit redshift and the evolution parameters from Boksenberg (1995), writes as for the Lyman forest clouds, and for the Lyman limit systems ( cm-2). The calculation of the first normalization constant accounts for the detection limit of 0.24 Å rest equivalent width and a velocity width of 30 km s-1 as in Miralda-Escudé & Ostriker (1990). The parameters for the Lyman forest clouds have been obtained for but the plot of their evolution up to (Boksenberg 1995) shows that our parameterization remains appropriate till the adopted limit at . As a numerical example, we find a transmission from to . Playing with the error bars given on the line densities per unit redshift and the evolution parameters (Boksenberg 1995) we find that this transmission does not change by more than 40%.
The background radiation calculated by equation (4) with the average transmission models discussed above is displayed in Fig. 1 as a function of the escape fraction. Although our evaluation is based on the measured photons density in the local universe and avoids therefore most of the uncertainties inherent to pure model calculations, it still depends on a few parameters, the index of the average spectral shape in the Lyman continuum, the evolution factor and the opacity of the intergalactic medium. The resulting uncertainties are illustrated in Fig 1. First, the impact of the ill-known index (values are used in Fig 1), is found to be reduced by the relation between and the luminosity density at the Lyman edge for a given photon density. Second, the effect of a larger intergalactic opacity as obtained with the upper limits given by Boksenberg (1995) on the density of Lyman clouds and Lyman limit systems per unit redshift is modest and comparable with a change of one unit of the index . In contrast, the calculation is sensitive to the amount of evolution as shown with the case of a milder evolution (till ) plotted in Fig 1. Selected as the variable against which the diffuse radiation has been plotted in Fig 1, the escape fraction is, as anticipated, the major source of uncertainty. We note, however, that the diffuse radiation does not decrease as fast as the escape fraction. The main reason is that galaxies at high redshift contribute to the diffuse radiation and may still be optically thin at Å while their nearby counterparts are optically thick at 912 Å. The issue of the uncertainty on the luminosity density itself at the Lyman edge will be addressed in the two following sections.
© European Southern Observatory (ESO) 1997
Online publication: April 28, 1998