3D Ly radiation transfer. III. Constraints on gas and stellar properties of Lyman break galaxies (LBG) and implications for high- LBGs and Ly emitters
Key Words.:Galaxies: starburst – Galaxies: ISM – Galaxies: high-redshift – Ultraviolet: galaxies – Radiative transfer – Line: profiles
Aims:The Aim of our study is to understand the variety of observed Ly line profiles and strengths in Lyman Break Galaxies (LBGs) and Ly emitters (LAEs), the physical parameters governing them, and hence to derive constraints on the gas and dust content and stellar populations of these objects.
Methods:Using our 3D Ly radiation transfer code including gas and dust (Verhamme et al. 2006), we fit 11 LBGs from the FORS Deep Field with redshifts between 2.8 and 5 observed by Tapken et al. (2007). A simple geometry of a spherically expanding shell of H i is adopted.
Results:The variety of observed Ly profiles is successfully reproduced. Most objects show outflow velocities of 150–200 km s; two objects are most likely quasi-static. The radial H i column density ranges from to cm. Our Ly profile fits yield values of 0.05–0.2 for the gas extinction. We find indications for a dust-to-gas ratio higher than the Galactic value, and for a substantial scatter. The escape fraction of Ly photons is found to be determined primarily by the extinction, and a simple fit formula is proposed. In this case a measurement of EW(Ly) can yield , if the intrinsic Ly equivalent width is known (or assumed). Intrinsic EW(Ly) 50–100 Å are found for 8/11 objects, as expected for stellar populations forming constantly over long periods ( 10-100 Myr). In three cases we found indications for younger populations. Our model results allow us also to understand observed correlations between EW(Ly) and other observables such as FWHM(Ly), , SFR(UV) etc.
We suggest that most observed trends of Ly, both in LBGs and LAEs, are driven by variations of and the accompanying variation of the dust content. Ultimately, the main parameter responsible for these variations may be the galaxy mass. We also show that there is a clear overlap between LBGs and LAEs: at approximately 20–25 % of the LBGs of Shapley et al. (2003) overlap with 23 % of the LAEs of Gronwal et al. (2007). Radiation transfer and dust effects should also naturally explain the increase of the LAE/LBG ratio, and a higher percentage of LBGs with strong Ly emission with increasing redshift.
Ly line radiation, often of the brightest emission lines in distant star-forming galaxies, is now frequently observed over a wide redshift range (Hu et al., 1998; Kudritzki et al., 2000; Malhotra & Rhoads, 2002; Ajiki et al., 2003; Taniguchi et al., 2005; Shimasaku et al., 2006; Kashikawa et al., 2006; Tapken et al., 2006; Gronwall et al., 2007; Ouchi et al., 2007, e.g.). Numerous narrow-band and other surveys use this line to search for galaxies at specific cosmic ages, and to hunt for the most distant objects in the universe (see Willis & Courbin, 2005; Cuby et al., 2007; Stark et al., 2007a). Furthermore extremely deep “blind” spectroscopic exposures have revealed very faint objects through their Ly emission (Rauch et al., 2007), illustrating also the discovery potential of future instruments such as the Multi Unit Spectroscopic Explorer (MUSE) for the Very Large Telescope (VLT) and extremely large telescopes.
Ly measurements are used to infer a number of properties such as redshift, star formation rates, constraints on the ionisation of the intergalactic medium and hence on cosmic reionisation, trace large scale structure at high redshift etc. (see e.g. Schaerer, 2007, for an overview, and references therein). However, given the physics of this generally optically thick resonance line quantitative interpretations of Ly are often difficult or even ambiguous, as shown by detailed studies of nearby starbursts carried out during the last decade (Lequeux et al., 1995; Kunth et al., 1998; Mas-Hesse et al., 2003; Hayes et al., 2005, 2007; Atek et al., 2008).
With the increased computer power and the availability of Ly radiation transfer codes (cf. Ahn et al., 2000, 2002; Cantalupo et al., 2005; Hansen & Oh, 2006; Dijkstra et al., 2006a; Tasitsiomi, 2006; Verhamme et al., 2006) time is now ripe not only to predict Ly in different astrophysical situations (e.g. galaxy and cosmological simulations) but especially also to confront observed Ly properties (line profiles, equivalent widths, etc.) of individual galaxies (nearby and distant ones) with detailed 3D radiation transfer calculations. This is one of the aims of our paper.
Two galaxy populations, the well known Lyman Break Galaxies (LBGs) and the Ly emitters (LAEs), represent currently the largest samples of distant galaxies, at least from to 6.5. Important questions remain unanswered about them, closely related to their Ly emission and absorption.
LBGs show a great diversity of Ly profiles and strengths, reaching from strong emission, over P-Cygni type profiles, to strong absorption (see e.g. Shapley et al., 2003). Furthermore the Ly properties are found to correlate with other quantities such as the strength of interstellar lines (IS), extinction, the star formation rate (SFR) etc. However, the origin of these variations and correlations remains largely unknown or contradictory at best. For example, Shapley et al. (2001) suggested, age as the main difference between LBG groups with different Ly strength, where objects with Ly in absorption would be younger and more dusty. However, it is not clear why older LBGs would contain less dust, especially since outflows, supposedly used to expel the dust, are ubiquitous in all LBGs and since precisely these outflows are the location where the emergent Ly spectrum is determined. Ferrara & Ricotti (2006) proposed that LBGs host short-lived ( Myr) starburst episodes, whose outflows – when observed at different evolutionary phases – would give rise to the observed correlations between IS lines and Ly. However, the ages obtained from SED and spectral fits of LBGs show older ages and fairly constant star formation histories (cf. Ellingson et al., 1996; Pettini et al., 2000; Shapley et al., 2001; Papovich et al., 2001; Pentericci et al., 2007). Pentericci et al. (2007) find that LBGs without Ly emission are on average somewhat older ( versus Myr) and more massive than those with Ly in emission. However, given the significant amount of ongoing star formation inferred for both galaxy subsamples (with/without Ly emission) types from their SED fits, intrinsic Ly emission is expected in both subsamples. The apparent differences in age and ratio of age over star formation time scale () can therefore not be a physical cause for the observed Ly variations. Erb et al. (2006), from an analysis of LBGs, also find that more massive galaxies have fainter Ly (or Ly in absorption); following the arguments of Mas-Hesse et al. (2003) they suggest that this is mostly due to an increased velocity dispersion of the interstellar medium, indicated by the increased strength of the saturated IS lines, which would increase the fraction of Ly being absorbed. However, from radiation transfer modelling we find that the behaviour of the Ly escape fraction is not linear with the velocity dispersion in the ISM (Verhamme et al., 2006, and below). Other physical parameters may play a more important role in the destruction of Ly.
Important questions remain also concerning the properties of LAEs, their similarities and differences with respect to LBGs, and about the overlap between these galaxy populations. Since a fraction of LBGs show strong enough Ly emission to be detected with the narrow-band technique mostly used to find LAEs there must definitely be an overlap. For example, at approximately 25 % of the LBGs of Steidel and collaborators (Shapley et al., 2003) have EW(Ly) 20 Å (restframe), sufficient to be detected in the LAE survey of Gronwall et al. (2007). However, what the properties of LAEs are e.g. in terms of stellar populations (age, star formation histories, ), mass, dust content, outflows, metallicity etc. is not yet well established, although first such analysis have recently become available (see Schaerer & Pelló, 2005; Lai et al., 2007, 2008; Gronwall et al., 2007; Gawiser et al., 2007; Pirzkal et al., 2007; Finkelstein et al., 2007b). Understanding the nature of LAEs and their relation to LBGs is also crucial since the contribution of the LAE population to the known starburst population seems to increase with redshift (Hu et al., 1998; Shimasaku et al., 2006; Nagao et al., 2007; Ouchi et al., 2007; Dow-Hygelund et al., 2007; Reddy et al., 2007).
Last, but not least, several types of theoretical models have been constructed during the last few years aimed at understanding LAE and LBG populations, the relation between the two populations, and to use them as constraints for galaxy formation scenarios, cosmic reionisation, and other topics (Thommes & Meisenheimer, 2005; Le Delliou et al., 2006; Mori & Umemura, 2006; Dijkstra & Wyithe, 2007; Kobayashi et al., 2007; Mao et al., 2007; Stark et al., 2007b; Nagamine et al., 2008). Observational constraints on crucial parameters such as the Ly escape fraction from LBGs and LAEs, and other insight from radiation transfer models are, however, badly needed to reduce uncertainties and degeneracies in these modeling approaches.
With these questions about LBGs and LAEs in mind, we have recently started to model a variety of starbursts with our new Ly radiation transfer code (Verhamme et al., 2006, paper I). First we have studied the well known LBG MS1512-cB58 (cB58 in short), whose spectrum is dominated by strong Ly absorption (Schaerer & Verhamme, 2008, paper II). In this third paper of the series we present an analysis of 11 LBGs observed with FORS2 at the VLT with sufficient spectral resolution () to allow detailed Ly profile fitting to constrain their properties. Taken together, the objects analysed in paper II and III cover a wide range of Ly strengths and also different morphologies, including absorption dominated Ly and Ly emission lines with equivalent widths between 6 and 150 Å (restframe). The variety of objects modeled in paper II and III covers thus in particular the entire range of Ly strengths defining the 4 spectral groups of the LBG sample of Shapley et al. (2003), the largest currently available at . Furthermore, several of the objects we model have strong enough Ly emission to classify as LAEs, according to the criteria used in many surveys. Our analysis represents the first modeling attempt of Ly line profiles of high redshift galaxies with a detailed radiation transfer code including gas and dust and treating line and continuum radiation.
The remainder of the paper is structured as follows. A description of the radiation transfer code, the assumptions, and input parameters is given in Sect. 2. In Sect. 3 we model the Ly profiles of the individual objects. Our main fitting results are discussed and confronted to other observations in Sect. 4. Other properties are derived in Sect. 5. In Sect. 6 we finally propose a unifying scenario for LBGs and LAEs and discuss several implications. Our main conclusions are summarised in Sect. 7.
2 Ly radiation transfer modeling
To fit the observations we used our 3D Monte Carlo (MC) radiation transfer code MCLya (Verhamme et al., 2006). The code solves the transfer of Ly line and adjacent continuum photons including the detailed processes of Ly line scattering, dust scattering, and dust absorption. The main assumptions required for the modeling concern the geometry, the choice of the input parameters, and the input spectrum. We discuss them now in turn.
For simplicity, and given empirical evidence in favour of a fairly simple geometry in LBGs discussed by Verhamme et al. (2006), we adopted first a simple “super-bubble” model to attempt to fit the observed Ly line profile. The assumed geometry is that of an expanding, spherical, homogeneous, and isothermal shell of neutral hydrogen surrounding a central starburst emitting a UV continuum plus Ly recombination line radiation from its associated H ii region. We assume that dust and H i are uniformly mixed.
The homogeneity and a large covering factor of the absorbing medium are supported by observations of LBGs and nearby starbursts. For example, the outflow of cB58 is located well in front of the stars and covers them almost completely, since it absorbs almost all the UV light from the background stars. Indeed, Savaglio et al. (2002) find only a small residual mean flux above the zero level in the core of the Ly absorption feature, whereas it is black for Pettini et al. (2002). Heckman et al. (2001) estimate an area covering factor for optically thick gas of 98% from the residual intensity at the core of the C ii 1335 line. A somewhat lower covering factor may be indicated for LBGs with strong Ly emission (Shapley et al., 2003). For simplicity, and in the absence of further observational constraints, we will assume a covering factor of unity.
2.2 Shell parameters
As described in paper II, the outflow is modelised by a spherical, homogeneous, and isothermal shell of neutral hydrogen and dust centered on a point source. Four parameters characterise the physical conditions in the shell:
the expansion velocity ,
the Doppler parameter ,
the neutral hydrogen column density ,
the dust absorption optical depth .
In principle is constrained by observations, either directly measured by the blueshift of low ionisation interstellar lines (hereafter LIS) compared to stellar lines (Pettini et al., 2002), or from the shift between absorption LIS lines and Ly in emission, , when the stellar lines are too faint to be observed (Shapley et al., 2003). Otherwise will be constrained by Ly line profile fits. In Verhamme et al. (2006) we showed that radiation transfer effects lead to in expanding shells with cm. For lower column densities the peak of the redshifted Ly emission may trace (leading to ), instead of twice this value (cf. Verhamme et al., 2006, and also Fig.17). For three of the 11 objects to be modeled here has been measured (see Table 1).
The Doppler parameter , describing the random motions of the neutral gaz possibly including microturbulence, is kept as a free parameter. For indication, km s corresponds to thermal motions for K; for the lensed LBG cB58 Pettini et al. (2002) derived km s from fits of LIS lines.
Although presumably the neutral column density and the dust amount are physically related, e.g. by a given dust-to-gas ratio, both parameters are kept free in our modeling procedure. The resulting values will later be compared to available observational constraints.
2.3 The intrinsic spectrum in the Ly region
The synthetic stellar spectrum of star-forming galaxies close to Ly is described in paper II. The stellar continuum presents an absorption feature around Ly whose strength varies in time, depending on the SF history, on the age of the star-forming galaxy, and less on its metallicity (Schaerer, 2003; Delgado et al., 2005).
The main H and He recombination lines created in the H ii region surrounding the starburst are also predicted by the models of Schaerer (2003, hereafter S03): for metallicities between 1/50 Z and solar the strength of Ly varies from EW(Ly) 250–360 Å at early time after the burst, and declines until zero for a burst whereas it reaches an equilibrium value of 60–100 Å for objects with a constant star formation rate (SFR) after Myr (see also Fig. 15).
In paper II, we showed that the fitting of the observed star-forming galaxy cB58 depends only very little on the details of the stellar continuum around Ly. Since the objects modeled here show stronger Ly emission than cB58, which is dominated by absorption, neglecting the detailed shape of the stellar continuum is even more justified here. Therefore we model the input spectrum as a flat continuum plus a Gaussian emission line described by two parameters:
the intrinsic equivalent width, EW(Ly), and
the intrinsic full width at half maximum, FWHM(Ly).
What “reasonable” values should we adopt for these parameters ? Our first approach, to reduce the number of free parameters in the model, was to test if a unique scenario was conceivable, i.e. if we could fit all data with the same intrinsic Ly spectrum, the differences in the observed spectra would then come from radiation transfer effects in the outflowing medium. These objects are likely starburst galaxies with a constant star formation (SF) history as derived from their UV low/medium-resolution spectra (Noll et al., 2004; Mehlert et al., 2006), so we fix the intrinsic equivalent width to EW(Ly) Å, as derived from the S03 models. We adopted the intrinsic value FWHM(Ly) km s, as it is comparable to the values measured from the velocity dispersion of H and CO lines in cB58(Teplitz et al., 2000; Baker et al., 2004), and the dispersion measured in 16 starbursts at by Erb et al. (2003).
|[M yr]||[M yr]||[km s]||[Å]||[km s]|
2.4 Description of the method
We ran a grid of models with varying physical conditions in the shell. The expansion velocity was varied from 0 to 400 km s in steps of 50 km s, the neutral column density from to cm, the dust amount from to 0.4 (0, 0.1, 0.5, 1., 2., 3., and 4.), and the Doppler parameter from to 200 km s (10, 20, 40, 80, and 200 km s).
In contrast to the four shell parameters, there is no need to run a new Monte Carlo simulation each time we want to change the input Ly spectrum. This can be made a posteriori without resorting to any simplifying assumption. To do so we run each simulation of the four-dimensional grid with a flat continuum as input spectrum, e.g. the same number of photons per frequency bin, and we memorise this input frequency for each photon. Once the simulation is done, we construct output spectra corresponding to the input frequency bins and we assign a different weight to each, in order to reconstruct any input spectrum shape (for example, a flat stellar continuum+a Gaussian centered on Ly, or a synthetic starburst spectrum as in paper II.
From each calculation we derive the integrated spectrum (Ly line profile) emerging from the expanding shell. For comparison to the observations, our synthetic spectra are convolved with a Gaussian with FWHM=150 km s corresponding to the experimental resolution. As shown below, moving in the space of these 6 input parameters, we can reproduce the whole diversity of observed Ly spectra, ranging from double-peaked profiles to broad absorption or asymmetric emission lines.
Finally, our calculations allow us also to derive the Ly escape fraction . This is computed from
where is the intrinsic Ly line profile computed from the MC simulation, and is the escape fraction in each input frequency bin computed from the MC simulation.
3 Fits of the FORS Deep Field sample
3.1 Description of the FDF sample
Our work uses the FORS Deep Field sample presented by Tapken et al. (2007, hereafter T07). Therefore, we give a brief overview of their work. T07 present the medium-resolution spectra (R=2000) of 16 high-redshift galaxies. The target selection for the objects of T07 was based on the FDF spectroscopic survey (Noll et al., 2004). The FDF spectroscopic survey aimed at obtaining low-resolution spectra (R200) of intrinsically bright galaxies with a photometric redshift (Bender et al., 2001; Gabasch et al., 2004) between 1 and 5 with a high signal-to-noise ratio (10). The spectra of 90 galaxies with redshift between 2 and 5 were analysed and published electronically by (Noll et al., 2004). The deep (up to 10h integration time with FORS1/FORS2) low-resolution spectra allowed them to derive the redshift with high accuracy and reliability. ad to search for even weak signs of AGN activity of the objects. Based on the low-resolution spectra, T07 selected starburst galaxies with strong Ly emission and/or with bright UV-restframe continuum for the follow-up medium-resolution spectroscopy.
These medium-resolution spectra were obtained with FORS2 at the VLT UT4 using the holographic grisms 1400V and 1200R. The spectral resolution of both grisms is 2000. The spectral range of the 1400V (1200R) grism is about 4500 to 5800 (5700 to 7300) Å. All data were collected in service mode using one single MXU mask for each grism. The total integration time of the 1400V (1200R) observations is 6.25 h (9.45 h). The data were reduced using the MIDAS-based FORS pipeline (Noll et al., 2004)). For more details see T07.
The spectra of all objects of T07 include the Ly profile. However, only eleven Ly profiles have a sufficient SNR (10), which allows a detailed comparison with our theoretical models. The properties of this sample are listed in Table 1. Note that except stated otherwise all equivalent widths are given in the restframe; we denote them by “observed” equivalent widths for distinction with “intrinsic” or “theoretical” values to be derived later.
Eight galaxies have redshifts around z3, while 3 galaxies have redshift with 4.5 5. While only a few Ly profiles of our sample show an absorption component (FDF5550), all our profiles display an emission component. The equivalent width of the emission component range between EW(Ly) = 6 and 150 Å. This Ly equivalent width is measured using the continuum redwards of the Ly emission line (at 1300 Å). Although the majority of LBGs have Ly equivalent widths lower than 20 Å, 8/11 of our galaxies have an equivalent width (of the total Ly line, including absorption and emission) higher than 20 Å. Therefore 70% of our sample would be detected in a typical narrow-band survey, searching for LAEs.
As described by T07 the profiles show a wide range of morphologies. For convenience mostly, we divide the galaxies in three groups according to their Ly profile: (A) Ly emitters with asymmetric profiles: FDF1337, FDF2384, FDF3389, FDF4454, FDF5550, FDF5812, and FDF6557, (B) double-peak profiles FDF4691 and FDF7539, and (C) asymmetric Ly plus a blue bump: FDF5215 and FDF1267.
3.2 The fitting procedure
For fits with our synthetic spectra the observed, non-normalised spectra were transformed to velocity space using the redshift listed in Table 1. If necessary was adjusted within the error bars cited. We then use the same normalisation as Tapken et al. (2007) to determine EW(Ly). Finally, we overlay synthetic spectra on observed ones and estimate fit qualities. The spectral parts we focus on are location of the peak, the shape of the peak and the extended wing, knowing that the blue side of the spectrum could be affected by the surrounding IGM.
The parameters of the best fits, as well as derived parameters from our model like the escape fraction are summarised in Table 2. Multiple entries correspond to multiple solutions of similar quality.
|ID||type||[km s]||b [km s]||[cm]||EW(Ly) [Å]||FWHM(Ly) [km s]|
3.3 The asymmetric profiles (group A)
Among the 11 objects, 7 present the characteristic Ly asymmetric emission line (FDF1337, 5550, 2384, 4454, 5812, 3389 and 6557). This line shape can be understood by radiation transfer effects through an expanding medium (Verhamme et al., 2006). Indeed, Tapken et al. (2007) were able to measure a velocity shift between the interstellar absorption lines and the Ly emission line for two of these objects, FDF1337 and FDF5550, because their UV continuum is bright. Both of them present a shift of km s (cf. Table 1), a clear sign of outflows, which is most likely related to 3 times the expansion velocity of the shell as shown in Verhamme et al. (2006).
FDF1337 and FDF5550
To model FDF1337 and FDF5550, we have therefore fixed km s, and we proceed to adjust the 3 remaining shell parameters. Our best fits and the corresponding parameters are presented in Fig. 1. Note that both spectra are fitted with an intrinsic EW(Ly) Å, which corresponds to the equilibrium value reached by a galaxy with constant star formation after 50–100 Myr (cf. Fig. 15). Even if the observed EW(Ly) is an order of magnitude lower, the intrinsic predicted EW(Ly) seems to be “standard” in this sense. Dust (we find for 1337 and for 5550, i.e. ) in a high neutral column density ( cm) outflow causes this attenuation. The derived Ly escape fraction is for FDF5550 and for FDF1337. A rather small Doppler parameter, km s, is derived compared to cB58 (km s) to reproduce the secondary peak on the elongated red wing. Note the extension of this red wing over km s.
FDF2384, FDF4454, and FDF5812
The Ly fits for these objects are shown in Fig. 2. The profiles differ from the former by their high EW(Ly) Å, and a less extended red wing ( km s). This leads to simulated neutral column densities an order of magnitude lower and higher escape fractions (, cf. Table 2).
To illustrate how well constrained the model parameters are we will present in detail the fitting of the object FDF2384. In Fig. 3 a fraction of the model grid we use is shown for a fixed value km s, and for several values of , and . The central profile of each 3x3 grid illustrates the best fitting profile for FDF2384. The observed profile of FDF2384 is overlayed on each cell. The overall behaviour of the spectra shown on this Fig. can be summarised as follows.
When increases, multi-peaks appear on the extended red wing. Indeed, the location of the second red-peak related to “backscattered” photons — photons which are reflected by the receding shell through the interior — is at , so when increases, the separation between this peak and the one at lower velocity (from the first-red-peak due to scatterings in the forthcoming part of the shell) increases (for more details, see Verhamme et al. (2006)).
When increases, the “extension” of the Ly line increases — the width between the sharp blue edge and the end of the red extended wing. Although very clear, this effect is relatively modest here, since changes only by a factor 2. This effect is easily understood by the increase of the optical depth in the medium, forcing Ly to reach frequencies further from line center to escape.
When increases, photons which undergo the highest number of scatterings will be destroyed, on average. This enhances the peaks made of back-scattered photons, whose escape is easier, thanks to this mechanism, than the escape of diffusing photons.
The Doppler parameter is fixed to km s, because higher values broaden the profiles too much, and secondary bumps are smoothed. Variations of are discussed in Sect. 3.6.2.
The overall shape of the Ly profile of FDF2384 is smooth, with a small bump visible in the extended red wing (at km s); its position corresponds to km s. Thanks to the “extension” of the line in FDF2384, we can fix cm. Finally, the relative height of the bump compared to the main peak will determine correlated values for and , and two combinations are possible: ( and ) provides a reasonable fit, but the best fit is obtained with ( and ), as shown in Fig. 2.
The derived best fit escape fraction is . There are big differences of the escape fraction, when increases from 0.5 to 2, going from to ; correspondingly the intrinsic strength of the emission line varies from EW to Å.
We proceed the same way to fit FDF4454 and FDF5812, which appear quite similar: the expanding shell has the same velocity km s, the column density in the shell is an order of magnitude lower than in FDF1337 and FDF5550, but the dust amount, , remains the same. Since the intrinsic EW(Ly) is EW and the latter values are already relatively large, one obtains quite large intrinsic Ly equivalent widths, between 100 and 280 Å, for FDF2384, 4454, and 5812. The highest values require fairly young ages (see Fig. 15); however since these objects are quite faint, their continuum placement may be uncertain and EW may therefore be overestimated. The Ly escape fraction of FDF4454 is high, , because the shell is less dusty.
FDF3389 and FDF6557
These two objects are both at . They have medium observed equivalent widths (EW(Ly) Å), compared to the former asymmetric profiles, and rather small extensions ( km s). We treat them separately, because their spectra seem more complex than a smooth asymmetric emission line. They present multi-peaks on the red extension of the line. However, they are also more noisy, and these secondary features are only twice the noise amplitude. Nevertheless, we assume that they are significant, and we derive best-fits, taking these features into account (see Fig. 4).
The parameters derived from our fitting are close to the 3 narrow asymmetric cases ( km s, cm, ) except for the intrinsic EW, which have “canonical” values — compatible with expectations for SFR=const — again, because the observed EW are lower for these objects. The escape fraction is high for FDF3389 (), and lower for FDF6557 () because of a higher column density and so a higher dust content.
3.4 The double-peaked Ly profiles (group B)
Two of the 11 spectra present double-peaked profiles (FDF4691 and FDF7539). Such line morphologies are a natural outcome of radiation transfer in a static medium Neufeld (1990), since in such media the Ly photons can only escape by diffusing into the red or blue wings, where the opacity decreases rapidly.
This galaxy has a high EW(Ly) Å. Tapken et al. (2004) fitted this object with a code using a finite element method (Richling et al., 2001), and proposed cm and km s in an almost static (=12 km s) and dust-free shell as best-fit parameters. The intrinsic spectrum they use is a Gaussian with FWHM=1000 km s, and no continuum. Using the same parameters we can reproduce their fit.
However, since their code is not suited to high column densities
we searched for other solutions.
We find a fit of better quality — the deep gap between the peaks
is better reproduced by a smaller , and the red wing is fitted
with more accuracy starting from an input spectrum with a continuum — with a
higher column density ( cm), and consequently
a smaller km s
A solution to reproduce the observed spectrum with a more realistic intrinsic spectrum (FWHM=100 km sand EW=80 Å) is to invoke two contributions from two different media: when we sum emergent spectra from two identical shells except for the column density ( km s, km s, no dust, EW(Ly) Å, and cm for one and cm for the other), we are able to reproduce a spectrum with narrow peaks close to the center and broad wings, starting from a “standard” value for the FWHM, FWHM(Ly) instead of 1000 km s (see Fig. 6). This could correspond to a physical situation where an initially thick shell has been stretched until a hole forms, and the diffuse medium in the hole is still opaque enough to imprint radiative transfer effects on Ly photons. The surfaces of the thick shell and the hole are of equal size in this first model. The parameters listed for FDF4691 in Table 2 are those of the homogeneous single shell model discussed above.
For this object the velocity shift between the LIS in absorption and Ly in emission was measured: km s, which implies a shell velocity km s, i.e. almost static as in the case of FDF4691, or at maximum km s, in case of a low column density. Indeed, the spectrum is also double-peaked as for FDF4691.
The large peaks separation ( km s, larger than for FDF4691) implies a high column density. Presumably, the rather low observed EW(Ly) also implies the presence of dust in the shell. Indeed, the best fit shown in Fig. 5 has a high column density ( cm), and dust (). It is compatible with the canonical value for the intrinsic Ly spectrum. The resulting escape fraction is .
For comparison, Tapken et al. (2007) proposed a fit of similar quality for this object, but the velocity of the shell they derive from their modelling is high ( km s), which is in contradiction with the observed small velocity shift between Ly and LIS ( km s), and surprising for a double-peaked profile. As their investigation is restricted to low column densities, the only solution they have to produce separated and broad peaks is with high values of and , and a huge intrinsic FWHM ( km s).
3.5 Other Ly profiles (group C)
This object presents an asymmetric emission peak plus a bump on the blue side of this peak with a strong observed EW(Ly) Å. The Ly profile, shown in Fig. 7, can be fitted by different scenarii.
On the left panel of Fig. 7, the bump is considered as a blue peak in an almost static ( km s) shell with a small column density ( cm) and a small amount of dust (). Values of higher than km slead to too separated peaks. The escape fraction is . On the right panel, we fit the profile with a fast moving (=300 km s), dense ( cm) and dusty() shell, leading to , and a very large intrinsic EW(Ly) Å. Dust is needed in this configuration to reproduce narrow and well separated peaks. Indeed, dust destroys more efficiently photons which undergo the highest number of scatterings, i.e. all but the backscattered photons, which leads “isolates” and “slims down” the peaks. Note that this solution requires an adjustment of the source redshift to instead of derived by Tapken et al. (2007). However, this object is the only one for which the redshift determination is only based on Ly, so this poses so far no difficulty.
Our favoured solution is the “quasi-static shell” picture. Indeed, the high EW inferred in the second fit seems unlikely. Furthermore, the large observed EW of 1267 would imply a rather low column density as discussed in Sect. 4.2.1. Finally, the SFR values derived from uncorrected UV and Ly fluxes () indicate a low dust content. An accurate redshift measurement of FDF1267, independent from Ly, should allow to distinguish between these two solutions.
FDF5215, shown in Fig. 8 presents the same spectral shape as FDF1267: a small bump on the blue side of the asymmetric strong emission, but the noise level is much lower, and this small bump has to be taken into account. Again, two different scenarii can reproduce the spectral shape. The bump is either considered as a blue peak in an almost static shell, or as the first red peak of Ly emission. The two solutions, differing by more than 5 orders of magnitude in , are listed in Table 2. In passing we note that FDF5215 was also modeled by (Tapken et al., 2007); their set of parameters is similar to our solution at low , except for a higher value of .
None of our fits are really satisfying. The solution at low- (left panel) reproduces well the observed profile, but the derived column density is very low: at least 4 orders of magnitude lower than the rest of the sample. As a consequence, the FWHM of the intrinsic Ly emission is huge (FWHM=700 km s) to reproduce a broad profile without efficient broadening due to radiation transfer in an almost transparent medium. Finally, this solution does not reproduce the absorption at km s discussed below. On the other hand, the solution with high and a standard intrinsic FWHM fits less well. Furthermore, the redshift derived from this fit is out of the error bars ( instead of ).
The black absorption component, found at km s in this object (see Fig. 8 left), is unaccounted for in the fit with low . How likely it is that this represents a chance alignment of an H i absorber? If fitted separately with a Voigt profile the absorption is well described by km s, and cm. Using the column density distribution from Misawa et al. (2007) we find that the probability to find an absorber with in a velocity interval of say 4000 km s is 5 %. It seems thus more likely that this feature is related to the galaxy. We conclude that none of our solutions is clearly favoured, and further observations are needed to help constraining the models for FDF5215.
3.6 Uncertainties and degeneracies in Ly fits
Uncertainties on each parameter
The parameters listed in Table 2 correspond to best-fits determined by eye, without resorting to minimisation techniques. We now attempt to indicate the approximate uncertainties of the derived parameters.
The characteristic uncertainty on the expansion velocity is estimated to be km s, which is the step in velocity in our grid of models. The sampling in , the dust absorption optical depth, is not linear (we have assumed values of 0, 0.1, 0.5, 1., 2., 3., and 4. for our grid), but it was refined when necessary. From the line fits the characteristic uncertainty on and on the neutral column density are estimated as . We refer to Fig. 3 to give an idea of the uncertainties on and . Other comparisons, e.g. with SED fits and by imposing a consistency between different SFR indicators, indicate that the uncertainty on the extinction may be somewhat larger, up to a factor 2 in some cases (see Sect. 4.8).
The Doppler parameter is set to the default value km s, which allows to reproduce the relative narrow peaks observed , and secondary features on the extended red wing of the profiles. Models were computed for other values of (10, 20, 40, 80, and 200 km s), but did not lead to better fits. We estimate the uncertainty on this parameter around 50%.
The uncertainty on the intrinsic equivalent width can be fairly large, and it mostly depends on the determination of the continuum. Indeed, for strong Ly emitters, like FDF5812, the continuum is so weak that it is poorly constrained, and the uncertainty on the continuum level is around 20%. As already mentioned, all the spectra presented above have been normalized to the same level as determined by Tapken et al. (2007) to derive observed Ly EW. However, choosing the continuum level by eye we may also obtain acceptable solutions with lower intrinsic EW(Ly) for the three stronger Ly emitters (FDF2384, FDF4454 and FDF5812), in better agreement with a scenario of a constant star formation over Myr. Only for FDF1267 do all solutions seem to imply a fairly high intrinsic Ly equivalent width.
The intrinsic FWHM was set to FWHM=100 km sas a default value, and good fits were obtained for almost all spectra; exceptions are the very low solution for FDF5215 and the very broad double-peaked profile of FDF4691 (except if we consider a shell with two components as described before).
Although degeneracies affect in principle our profile fits, it turns out that asymmetric line profiles provide fairly undegenerate solutions. We briefly describe how the influence of the main parameters (, , , FWHM, and ) can be discerned.
Doppler parameter : After several attempts to fit the data with large values ( km s), we adopted a typical value km s. The Doppler parameter has a complex influence on the Ly profile: small lead to asymmetric emission lines on which the potential multi-peaks due to a high expansion velocity would be visible (see Fig. 3). Large values of ( km s) lead to a smoothed red peak (the multi-peaks are not visible any more), whose location is redshifted. Furthermore a blue emission component appears whose strength increases with , reproducing a kind of “double-peaked profile” like in static media, but with asymmetric peaks — the two sides of each peak don’t have the same slope (see Fig. 9)— even in shells with high velocities ( km s).
In our sample, the spectra from group A (asymmetric profiles) do not show blue components, and faint multi-peaks (better said bumps on the extended wing) may be visible. Therefore small values of ( km s) are required to fit our spectra. On the contrary, other spectral types (B and C) may be fitted with larger values of .
Column density : With increasing neutral hydrogen column density, the blue edge of the Ly emission is progressively redshifted with respect to the systemic galaxy redshift. Thus, if is known accurately enough, the neutral column density in the shell is well constrained. The full width of the line increases with increasing too. Indeed, it is impossible to fit narrow lines ( km s) with high column densities ( cm), and extended lines ( km s) with low column densities ( cm), if the expanding shell model applies (cf. Sect. 4.6 and Fig. 13).
Expansion velocity: The overall shape of the line profile — presence of secondary peaks or not — constrains the velocity of the shell: fast moving shells ( km s) lead to multi-peaks in the Ly profile, whereas static or almost static shells lead to double-peaked profiles with symmetrical peaks — the two sides of each peak have the same slope.
Dust content: Finally, the dust content is adjusted to fit the peak width and the relative height of the bumps – if any – compared to the main peak.
In conclusion, few degeneracies appear in the modeling of asymmetric Ly line profiles (group A). Thanks to the location of the blue edge and to the full width of the line is well constrained for asymmetric spectra. Furthermore, the global line shape (one single peak) implies low values of ( km s) and ( km s). On the contrary, the 4 spectra with more complex profiles (1267, 5215, 4691 and 7539) present degeneracies. Tapken et al. (2007) proposed Ly fits for 3 of them (4691, 5215, 7539). We can reproduce their fits, but propose fits with other sets of parameters, as our code allows for higher column densities than the code Tapken et al. (2007) used. Automated fitting methods and a thorough examination of the uncertainties and possible degeneracies in Ly line profile fits will be useful in the near future, also when larger samples of spectra of sufficient S/N and resolution become available.
Possible limitations of the model
As described and motivated in Sect. 2, our modeling makes some simplifying assumptions, including in particular geometry and the homogeneity of the shell. How far these assumptions would alter our results is presently unclear and remains to be explored in the future.
Inhomogeneous/clumpy geometries have e.g. been explored by Hansen & Oh (2006); the line profiles obtained from such models do not seem to change significantly. However, how much our model parameters would be modified remains to be examined. For the time being it seems clear that few if any cases are know, where clumpy geometries would favour Ly transmission with respect to the continuum (cf. Neufeld, 1991; Hansen & Oh, 2006). This can e.g. be concluded from the comparison of H and Ly in local starbursts (Atek et al., 2008), and from the comparison of UV and Ly SFR indicators. Indications for one possible case of such a Ly boosting have been found among 4 objects analysed by Finkelstein et al. (2007a). Other geometries have e.g. been considered in paper II, where deviations from the constant velocity shell have been necessary for the analysis of cB58.
The effect of the intergalactic medium (IGM) has been neglected in our approach. Even if the IGM is almost fully ionised at , the redshift of the bulk of our objects, the effect of the intervening Ly forest corresponds statistically to a transmission of 70 and 40 % ( 0.3–1) between 3 and 4 (Faucher-Giguere et al., 2007). In our modeling we find no need to account for such an IGM reduction within 1000-2000 km s of the Ly line; not even in the two highest redshift ( 4.7–5) objects. No individual Ly forest absorption components are found in this interval; furthermore for most objects, except possibly FDF 1337, 5550, 7539, and maybe also 3389, the predicted continuum flux blueward of Ly agrees within the uncertainties with the observed continuum. In any case, less importance has been given to the line fits on the blue side of Ly. Also, in a detailed analysis of the Ly forest along the line of sight of the LBG cB58, Savaglio et al. (2002) found no indication for neutral gas within 4000 km s of the systemic velocity of the galaxy. From these considerations we conclude that our Ly line fits are probably unaffected by additional matter beyond the expanding shell included in our models.
4 Discussion and implications from our model fits
We now discuss the values of the parameters determined for the 11 objects, possible correlations among them, and we compare them with other measurements from the literature.
4.1 The neutral column density
The neutral column density we derive from the fitting of 11 Ly spectra from the FORS Deep Field is radial — along a line of sight, from the center of the shell to the end of the simulation volume — and ranges over more than an order of magnitude, from to cm. However, the majority of objects have a small column density (8/11 have cm).
How do our determinations from Ly fitting compare with other determinations in starbursts?
Carrying out such a comparison is difficult
for LBGs, since H i column densities are usually not measured.
Even for nearby starbursts the available data is scarce.
Calzetti (2007, private communication) has kindly determined for us
from published 21cm RC3 radio observations
and assuming sizes given by
Despite some overlap, is lower in our objects than the column density observed by Calzetti and collaborators, and more similar to the small sample of Kunth et al. An attempt to explain this can be the different ways of determining : for the Calzetti sample the determination of the neutral column density was achieved by radio observations of the whole galaxy, whereas in the case of Kunth et al., is derived from Voigt fitting of the Ly profile, so it only takes into account the neutral gas which influences Ly radiation transfer, even if a Voigt fitting may lead to an underestimate of (Verhamme et al., 2006, see Sect. 4). This may explain why the determination from Kunth et al. is closer to our values than those of Calzetti et al. In any case, to compare our column densities with those of Calzetti one needs to increase our values typically by a factor to convert the radial shell column density to a total one.
The range of found for the FDF objects is also compatible with our confirmation of the neutral column density of the gravitationally lensed LBG MS1215-cB58 (cB58, shown as the blue cross) that we fitted previously (paper II). Indeed, cm is slightly higher than of the FDF Ly emitters, as expected for a Ly spectrum in absorption.
4.2 Ly equivalent widths
The observed Ly EWs range from to Å in the rest frame. Does this range reflect intrinsic differences, or is it somehow related to the physical conditions of the ISM in which Ly radiation transfer takes place ? We examined how EW correlates with other parameters, but no clear correlation is seen. We found a trend in EW with respect to the neutral column density in the shell (see Fig. 11): EW seems to decrease with , at least for the asymmetric profiles (filled circles). In fact, the objects with a low EW ( Å, i.e. FDF1337, 5550, and 7539) can only be fitted with a high value of , since their profiles are very broad. On the other hand, narrow lines with large EW can only fitted with small values of . There may be three exceptions to this trend, FDF4691, 5215, and 1267. The double-peaked profile of 4691 seems peculiar, as it is static and dust-free, as suggested by the SFR(UV) and SFR(Ly) which are almost identical. We imagine that this trend breaks down in dust-free media, or in media with a very small amount of dust, because dust is needed to absorb radiation and decrease the intrinsic EW value. FDF1267 and 5215 are peculiar/degenerate for the reasons discussed above (Sect. 3).
An anti-correlation of EW vs can be understood by radiation transfer effects if the intrinsic EW is approximately constant. When increases, the path length of Ly photons increases, and so does their chance to be absorbed by dust: the Ly escape fraction decreases with increasing , as mentioned above. Furthermore, if we assume a constant dust-to-gas ratio, an increase in naturally leads to an increase of the dust quantity (the optical depth). These two effects explain the decrease of the observed Ly EW with increasing from a theoretical point of view.
Three objects (2384, 5812, 1267) have clearly very large intrinsic EW(Ly) Å, for one (5215) the two solutions give quite disparate results, and the remaining 7 objects have all intrinsic EW of 50–100 Å. Taking the uncertainties in the continuum placement into account (cf. above) we consider that this latter group (7 of 11 objects) have intrinsic EW(Ly) compatible with expectations for star-forming galaxies with a constant star formation history over periods 10-100 Myr, as seen in Fig. 15. The strength of Ly in three high EW objects requires younger ages, irrespectively of their star formation history.
4.3 Dust extinction
For the first time we have derived here constraints on the dust content of galaxies using the Ly profile only. It is therefore of interest to examine how this determination compares with other methods.
Using a version of the SED fit and photometric redshift code Hyperz described in Schaerer & Pelló (2005) and the UBgRIJKs photometry published by Heidt et al. (2003), we have modeled the SED of our objects, assuming the Calzetti attenuation law (Calzetti et al., 2000). Three objects, 5812, 3389, and 6557, have insufficient data (photometry in 3 or less bands) which does not allow meaningful SED fits. 2384 is also excluded, since it appears to be a multiple source, where the Ly emission is clearly displaced from the continuum. Results for the remaining objects are shown in Fig. 12, where we compare the extinction derived assuming models with constant star formation (red filled triangles) or exponentially decreasing SF histories (blue open circles) described by the Bruzual & Charlot templates with derived from our Ly line fits. Note that these values, denoted here as , measure the extinction suffered by the stars, whereas measures that of the Ly emitting gas. The two may differ, as e.g. known to hold empirically for local starbursts between the stellar extinction and the one measured from the Balmer decrement (Calzetti et al., 2000).
Figure 12 shows a good correlation between the different extinction measures, especially when different SF histories are allowed for. Indeed, for the bulk of the objects the extinction derived from Ly profile fitting is between the gas extinction expected from Calzetti’s empirical relation, , and a somewhat lower value of .
We can also estimate the extinction from the UV slope. Excluding again the multiple source 2384, the observed UV slopes show basically two groups, whose extinction, estimated following Calzetti et al. (2000), is 0.–0.02 and 0.16–0.18 shown by the green squares in Fig. 12. These values cover a similar range as , although with a poor correlation. This could be due to the statistical nature of the underlying correlation between attenuation and . We conclude that overall Ly line profile fits allow us to obtain quite consistent extinction values compared to broad band photometry fits of the individual objects. Our derived extinction values, corresponding to 0–0.2 are also in good agreement with the values found for LBGs by Shapley et al. (2003) and others (e.g. Papovich et al., 2001).
4.4 Gas to dust ratio
How do the gas-to-dust ratios obtained from our line profile fits compare with other values observed in starbursts? To address this we turn again to our comparison samples shown in Fig. 10.
Compared to the Galactic average of cm mag and its scatter (cf. Bohlin et al., 1978), our results show somewhat lower gas-to-dust ratios and apparently a larger scatter. The scatter is, however, similar to the one found among the local starbursts also shown in this Fig. Pettini et al. (2000) have already noted this difference in gas-to-dust ratio for cB58, which they suggest could indicate that a significant fraction of the gas is not in atomic form, i.e. is either ionised and/or in molecular hydrogen.
The Calzetti objects have a median of similar to the “classical” Galactic value but showing a wide dispersion. Approximately half of our objects show values lower than those of Calzetti’s local starbursts. For the other half of the sample, and for the LBG cB58 modeled in paper II, the values of the gas-to-dust ratio overlap with those for the Calzetti sample. In any case the comparison may be hampered for the reasons affecting also the comparison (cf. Sect. 4.1). Overall we conclude that the gas-to-dust ratios determined purely from Ly line profile fitting yield values reasonably consistent with local starbursts or somewhat lower. For comparison Vladilo et al. (2007) find higher gas-to-dust ratios in DLAs compared to the Galactic value.
If some LBGs show truly lower gas-to-dust ratios, this may be due to a higher degree of metal enrichment in the outflowing H i gas, and/or an overall smaller fraction of neutral gas being “polluted” in nearby starbursts, and/or a smaller fraction of atomic hydrogen. However, the column densities found in LBGs are also consistently lower than in local starbursts (cf. Fig. 10), which may indicate a different “regime”. Other, independent determinations of the gas-to-dust ratio in LBGs and other starbursts and more detailed examinations would be required to clarify these issues and to understand the possible physical causes for these apparent differences.
4.5 Velocity of the outflow
Assuming a galaxy-scale outflow surrounding our 11 objects (the relevance of this model is discussed in Sect. 2), we have derived the expansion velocity of the expanding shell from the Ly spectral shape for 8 objects, and deduced it for the 3 other objects (1337, 5550, 7539) from the measurement of a shift between the LIS absorption lines and the Ly emission. Overall 9 objects out of 11 have shell velocities around 150-200 km s, and 2 objects present almost static shells. No correlation between the expansion velocity and other parameters is found. We now briefly discuss the high velocity outflows and the few nearly static cases. Beforehand we can already mention that we have no explanation for the causes leading to low ISM velocities or allowing to distinguishing these objects from the more common cases showing outflows.
High velocity objects
All objects presenting an asymmetric emission line are reproduced with km s. This velocity range is very similar to the determination of Shapley et al. (2003), from the blueshift of LIS absorption lines with respect to stellar lines in a sample of LBGs at redshift , as well as in cB58, where the outflow velocity is estimated to km s (Pettini et al., 2002).
Two objects with a more peculiar spectral shape, FDF 1267 and 5215, can also be fitted with higher expansion velocities ( km s), a high , and high dust content. However, 1267 has such a large observed equivalent width ( Å) that it may rather have a small column density to fit better in the plot showing a correlation between the observed Ly EW and (see Fig. 11). In passing we note that this peculiar spectral shape seems also to be found in observations of an LBG at redshift by Vanzella et al. (2008) in the GOODS-South field.
Low velocity objects
The two double-peaked spectra (4691 and 7539) are characterised by a static (or almost static, km s) surrounding shell. In the case of 7539, the velocity shift between the LIS absorption lines and Ly is even measured ( km s), so the shell velocity is here an observational constraint (). This peculiar spectral shape (double peaks) is predicted by theory, arising from Ly resonant scattering through static H i media (Neufeld, 1990), but surprisingly, observed double-peaked profiles are in general not interpreted as a signature of Ly radiation transfer through static media (Fosbury et al., 2003; Christensen et al., 2004; Wilman et al., 2005; Venemans et al., 2005; Vanzella et al., 2008), even if they appear much less common than the asymmetric emission observed in all high-z LAEs. In the static case the separation of the peaks is not only related to the thermal velocity of the H i gas, but depends also on the H i column towards the Ly source, . For a homogeneous slab one has:
where is the light speed, is the restframe wavelength of Ly, is the Doppler parameter, and is the location of the peaks in units of the Doppler width (Neufeld, 1990; Dijkstra et al., 2006b). For example, for km sand cm one obtains Å. If applied to the double peaked Ly object of Vanzella et al. (2008), the observed velocity shift of 13 Å would indicate cm for km s. Establishing accurate enough galaxy redshifts for these objects is important to be able to assert if one is truly dealing with a nearly static case (in which case zero velocity is between the two Ly peaks), outflows (with both Ly peaks redshifted), or other situations.
4.6 Observed Ly Fwhm
A possible correlation may be found between the observed FWHM and the neutral column density of the expanding shell, as shown in Fig. 13. If real, it may be used to estimate the H i column density in starbursts from a simple measurement of FWHM(Ly). Such a correlation can easily be explained by radiation transfer effects: the Ly optical depth increases with so that Ly photons have to diffuse further in the wings to be able to escape the medium, which naturally broadens the Ly red wing.
Another point worth noticing about FWHM is that we were able to fit all observed Ly profiles with an intrinsic FWHM of km s, except for 4691, for which we have to start with an already very extended intrinsic Ly line to reproduce the very broad double-peaked profile. Otherwise, more exotic scenarii, like a shell with a hole, have to be invoked.
Finally, the two correlations presented above (FWHM vs and EW vs , Figs. 11 and 13) explain the observed anti-correlation shown by T07 between FWHM and EW. Indeed, they are both related to the neutral column density which surrounds the starburst. The EW increases with decreasing and the FWHM increases with increasing . Therefore, we should not observe objects with a large EW and very broad lines, what is also confirmed by the observed FWHM of LAEs which are always below km s (Rhoads et al., 2003; Dawson et al., 2004; Venemans et al., 2004). If the tentative correlation between FWHM and really holds it would imply a maximum column density of cm in LAEs.
4.7 Escape fraction
The Ly line escape fractions derived for our FDF objects range from for the dust-free object FDF4691 to for objects with the highest extinction. From our modeling we find that the main parameter determining the escape fraction is the dust amount in the shell. Furthermore, from all our modeling results already discussed above, it is quite clear that no single value of the Ly escape fraction is expected from LBGs and LAEs, in contrast to simplifying assumptions made in some models (e.g. Le Delliou et al., 2006).
Our model grids predict the following behaviour for the escape fraction:
increases with increasing , because the Ly optical depth decreases in a fast moving medium compared to the static case. Therefore the mean path of Ly photons in the medium decreases, so their chance to be absorbed by dust too.
decreases with increasing , because the Ly optical depth is proportional to .
decreases with increasing , obviously since with a larger number of absorbers in the medium, Ly photons increase their chance to interact with them.
However, the only clear correlation we found in the data between and other parameters is with (see Fig. 14). No correlation is found in particular with (our objects cover probably a too small range in velocity range), and with (the variation in the dust-to-gas ratio is probably more dominant). On Fig. 14, we see also that, as expected, Ly photons are more attenuated by dust than the continuum; indeed all LBG data points (circles) are located below the solid line corresponding to . The reason is that multiple resonant scattering of Ly photons increases their path through the medium, and hence their chance to be absorbed by dust, compare to the continuum. Note that other dependences of on could be expected with other geometries. For example in clumpy media, the reflection of Ly on the clump surfaces could ease the Ly transmission for the same amount of dust (Neufeld, 1991; Hansen & Oh, 2006).
We propose a fit to predict the escape fraction of Ly photons knowing the dust extinction (dashed curve on Fig. 14):
Note that E(B-V) in the precedent formulae is the extinction in the gas, which may be different from the extinction of the stars (Calzetti et al., 2000), as already mentioned above. Interestingly, the two static objects are also fitted by this formulae, which illustrates that dust is really the dominant parameter which governs the Ly escape in our objects. One of these (4691) is dust-free, so its escape fraction is , but in the other object (7539), 30% of the Ly photons escape the medium. For the same extinction, moving media present an escape fraction of 40-45%, which is coherent with the theoretical prediction that increases with .
Empirical Ly escape fractions have recently been measured by Atek et al. (2008) from imaging for a sample of 6 local starbursts. Their values are compared to our data for LBGs in Fig. 14. For , our results are in good agreement with three local objects. SBS 0335-052 with an integrated extinction of shows no Ly emission, it is a net absorber. For larger values, the Ly escape fraction of two local starbursts (Haro 11 and NGC 6090) are higher than predicted by our fit to the LBGs studied here. Deviations from a simple homogeneous shell geometry are the most likely explanation for this difference. This will be testable through detailed modeling both of the spatially resolved and integrated properties of the local objects.
Since the Ly line flux is more strongly reduced (due to multiple scattering effects) than the adjacent continuum, the Ly equivalent width depends on the extinction. This phenomenon is added to the one already known to result from the extinction difference between the gas and the stellar continuum. In principle, a measurement of EW(Ly) could thus be used to determine the extinction, provided the intrinsic equivalent width EW is known. Concretely, the fit-relation between and proposed above (Eq. 3) translates to the following behaviour of the Ly equivalent width with extinction:
where and E(B-V)/E(B-V) according to Calzetti (2001). Adopting reasonable values for EW (e.g. from Fig. 15), this formula may be used to obtain a crude estimate of the extinction in LAEs based on a pure equivalent width measurement. An extinction corrected SFR(Ly) value can then be obtained from the Ly luminosity using an appropriate SFR calibration from Fig. 15, consistently with the assumed value of EW.
4.8 SFR indicators
Given our quantitative analysis of Ly radiation transfer, the determination of the Ly escape fraction and of the extinction, we are now able to examine to what extent Ly and the UV continuum provide consistent measures of the star formation rate. The main results of this exercise are shown in Fig. 16
First we note that three of our objects (1267, 4454, 5812) show observed, i.e. uncorrected SFR values corresponding to SFRSFR. Such a result need not be inconsistent; this behaviour is indeed expected for young bursts or objects where constant star formation has not yet proceeded over long enough timescales, i.e. for timescales 10–100 Myr as shown in Fig. 15. In this case values up to SFRSFR can be obtained; this allowed range of values for SFRSFR for constant SF is shown by the shaded region on Fig. 16. Indeed these objects also show among the largest EW(Ly), as expected from Fig. 15 for relatively young, but on average constantly star-forming, objects. Second, 4691 shows SFR(Ly) SFR(UV) (observed), indicative of little or no dust, and confirmed by our modeling. Finally the remaining 7 objects show with UV star formation rates up to 14 times larger than Ly, a result found for the majority of LBGs and LAEs (e.g. Yamada et al., 2005; Gronwall et al., 2007).
We now correct these SFR indicators for the effects of dust and radiation transfer. O