© ESO, 2011

1. Introduction

Interstellar dust plays a key role in a variety of physical and chemical processes relevant to galactic evolution. The presence of dust affects the observed spectral energy distribution of astronomical sources and diffuse extragalactic backgrounds. Characterizing the properties of dust at high redshift, even if challenging, is an important step to improve our understanding of the early Universe. In particular, models of dust production and evolution in galaxies (e.g. Dwek 1998; Calura et al. 2008; Zhukovska et al. 2008) would greatly benefit from new observational constraints on the dust properties at early stages of chemical evolution.

Quasar spectroscopy is an excellent tool for probing the dust at high-redshift: the quasar absorption line systems with neutral hydrogen column density¹N(H   i) ≳ 10^20.3 atoms cm^-2, known as damped Ly α (DLA) systems, originate in the interstellar medium of foreground galaxies (Wolfe et al. 1986, 2005) and probe the dust embedded therein.

Evidence of dust in individual DLA systems has been found as a result of searches for classical dust indicators, such as the 217.5 nm extinction “bump” (Junkkarinen et al. 2004; Wang et al. 2004; Srianand et al. 2008a), the 9.7 μm silicate absorption (Kulkarni et al. 2007, 2011), and the quasar reddening due to foreground absorption (Vladilo et al. 2006; Jiang et al. 2010). The DLA systems investigated in these studies offer unique opportunities for characterizing the dust in distant galaxies. However, these dust-rich absorbers are the exception rather than the rule: the bulk of the DLA population is believed to be dust-poor. Evidence of this comes from the extremely weak signal of mean reddening in quasar samples with intervening absorbers (Murphy & Liske 2004; Vladilo et al. 2008; Frank & Peroux 2010) and from studies of radio-selected quasar samples unbiased by dust extinction (Ellison et al. 2001; Jorgenson et al. 2006). The low level of dust is broadly consistent with the low level of metal enrichment of DLA systems (Pettini et al. 1994).

In this work, we are interested in characterizing the properties of dust in the metal-poor population of DLA systems to improve our understanding of the early build-up and chemical evolution of dust in galaxies. In particular, we wish to trace the presence of silicon in dust, as a preliminary step for studies of silicates at high redshift. Silicates are a widespread component of interstellar dust, representing up to  ~ 70% of the core mass of dust grains (Draine 2003; Henning 2010). The first detections of silicate absorption at 9.7 μm in dust-rich DLA systems at low redshift (Kulkarni et al. 2007, 2010) indicate that times are ripe for understanding whether or not silicates are pervasive in the bulk of the DLA population at z_abs ≳ 2.

To accomplish our goal, we rely on an observational technique that is effective in finding the signature of dust even in metal-poor systems, i.e. the study of abundance ratios of elements with different affinities with dust. Chemical abundances of DLA systems can be obtained with high accuracy from the analysis of optical/UV absorption lines. Atoms locked into dust grains do not contribute to the optical/UV absorption-line cross-section² and are not counted in the column density determinations. For this reason, the abundances of refractory elements, which tend to stay in the dust, appear to be depleted relative to the abundances of volatile elements, which tend to stay instead in the gas. Therefore measuring the abundance ratios of volatile and refractory elements allows us to probe the dust in DLA systems.

The first application of this technique to DLA systems was based on the Zn/Cr ratio (Pettini et al. 1994, 1997). The apparent overabundance of the Zn/Cr ratio in DLA systems is similar to that found in the Galactic ISM, where we know that zinc is volatile while chromium is refractory. Starting from this study, it became clear that it is unsafe to trace the metallicity³ of DLA systems using refractory elements, such as iron. For this reason, zinc has become the first choice in studies of DLA metallicity, yielding a typical value  [Zn/H]  ≃  −1.1 dex (Pettini et al. 1994; Akerman et al. 2005), with a large dispersion at each redshift and a gradual rise with cosmic time (e.g. Vladilo et al. 2000; Prochaska et al. 2001).

Zinc has also become the reference element for estimating the differential depletion of refractory elements, using abundance ratios such as Zn/Fe or Zn/Cr. To measure the absolute depletion of the refractory element from the observed ratios, we need to know in advance the intrinsic total ratios (gas plus dust) and also to take into account that zinc can be slightly depleted. By adopting a solar, or slightly supersolar, intrinsic Zn/Fe ratio, as observed in Galactic metal-poor stars, and by modeling zinc depletion using Galactic ISM observations, in a previous work we were able to measure iron depletions in 38 DLA systems (Vladilo 2004). In this way we found evidence for a rise of iron depletion with metallicity. This rise disagrees with the expectations, based on studies of the local Universe, that the dust-to-metal ratio should be roughly constant in the course of chemical evolution.

In the present work, we wish to study the depletion of silicon. Measuring silicon depletion in metal-poor galaxies is rather challenging because silicon is less depleted than iron and is expected to be hardly depleted at all in metal-poor gas. This probably explains why searches for silicon depletion in the Small Magellanic Cloud (SMC) have provided contrasting results (Welty et al. 2001; Sofia et al. 2006). In DLA systems, Si depletion has only been estimated in a few dust-rich molecular absorbers (Noterdaeme et al. 2007; Jiang et al. 2010). More often, the very existence of silicon depletion has not been considered, to the point of treating silicon as a dust-free metallicity indicator in the absence of abundances of volatile elements.

The challenge of measuring silicon depletion in DLA systems must be confronted with new tools. Here we propose a method specifically developed for this purpose that has two main novelties relative to previous work. The first is that we use sulfur, in conjunction with zinc, to define a pair of volatile elements to be used as a dust-free reference for abundances. The second is that we use models of galactic chemical evolution designed for DLA systems to estimate the total abundance ratios (gas plus dust) used to derive the depletions. In practice, we use the S/Zn ratio as a dust-free diagnostic of chemical evolution to constrain the chemical history of DLA systems. This work has been possible thanks to the continuous accumulation of new S ii column density measurements in DLA systems, starting from the first work designed to study the S/Zn ratio as a dust-free indicator of chemical evolution of DLA host galaxies (Centurión et al. 2000).

Our adopted method is described in the next section. The observational database, which includes a few new measurements of S ii column densities, is presented in Sect. 3. The models of galactic chemical evolution for DLA systems are discussed in Sect. 4. The depletions are presented in Sect. 5 and discussed in Sect. 6.

2. The method

2.1. Assumptions

The abundance ratios that we measure in DLA systems are determined by three processes operating in the gas of the host galaxy: (1) chemical evolution, due to the enrichment of the gas by stellar ejecta; (2) elemental depletion, caused by the incorporation of part of the metals into dust grains; and (3) ionization effects, produced by radiation fields penetrating the neutral gas. To the best of our knowledge, these are the only processes that can affect the measured abundances. In DLA systems, ionization corrections are generally smaller than the column density errors (e.g. Vladilo et al. 2001), with some exceptions (e.g. Prochaska et al. 2002) that we exclude from our samples. In our analysis, we therefore assume that the abundance ratios are uniquely determined by chemical evolution and elemental depletion, other effects being negligible.

2.2. Definitions from Galactic ISM studies

2.2.1. Depletions in the Galactic ISM

The elemental depletion of an element X is usually defined in logarithmic notation, with the expression (1)where (X/H)_gas is the number ratio of atoms in the gas, and (X/H)_ref a reference value for the total number ratio (gas plus dust). In practice, the ratio in the gas is measured from the column densities of the dominant ionization stages in H i regions⁴, i.e. (X/H)_gas = N(X_dom)/N(H   i). In Galactic ISM studies, the total ratio (gas plus dust) is assumed to be solar. This gives the empirical expression (2)The depletions measured in this way are negative for most elements. This indicates that a fraction of the atoms is missing from the gas (Savage & Sembach 1996; Jenkins 2009). The strength of the depletion correlates with the condensation temperature, a quantity obtained from thermodynamical computations of particle condensation in a cooling gas of solar composition (Field 1974; Lodders 2003). The trend with condensation temperature suggests that the fraction of atoms missing from the gas is locked into dust grains.

2.2.2. Volatile and refractory elements

Refractory and volatile elements can be distinguished on the basis of either the depletion strength (Savage & Sembach 1996; Jenkins 2009) or the condensation temperature (Lodders 2003). For the purpose of our discussion, we recall that magnesium, silicon, and iron are classified as refractories; they have similar condensation temperatures (T_cond ≈ 13  ×  10² K) and all of them are depleted; however, the depletions of silicon and magnesium are milder than the depletion of iron (see Table 3). Zinc and sulfur are classified as volatile elements because they display negligible or small depletion and because they have low condensation temperatures (T_cond  ≈  7  ×  10² K).

2.3. Equations of the method

Equation (2) cannot be used to measure depletions in extragalactic studies because the assumption that the reference abundances are solar is not in general acceptable. A starting point in tackling this problem is to substitute hydrogen with a volatile element V, in Eq. (1). If the depletion of V is negligible, then (V/H)_gas = (V/H)_gas + dust = (V/H)_ref and Eq. (1) is equivalent to (3)In this way, the measurement of the depletions relies on a set of relative abundances X/V (metal-to-metal), rather than absolute abundances X/H (metal-to-hydrogen). Relative abundances are less sensitive to chemical evolution than absolute ones and, in some cases, one can assume that relative abundances are solar even when the absolute abundances are not⁵. In general, however, this will not be true. In our method, we calculate sets of reference abundances (X/V)_ref for DLA galaxies, without relying on the solar abundance pattern.

The ratio (X/V) will vary in the course of chemical evolution. Since DLA host galaxies are observed at different stages of their history, we need to estimate a specific value of (X/V)_ref for each DLA system. We assume that the evolutionary state of the ith DLA galaxy can be deduced from the absolute abundance of the gas sampled by the line of sight. The volatile element is used to measure an absolute abundance unaffected by dust depletion . To estimate (X/V)_ref, we use a model of galactic chemical evolution developed for DLA galaxies, ℳ, that calculates (X/V) as a function of (V/H). The value predicted by this model at is adopted as the reference value of Eq. (3), i.e. (X/V)_ref,i = (X/V)_ℳ,i. The depletions are calculated with the expression (4)where is the ratio in the gas measured from the observed column densities.

2.4. Model constraints

A key point of the method is the choice of the models of galactic chemical evolution. To this end, we consider two volatile elements, V and V′, that have a large number of column density measurements in DLA systems. We then use the observational diagram  versus as a dust-free diagnostic of the chemical evolution of DLA host galaxies. In practice, starting from a grid of models of galactic evolution, we select the model that minimizes the differences between the observed ratios,  and the ratios (V′/V)_ℳ,i predicted at . Once a reference model is selected, we use its predictions (X/V)_ℳ,i to measure the depletions of refractory elements X with Eq. (4).

In principle, we can choose any pair of volatile elements when applying the method. In practice, zinc and sulfur are the only suitable choice to ensure a sufficient statistics (see Sect. 3). We therefore use the observed evolution of S/Zn versus Zn/H ratios to constrain the models. In the absence of other dust-free diagnostic tools of this kind, we also introduce an age test to restrict the choice of the models. This test exploits information on the age of DLA galaxies that can be obtained by their absorption redshifts. The detailed implementation of the method is presented in Sect. 5.1.

2.5. Caveats from Galactic ISM studies

2.5.1. Zinc and sulfur depletion

No element is perfectly volatile and zinc and sulfur are no exception. In highly depleted Galactic lines of sight, these elements also appear to be depleted (Savage & Sembach 1996; Jenkins 2009). Since DLA systems are generally dust-poor, zinc and sulfur will be undepleted in most cases. However, they can be depleted in some DLA systems and we must assess the consequences of this effect on our method. To identify dust-rich systems, we use the molecular fraction f(H₂) ≡ 2   N(H₂)/ [N(H   i + 2   N(H₂)] . This quantity correlates with depletion in both the ISM (e.g. Savage & Sembach 1996) and DLA systems (e.g. Noterdaeme et al. 2008). Even zinc and sulfur could be depleted in lines of sight with high f(H₂). Chemical pathways that may lead to sulfur depletion in molecular clouds were discussed by Scappini et al. (2003).

If present, zinc and sulfur depletion would affect our procedure at two stages: first in the selection of the models of galactic chemical evolution and then in the measurements of the depletions.

In the first stage, the presence of zinc and sulfur depletion would affect the process of minimization of the χ² defined in Eq. (6). Since the critical quantity in this process is the ratio S/Zn, the procedure would be mainly affected by the differential depletion between zinc and sulfur. To the extent at which the depletions of these two elements track each other, the differential depletion may be negligible. In any case, to minimize the impact of zinc or sulfur differential depletion on our procedure, we excluded molecular DLA systems during the stage of model selection. More details are given in Sect. 5.1.

After the models are selected, the presence of zinc and/or sulfur depletion would affect the measurements of the depletions of the refractory elements. If the depletion of the volatile element, δ_V,i, is not negligible, Eq. (4) should be re-written as (5)To derive δ_X,i, one could model the unknown value of δ_V,i by using a scaling law between δ_V,i and δ_X,i based on Galactic ISM studies, and solving the equation by iterations (e.g. Vladilo 2004). However, in the present work we prefer not to use relations between the depletions obtained from Galactic ISM studies. These relations might not hold in the very first stages of dust production, when the chemical processes determining the dust composition might differ from the present-day ones. In any case, we expect δ_Zn and δ_S to be small in most cases, with the exception of molecular DLA systems. In these cases, zinc and sulfur could be depleted and the depletions of refractories calculated with Eq. (4) should be considered a lower limit to the true value.

2.5.2. Sulfur abundances in partially ionized regions

In some Galactic lines of sight, the interstellar S/H ratio has peculiar values, apparently in excess of the solar ratio (Jenkins 2009). Before using sulfur as a reference element for our analysis, we investigated the reason for this behavior. An explanation can be found in terms of ionization effects. Owing to the large difference between the ionization potentials of S ii (23.3 eV) and H i (13.6 eV), S ii can be the dominant state of sulfur in partially ionized regions, where a fraction of H i is converted to H ii. Therefore in partially ionized regions the total H, calculated from H i, is underestimated and the ratio S/H overestimated. According to this explanation, the apparent S/H excess should only appear along lines of sight with low N(H i) column density, typical of partially ionized regions. Indeed, by using Jenkin’s database (Tables 2 and 13 of that paper), one can see that the anomalous ratios (e.g.  [S_gas/H]  ≳  + 0.2   dex) are only found when N(H   i) ≲ 10^19.5 cm^-2. We conclude that this effect is not a reason of concern in our analysis because we consider absorption systems with high H i column density, N(H   i) ≥ 10^20.1 cm^-2, which are typical of neutral regions.

3. The data

The present study requires a large database of DLA column densities: for the volatile elements, we need a good coverage in metallicity to constrain the models of chemical evolution; for the refractory elements, we need many measurements to characterize the depletion properties as a function of metallicity and redshift. It is only thanks to the accumulation of high quality literature data that this task has become possible.

The number of column-density measurements that has been accumulated for the refractory elements is generally much higher than that of the volatiles. Among the refractories, we consider here silicon, the focus of the present investigation, and iron, for the sake of comparison with previous studies of depletion in DLA systems. In addition, we consider magnesium, another important ingredient of silicates poorly investigated in DLA systems. Silicon and magnesium are expected to behave similarly because they have similar condensation temperatures (Lodders 2003) and Galactic depletions (Savage & Sembach 1996). Therefore, magnesium results, even if scarce, can be used as an independent cross check of silicon results.

The dominant ionization states of magnesium, silicon, and iron are Mg ii, Si ii and Fe ii. The number of Si ii and Fe ii measurements is large because these ions have enough transitions to cover a wide range of wavelengths and oscillator strengths. The wavelength coverage translates into a good coverage of absorption redshift; the range of oscillator strengths yields accurate measurements, unaffected by saturation, over a wide interval of metal column density. The number of Si ii and Fe ii DLA column densities currently available is 131 and 167, respectively, with the selection criteria specified in Sect. 3.2. The number of reliable measurements of Mg ii is very small because the resonance doublet at  ≃280 nm is usually saturated. We therefore used measurements obtained from the doublet at 124 nm, which is hard to detect because it is weak and falls inside the Ly α forest of the quasar spectrum. We only found data for ten DLA systems. This small number is still sufficient for comparison purposes.

The bottleneck in our procedure is the number of available column densities of the volatile elements, which are essential for constraining the models and measuring the depletions. The two dominant ions of volatile elements with the largest number of accumulated measurements are Zn ii and S ii, with a total of 94 and 64 data, respectively (excluding our S ii data presented below). The only Zn ii usable transitions fall at λλ202.6,206.2 nm in the rest frame. These lines are generally unsaturated, but can be too faint in DLA systems of low metal column density. The S ii lines fall at λλ125.0,125.3,125.9 nm. The presence of these three lines in the same portion of the spectrum, with a good coverage of oscillator strengths, makes it possible to obtain accurate S ii column densities despite these lines falling inside the Ly α forest.

The combined zinc-sulfur sample (33 DLA systems without our S ii measurements) does not have a perfect match in redshift because the Zn ii lines are easier to detect at z_abs ≲ 3, while the S ii lines can only be detected at z_abs ≳ 2 with ground-based instrumentation. This non-ideal coverage in redshift is not a severe limitation for the application of our method, since the models are constrained in metallicity rather than in redshift. The metallicity interval spanned by the combined Zn ii + S ii sample covers most of the metallicities observed so far in DLA systems, which are concentrated in the interval  − 2 ≲  [Zn/H]  ≲ 0, with a few exceptions.

Since S ii is currently the bottleneck, we performed an extensive analysis of archive spectra to obtain new measurements of N(S   ii). We report below the results of this search. In Sect. 3.2, we present the literature database.

3.1. New measurements of S ii column densities

In our search for new measurements of S ii column densities, we used the ESO archive of spectra collected with the UVES spectrograph fed by the unit 2 of the ESO VLT (Very Large Telescope) at Cerro Paranal (Chile). The high spectral resolution (R = λ/Δλ ≃ 5  ×  10⁴) combined with high spectral coverage and high efficiency in the visible ultraviolet makes this instrumentation particularly well suited to searching for lines in the Lyman α forest, as in the case of S ii lines at λ ≃ 125 nm.

Fig. 1 Multi-component Voigt profile fits of S ii and Fe ii lines for the DLA systems at zabs = 2.2930 in Q0216+080 (left panels) and at zabs = 1.9200 in Q2206-199 (right panels). In each spectral region, we indicate the ion and rest-frame wavelength of the transition; the continuum and zero levels are indicated with solid and dashed lines, respectively. The radial velocity scale is obtained from the wavelength scale using the redshift of the strongest absorption component of each system as a reference. The velocity components used in the fits are marked with vertical ticks. The positions of the S ii velocity components (top panels) were constrained from the analysis of Fe ii lines outside the Ly α forest (bottom panels). The spectral region around the redshifted 2249 Å Fe ii lines in Q2206-199 (bottom right panel) is disturbed by a few cosmic events (emission spikes).

From our search of the ESO archive, we found UVES spectra relative to 18 DLA systems with unpublished S ii data. We analyzed these spectra with the MIDAS software package developed at ESO. An average spectrum was derived from the multiple spectra present in the archive for each DLA system. The average spectra were normalized to the local continuum of the quasar. For more details on the analysis of these UVES spectra we refer to previous work of ours (e.g. Centurión et al. 2003). The column densities were derived by fitting the normalized spectrum with the Voigt-profile fitting routines FITLYMAN (Fontana & Ballester 1995) implemented in MIDAS. Oscillator strengths for the transitions of interest were taken from Morton (2003).

We detected S ii in 3 of the 18 systems analysed. The results of these systems are shown in Table 1. For two of them, we found a clear detection in at least two lines of the S ii triplet, as we show in Fig. 1. The column densities were obtained by fitting the multiple absorption components seen in each DLA system. The quoted column-density errors, provided by FITLYMAN, do not include uncertainties associated with the tracing of the continuum. The redshift position of each S ii component was constrained from the analysis of Fe ii lines outside the Ly α forest. The comparison between S ii and Fe ii normalized profiles was also used to check for potential contamination of Lyα forest. The Fe ii column density was also measured as a cross-check with previous work. The Fe ii measurements that we find are in good agreement with previous measurements in the same systems.

For a third DLA system, z_abs = 2.0950 in Q2359-0216, we did detect the S ii line at 125.0 nm, but not the other two lines in the same range, owing to contamination by the forest. In this case, the quality of the spectra was also lower. The S ii measurement of this system was not considered in the subsequent analysis.

3.2. The column density database

The data used in this work were selected from 293 DLA systems with published measurements of column densities. These include the data presented in Table 1. Only measurements were considered, rather than lower or upper limits. Most of the original observations were obtained at high spectral resolution, R ≈ λ/δλ ≃ 5  ×  10⁴, with the UVES spectrograph, at the unit 2 of the ESO Very Large Telescope, and with the HIRES spectrograph at the Keck telescope. Data obtained from low-resolution spectra (e.g. SDSS spectra) were not considered. More details on the observational data can be found in the original papers listed in Table 7. Here we explain the criteria followed to build the database.

We considered all quasar absorption systems with N(H   i) ≥ 10^20.1 cm^-2. This threshold is slightly lower than the classical limit that defines DLA systems, i.e. N(H   i) ≥ 10^20.3 cm^-2 (Wolfe et al. 1995). By relaxing this limit, we have been able to include in our sample eight “sub-DLA” systems that allow us to probe the high end of the metallicity distribution (Péroux et al. 2007; Prochaska et al. 2006). Most sub-DLAs in our sample have N(H   i) consistent with the classic DLA limit within the errors. For simplicity, we refer to as DLA systems all the quasar absorbers of our sample.

A reason of concern in our analysis is the possible presence of ionization effects that could alter the measured abundances. Therefore, while collecting the column density data, we searched the literature for reports of ionization studies. As a result, we discarded from our sample two cases affected by ionization, namely the systems at z_abs = 2.625 in GB 1759+7539 (Prochaska et al. 2002) and at z_abs = 2.067 in Q0450-13 (Dessauges-Zavadsky et al. 2006). We checked with particular attention the systems with lowest values of N(H   i), for which ionization could be more relevant. However, we did not find any evidence of enhanced ionization effects in our sample of sub-DLA systems. In constrast, for the system at z_abs = 1.768 in Q0953+5230, the measured Al⁺⁺/Si⁺ ratio is consistent with predominantly neutral gas (Prochaska et al. 2006). This system is particularly important for constraining the high end of the metallicity evolution in our models of galactic chemical evolution.

While assembling the database, we also checked for proximate DLA systems (PDLAs) with z_abs ≃ z_qso, which in principle could be severely ionized by the radiation of the nearby quasar. Detailed studies of individual PDLAs have not revealed peculiar ionization effects (Møller et al. 1998; Rix et al. 2007). The most recent study of abundances in PDLAs reports some evidence of enhanced ionization, particularly at low N(H   i); however, no distinct trends are found in Zn/Fe and Si/Fe abundance ratios compared with intervening DLAs (Ellison et al. 2010). In our accumulation of literature data, we found several PDLAs with relative velocity separation from the quasar ΔV < 3000 km s^-1. For our analysis, we kept five cases with ΔV > 1000 km s^-1 and log N(H   i) ≳ 20.7 and rejected the other cases more prone to ionization effects. The five PDLAs in our sample yield results generally in line with intervening DLA systems (see e.g. Fig. 5).

The metal column densities that appear in the literature are measured with two methods: Voigt profile fitting and the apparent optical depth (AOD) method (Savage & Sembach 1991). In the absence of saturation, these two methods yield identical values of total metal column density. With the Voigt method, it is possible to solve the multiple-component velocity structure generally present in the absorption profiles of the metals. However, this structure cannot be solved in the H i profiles and for this reason we are interested in the total column density of the metals, the only one that can be compared to N(H   i) in the abundance ratios. When a detailed multiple component analysis was reported by the authors, we checked that the bulk of the column density of interest originated in the same velocity components.

The errors in the metal column densities quoted in the literature, σ_m, generally take into account the statistical noise of the measurements, but no other uncertainties, such as the error in the continuum tracement. For this reason, values of σ_m ≃ 0.01 dex, or even smaller, are sometimes reported. These errors are sometimes unrealistic, in which case they tend to overestimate the true χ² (Eq. (6)). To reduce this effect, we used the published error when σ_m ≥ 0.02 dex and adopted instead a minimum value of 0.02 dex in the other cases. This criterion was also consistently applied to our new measurements shown in Table 1.

3.3. Observational samples

We refer to as the “(Zn+S) sample” the set of DLA systems with both S and Zn column density measurements. These are the systems for which we can fully apply our method. In addition to this sample, we also considered DLA systems with either Zn or S column densities available, but not both of them. For these systems, it is still possible to apply our method with some limitations described below.

For brevity, we refer to as the “ sample” the set of DLA systems with Zn ii (but not S ii) data and the “ sample” the set of DLA systems with S ii (but not Zn ii) data. The advantage of using the  and  samples is twofold: we can increase the statistics and probe a larger redshift interval (Fig. 6). The (Zn+S), and  samples are listed in Tables 4 − 6, respectively. These samples contain 35, 47, and 29 DLA systems, respectively, including our S ii data. For each absorber, we give the column densities of the volatile elements and references for the literature sources consulted for both volatile and refractory elements.

4. Models of galactic chemical evolution

Our choice of chemical evolution models was based on previous studies of the nature of DLA systems. This has been the subject of intense debate in the literature owing to the difficulty of revealing the nature of the absorbing galaxies on pure observational grounds.

Direct imaging of the field of the background quasar should ideally reveal the morphology of the intervening galaxies. Unfortunately, it is extremely difficult to solve the galaxies from the quasar light and to choose between different candidate galaxies present in the same field. At intermediate redshift (z_abs ≲ 1), where this task is more feasible, the DLA candidate galaxies show a variety of morphological types, spirals not being predominant (Le Brun et al. 1997). At higher redshifts (z_abs ≳ 2), where most known DLA systems are found, the observations are rather sparse and generally consistent with these systems being objects of low emissivity and low rates of star formation (Weatherley et al. 2005; Péroux et al. 2011).

Studies of high-resolution absorption-line spectra cast light on the kinematics and the chemical abundances of DLA galaxies. However, kinematical studies are not conclusive: the statistics of the radial-velocity profiles is consistent with a variety of possible origins, ranging from progenitors of present-day spirals (Prochaska & Wolfe 1997) to proto-galactic clumps (Haehnelt et al. 1998).

Studies of chemical abundances have tried to solve this issue by comparing DLA abundance ratios with ratios observed in stars or galaxies of similar metallicity. In particular, the ratio of α-capture to iron-peak elements has been used to constrain the chemical history of DLA galaxies. The high values of the Si/Fe ratio at low metallicity observed in DLA systems, if taken at face value, are consistent with these systems being possible progenitors of the Milky Way (e.g. Lu et al. 1996). However, differential depletion is likely to enhance the Si/Fe ratios and the study of dust-free α/Fe ratios suggests that DLA systems behave differently from metal-poor Milky Way stars (e.g. Centurión et al. 2000; Vladilo 2002).

Models of galactic chemical evolution have been used to cast light on the history of the chemical enrichment of DLA galaxies (e.g. Matteucci et al. 1997; Jimenez et al. 1999; Prantzos & Boissier 2000b). To cope with the problem of differential depletion, dust-corrected abundances have been used in some studies (e.g. Calura et al. 2003). These studies together imply that DLA systems originate in galactic regions with low rates of star formation, such as dwarf galaxies or external regions of spirals (e.g. Dessauges-Zavadsky et al. 2004). An origin in elliptical galaxies is ruled out.

The motivation behind the present work is not to determine conclusively the nature of DLA galaxies but to calculate the implications in terms of depletions of different, plausible models of chemical evolution of DLA galaxies. In our choice of the models, we considered both dwarf irregular galaxies, with a broad range of parameters, and spiral galaxies, without making any selection a priori. We then used our methodology to select the models that most closely reproduce the observational trend S/Zn versus Zn/H. The models differ in their star formation histories, but share the same, up-to-date nucleosynthetic prescriptions: stellar yields were taken from François et al. (2004). We chose these yields because they were tuned to reproduce at best the abundance patterns in the Milky Way. In the case of sulfur, we calculated the yield by using the same procedure followed by François et al. (2004). We now introduce the main features and parameters of the models. The process of model selection is explained in Sect. 5.1.

4.1. Models of spiral galaxies

The models for spiral galaxies are based on the chemical evolution model of the Milky Way (Chiappini et al. 1997). The Galaxy is assumed to have formed by means of two main infall episodes: the first forms the halo and the thick disk, the second the thin disk. The formation of the thin disk is assumed to be a function of the galactocentric distance, leading to an inside-out scenario for the Galaxy disk build-up (Matteucci & François 1989). The Galactic disk is approximated by several independent rings, 2 kpc wide, without any exchange of matter between them, and for each ring we compute the chemical evolution.

This model has been revised with time to reproduce an extended set of observational constraints, in particular the G-dwarf metallicity distribution and various relations between the abundances of metals (C, N, O, α-capture elements, iron-peak elements) as functions of the [Fe/H] abundance (see François et al. 2004; Chiappini et al. 2006; and Cescutti et al. 2009). Although this model is not the only one capable of reproducing the data, it does reproduce the majority of the Milky Way properties, and most of its assumptions are shared by other authors (e.g. Prantzos & Boissier 2000a; Alibés et al. 2001; Chang et al. 1999). By adopting this model for other spiral galaxies, most of the parameters are fixed by the observational constraints of the Milky Way. The advantage is that many features of the model are well constrained.

In our application of the Milky Way model to DLA host galaxies, we took into special consideration the assumptions that may influence the abundance ratios in the external parts of the disk. In fact, if some DLA galaxies are spirals, geometrical considerations indicate that the random line of sight to the quasar will intersect the external regions more probably than the internal ones. As Chiappini et al. (2001) have shown, the chemical evolution of the halo can have an impact on the abundance gradients in the outer parts of the disk. In their model A, the halo surface mass density is assumed to be constant (=17   M_⊙ pc^-2) for galactocentric distances R_GC ≤ 8 kpc and to decrease outwards as . Their model B has instead a constant value (=17   M_⊙ pc^-2) for all the galactocentric distances. The differences among model A and B arise primarily in the predicted evolution for the outermost regions of the disk. In these regions, the model B predicts the flattest gradients and provides the best fit to recent observations of Cepheids stars (see Cescutti et al. 2007) and distant open clusters, whereas model A is in good agreement with the steeper gradients traced by planetary nebulae (see Chiappini et al. 2001). In the present work, we decided to compare the DLA data with the results of both models for the evolution of external rings. We also considered two different types of stellar initial mass function (IMF): the Scalo IMF (Scalo 1986), which gives the best results for the Milky Way observational constraints, and the Salpeter IMF (Salpeter 1955), for the sake of comparison with the models of dwarf galaxies discussed below.

In summary, for each of the two Milky Way models A and B we calculated the abundance ratios of interest in eight rings of galactocentric distance between 4 and 18 kpc. These computations were performed with the Scalo IMF and repeated with the Salpeter IMF. In this way, we obtained 32 sets of model predictions of spiral galaxies. More details on the adopted Milky Way model can be found in Chiappini et al. (2001).

4.2. Models of dwarf irregular galaxies

The chemical evolution model of dwarf galaxies is an updated version of the one-zone model developed by Bradamante et al. (1998) to study the formation and evolution of late-type dwarf galaxies, dIrrs, and BCDs. The model assumes that the galaxy is built up by continuous infall of primordial gas.

Stars form and later contaminate the ISM with their products of nucleosynthesis, which mix instantaneously and completely with the ISM. Stellar lifetimes are taken into account in detail, i.e. the instantaneous recycling approximation (IRA) is relaxed. Part of the energy released by supernovae (SNe) and stellar winds is deposited into the ISM, and galactic winds develop when the thermal energy of the gas exceeds its binding energy. The wind expels metals from the galaxy, hence it has a significant influence on the history of chemical enrichment of the galaxy.

In the present application of the model, the rate of gas infall is assumed to exponentially decrease with time, with a time scale τ = 1 Gyr. A Salpeter IMF with index x = 1.35 is adopted. The chemical evolution is calculated as a function of time and the computations are stopped after 13 Gyr. The total quantity of infalling gas accreted throughout the lifetime of the galaxy is fixed by the parameter M_infall. Changes in M_infall are expected to affect the absolute value of gas and stellar mass, but have a negligible effect on the abundance ratios. In this respect, models with different values of M_infall are self-similar before the wind blows. Even if abundance ratios depend weakly on M_infall, we considered two possible values of M_infall = 1  ×  10⁹   M_⊙ and 5  ×  10⁹   M_⊙. If we only adopt the smaller value, the galaxy mass will become too low in the case of a strong galactic wind. To cover the whole mass range of dwarf galaxies, we chose to also experiment with the higher value.

The evolution of the abundance ratios is mostly affected by the efficiency of star formation and the possible presence of galactic winds. We therefore considered a broad range of values for the parameters that describe these effects. The star formation efficiency (SFE) is specified by the parameter ϵ, while the strength of the wind is given by w_iλ_w, where w_i is a weight parameter that is specific to each chemical element. By changing w_i, we can consider normal winds, with w_H,He = w_i = 1, and metal-enhanced winds, where the wind efficiency of metals is enhanced for some specific elements (for example metals), i.e. w_i > w_H,He. Both parameters ϵ and λ_w are expressed here in units of Gyr^-1 (Yin et al. 2010).

The combinations of model parameters that we adopted are summarized in Table 2. A total of 11 star formation efficiencies and 21 wind strengths were considered, in conjunction with two possibles values of M_infall and two types of wind. From these, we computed a grid of 924 models. The broad range of star formation efficiencies and wind strengths allow us to explain the possibility that DLA systems originate in a variety of dwarf galaxies, including dwarf irregulars, low surface brightness, and dwarf spheroidals. A complete description of the model equations and parameters can be found in Yin et al. (2010).

Fig. 2 Abundance ratios S/Zn versus Zn/H in DLA systems. Symbols with error bars: measurements for the samples of DLA systems listed in Table 4; curves: predictions of the models of galactic chemical evolution presented in Sect. 4. Blue diamonds: sample used to select the models of galactic chemical evolution (first part of Table 4); red circles: molecular DLA systems (second part of Table 4); black square: DLA system with peculiar nucleosynthetic abundances (last row in Table 4). Top panel: thin, cyan curves: models that pass the  selection condition (Eq. (8)); thick, magenta curve: best-fit reference model. Bottom panel: solid, red curves: dwarf models rejected by the  selection process; black, dotted curves: Milky Way models rejected by the  selection process.

5. Application of the method

We present the main steps followed in the application of our method: (1) the selection of chemical evolution models based on the abundances of volatile elements; (2) the comparison of measured and predicted abundance ratios of refractories and; finally, (3) the calculations of depletions. The resulting depletions are discussed in Sect. 6.

5.1. Selection of the chemical evolution models

5.1.1. Data used for the model selection

The DLA systems used to constrain the models are shown with filled diamonds in Fig. 2 and listed in the first part of Table 4. The other systems displayed in the same table were temporarily excluded from the process of model selection but used in the rest of the analysis. One case (last row in Table 4, square symbol in Fig. 2) was excluded because it shows a peculiar nucleosynthetic behavior, with Ti, Ni, and Zn abundances anomalous relative to any other DLA or metal-poor Galactic star (Cooke et al. 2010). Molecular DLA systems (second part of Table 4, filled circles in Fig. 2) were excluded because they may be affected by zinc/sulfur differential depletion, as we explained in Sect. 2.5.1. To this end, we used the results of the VLT/UVES DLA molecular survey (Noterdaeme et al. 2008, and refs. therein). Systems with f(H₂) > 10^-4.5 were not used to constrain the models. This threshold was proposed by Noterdaeme et al. (2008) to separate systems with high molecular content. The majority of the data sample that we use has been surveyed for H₂ and is proven to have low molecular content, i.e. f(H₂) < 10^-4.5. However, for 11 systems of this sample there is no measurement of f(H₂). Given the low rate of incidence of molecular systems with f(H₂) > 10^-4.5 ( ≃ 14% in the VLT/UVES survey), we expect only one or two out of these 11 systems to be molecular. In order to have a larger statistics, we kept these DLA systems in the sample.

5.1.2. Criteria adopted for the model selection

To select the models, we calculated the reduced χ² obtained by comparing observations and models in the diagram S/Zn versus Zn/H given by (6)where is the observed column-density ratio, (S/Zn)_ℳ,i the ratio predicted by the model at , and  is the measurement error in . The number of degrees of freedom is ν = n − n_par, where n is the number of DLA systems of the data set that satisfy the age test described below, and n_par is the number of parameters that are changed in the process of model selection.

The age test can be explained as follows. Each model ℳ gives the abundances X/H as a function of the time elapsed since the beginning of star formation, Δt_ℳ. From this, we calculate the Δt_ℳ,i predicted by the model at the observed abundance Zn/H or S/H of the ith DLA system. We then calculate the look back time of the system, t_i(z_abs), from its absorption redshift and a set of cosmological parameters. At this point, we estimate the look back time of the beginning of star formation implied by the model, t_ℳ,i(0) = t_i(z_abs) + Δt_ℳ,i. Finally, we impose that t_ℳ,i(0) ≤ t_reion, where t_reion is the look back time of the epoch of cosmic reionization. From this condition, we obtain the constraint (7)In the calculations of look-back time and t_reion, we adopt H₀ = 71.9 km s^-1 Mpc^-1, Ω_m = 0.258, and z_reion = 11.0 (Dunkley et al. 2009), and we allow for an uncertainty of 0.5 Gyr in the cosmic age scale. Any representative model of DLA galaxies should satisfy the condition given in Eq. (7), within the errors, for most of the DLA systems of the sample. In practice, we selected models for which  ≥ 95% of the DLA systems satisfy the test.

5.1.3. Results of the model selection

With the above discussed criteria, we selected the models that satisfy the condition (8)where is defined in such a way that the probability of  exceeding  is (see e.g. Bevington & Robinson 1992). The models that do not satisfy this condition have a probability  > 95% of not being representative of the observational sample.

From this selection, we found 322 models of dwarf galaxies; their tracks are shown as thin, cyan curves in the top panel of Fig. 2. The tracks of the rejected models are shown in the bottom panel of the same figure; these include the remaining 602 dwarf models (red, solid curves) and all the 32 Milky-Way models (black, dotted curves).

The SFE of the selected dwarf models lie in the range 0.3 ≤ ϵ ≤ 1.0 Gyr^-1, with a median value ϵ = 0.4 Gyr^-1. The wind strength covers the full range considered, i.e. 1.0 ≤ λ_w ≤ 5.5 Gyr^-1, with a median value λ_w = 3.5 Gyr^-1. The wind type is generally normal rather than metal-enhanced. Most of these models have M_infall = 1  ×  10⁹   M_⊙, and only two of them have M_infall  =  5  ×  10⁹   M_⊙.

Among the selected models, the model with minimum  is shown as a thick magenta curve in Fig. 2; its reduced chi-square is . This best-fit model is that of a dwarf galaxy with parameters M_infall = 1  ×  10⁹   M_⊙, ϵ = 0.4 Gyr^-1, and normal wind with λ_w = 3.5 Gyr^-1.

We adopt the best-fit model as a reference model to calculate the reference abundances (X/V)_ℳ,i and derive the depletions with Eq. (4). We use the rest of the selected models to estimate the spread of the reference abundances (X/V)_ℳ,i at the value of metallicity of the ith DLA system. By taking into account the spread of the selected models in the error propagation (Eq. (9)), we acknowledge that DLA galaxies may undergo different chemical evolution histories. The selected models cover a broad range of physical parameters representative of quite different types of galaxies.

Fig. 3 Abundance ratios of the refractory elements Mg, Si, and Fe over the volatile element Zn (left panels) and S (right panels), plotted versus the absolute abundance of the volatile element. Filled symbols with error bars: measurements in DLA systems obtained from the (Zn+S) sample (black circles), the  sample (red diamonds), and the  sample (blue squares); see Sect. 3.3. Curves: predictions obtained from the models of galactic chemical evolution selected by our procedure (same legenda as in Fig. 2, top panel).

The reference model fits 60% (21/35) of all the DLA systems of Table 4 at  ≤ 1σ level and 94% (33/35) at  ≤ 2σ level. These figures indicate the extent to which this best-fit model is representative of DLA systems selected with our criteria. The reference model is certainly not the unique answer about the nature of DLA host galaxies, but it is a practical tool useful for the calculations of depletions. In this spirit, we present in Table 8 the evolutionary tracks predicted by the reference model for the abundances of Mg, Si, S, Fe, and Zn. The full version of the tracks is available in electronic form.

5.1.4. Comments on the model selection

The seven molecular DLA systems not used in the χ² minimization are plotted with red circles in Fig. 2. Most of their deviations from the reference model (6 out of 7) are negative, which hints at the presence of some systematic effect (possibly differential sulfur/zinc depletion, as we later discuss in Sect. 6.5). The possibility of a systematic effect supports our choice of excluding molecular systems during the χ² minimization. However, the magnitude of this effect is so small that molecular systems are almost indistinguishable from the rest of the sample. The reference model fits 43% of molecular systems (4/7) at the 1σ level and 100% (7/7) at the 2σ level. Since the differences between molecular and non-molecular systems are small, we expect differential sulfur/zinc depletion to be consequently small. This conclusion is true a fortiori for the non-molecular sample that we used to model the chemical evolution of the S/Zn ratio. We conclude that differential sulfur/zinc depletion does not affect our process of model selection via the S/Zn ratio.

The DLA system with peculiar nucleosynthetic abundances (square symbol in Fig. 2) is the only case of deviation from the reference model at almost the 3σ level. This is an example of a DLA system that does not follow the chemical evolution history described by the models considered here. However, the frequency of these systems in the DLA population is very small, as we also expect their impact to be on our methodology for calculating depletions.

5.2. Ratios of refractories to volatiles

After using the volatile elements to select the models of chemical evolution, we checked the ratios of refractory to volatile elements, X/V, predicted by these models. These ratios can be used to test the assumption that the measured abundances are uniquely determined by nucleosynthesis and depletion (Sect. 2.1). If the assumption is correct, the measured X/V ratios, influenced by both processes, must be lower than the predicted ratios, representative of the total abundance, due to the depletion of the refractories. Only in lines of sight particularly poor of dust, or for undepleted elements, should the measurements overlap with the model predictions. In any case, no measured X/V ratios should lie above the predictions: data points above the evolutionary tracks indicate either a failure in our assumption (e.g. presence of ionization effects) or the inability of the adopted model of chemical evolution to describe the intrinsic abundance ratios.

In Fig. 3, we show the X/V ratios of Mg, Si, and Fe over Zn (left panels) and S (right panels). One can see that the measurements lie below, or close to, the model predictions for all the six ratios considered. Data points above the curves, which would contradict our assumptions, are almost absent: out of a total of 189 measurements, only a couple of them lie above the models at more than the 1σ level. With these exceptions, this comparison supports the assumption underlying our method. It is worth recalling that in our model calculations we fixed the stellar yields, which are quite effective in changing the level of the predicted ratios. It is remarkable that the same set of stellar yields that we adopted for all of our models (François et al. 2004) is capable of reproducing the ratio of volatile elements in Fig. 2 and, at the same time, to give consistent results for all the six ratios in Fig. 3. It is encouraging that the point with the largest positive deviation from the model tracks (bottom left panel in Fig. 3) corresponds to the system at z_abs = 1.6265 in QSO 0142-100, which was found to have peculiar nucleosynthetic abundances (Cooke et al. 2010). With the exception of rare cases of this type, we interpret the deviations between data and models in Fig. 3 as being caused by depletion effects.

The spread in the X/V ratios predicted by the models in Fig. 3 is larger for Mg than for either Si or Fe. This is because Mg is produced only by massive stars (core-collapse SNae), whereas Si, S, Fe, and Zn are produced in large amounts by type Ia SNae, whose progenitors are low and intermediate mass stars. As a result, the ratios Mg/Zn and Mg/S are more sensitive to variations in the star formation histories than the other ratios Si/Zn, Si/S, Fe/Zn, and Fe/S shown in the figure. This spread in the X/V ratios is included in the calculation of depletions as we explain now.

5.3. Calculation of depletions

After selecting and testing the models, we finally calculate the depletions. We proceed in two steps. We first measure δ_(X;Zn) and/or δ_(X;S) with Eq. (4). We then follow two different procedures for the DLA systems of the (Zn+S) sample, which have both measurements δ_(X;Zn) and δ_(X;S), and for those of the  and  samples, which have only one of the two. The errors in δ_(X;Zn) and/or δ_(X;S) are calculated with the expression (9)where is derived by propagating the measurement errors in  and ϵ_ℳ,i is the standard deviation in the predictions of the selected models calculated at the value of metallicity of the ith DLA system.

Fig. 4 Comparison of Mg, Si and depletions of the (Zn+S) sample of DLA systems obtained from different refractory/volatile abundance ratios. Abscissae: depletions obtained from ratios of refractories to Zn (see left panels in Fig. 3). Ordinates: depletions obtained from ratios of refractories to S (see right panels in Fig. 3).

In additions to the error given in Eq. (9), the depletions are affected by a systematic error induced by the imperfect matching of individual measurements and reference model in S/Zn (Fig. 2). We can quantify this effect as follows. We define the deviation between the observed and the predicted ratio in Fig. 2. By applying Eq. (4) twice, first with V = Zn and then with V = S, it is easy to see that the difference between the depletions calculated in the two cases is⁶δ_(X;Zn) − δ_(X;S) ≃ Δ_S/Zn. Therefore the deviation Δ_S/Zn yields a systematic error  ≃  on δ_(X;Zn) and  ≃  on δ_(X;S). We treat these errors in two different ways in the (Zn+S) sample and in both the  and  samples.

5.3.1. The (Zn+S) sample

In this sample, we know both δ_(X;Zn) and δ_(X;S). We average these two values and derive the mean depletion  ⟨ δ_X ⟩  = 0.5    [δ_(X;Zn) + δ_(X;S)] . The average approximately cancels the systematic errors  ≃  and  ≃ . Therefore the average depletions derived for the (Zn+S) sample are almost unaffected by the mismatch between data and reference model in Fig. 2. To assign an error to  ⟨ δ_X ⟩ , we cannot use the common formulae of error propagation since δ_(X;Zn) and δ_(X;S) are not independent⁷. We instead adopt the average of the errors ϵ(δ_X,i) previously measured with Eq. (9).

In Fig. 4, we compare δ_(X;S) with δ_(X;Zn) the DLA systems of the (Zn+S) sample. In the three panels of the figures, we show the three refractory elements considered, i.e. Mg, Si, and Fe. One can see that the differences between δ_(X;Zn) and δ_(X;S) are generally within the errors. Systematic differences between the two are virtually cancelled by averaging δ_(X;S) and δ_(X;Zn).

5.3.2. The and samples

For the DLA systems of these samples, we know either δ_(X;Zn) or δ_(X;S), but not both of them, the deviations Δ_S/Zn of individual systems being unknown. In this case, we assume that these unknown deviations behave similarly to the deviations in the (Zn+S) sample. We calculate the standard deviation of these Δ_S/Zn values, ϵ_S/Zn, and finally estimate the error (10)By assigning this error to the depletions of the and  samples, we incorporate into the error budget the effect of the mismatch between data and models in Fig. 2.

Fig. 5 Depletions of Mg, Si, and Fe in DLA systems plotted on a common scale of [Fe/H] metallicity defined in Sect. 6.2. Blue diamonds: results obtained by merging all the observational samples (Tables 4 − 6); molecular DLA systems are marked with red circles; proximated DLA systems with red triangles; the peculiar system at zabs = 1.6265 in QSO 0142-100 is marked with a red square. The labels CD, WD, and WH indicate the typical values of Galactic ISM depletions in cool disk gas, warm disk gas, and warm halo gas, respectively (see Table 3).

Fig. 6 Depletions of Mg, Si, and Fe versus absorption redshift in DLA systems. Symbols with error bars: data obtained from the (Zn+S) sample (black circles), the  sample (red diamonds), and the  sample (blue squares); see Sect. 3.3. The labels CD, WD, and WH indicate the typical values of Galactic ISM depletions in cool disk gas, warm disk gas, and warm halo gas, respectively.

6. Discussion

We now discuss the DLA depletions derived with our method. Depletions of individual systems are shown in Figs. 5 and 6. Mean values of δ_X are listed in Table 3. The statistical properties that we present are based on the merged sample, i.e. the set of data obtained by merging the depletions of the (Zn+S), , and  samples (Sect. 3.3). We do not include in the sample the DLA system with peculiar abundances at z_abs = 1.6265 in QSO 0142-100; statistical results derived by including this system in the analysis are generally identical (see e.g. Table 3).

In the course of the discussion, we compare our results with typical values of Galactic ISM depletions. In performing this comparison, we must take into account that the strength of ISM depletion varies with the physical conditions of the gas. Following Savage & Sembach (1996), we consider, for each element of interest, three representative levels of depletions, corresponding to Galactic warm-halo (WH), warm-disk (WD), and cool-disk (CD) gas⁸. These reference values of Galactic ISM depletions, updated following Jenkins (2009), are listed in Table 3.

The depletions are negative by definition; we follow the convention of saying that a depletion is high (or low), when its absolute value  | δ_X |  is high (or low).

6.1. Mean values of depletion

Before studying trends between depletions and other quantities we consider the mean statistical properties shown in Table 3. The first result to point out is that the mean depletion of Mg and Si is detected, in spite of these elements being mildly depleted in the ISM and DLA depletions generally being low. The case of Si is quite strong: the mean depletion of 74 DLA systems is detected at a level of 14 times the standard error of the mean. The case of Mg is less strong: the mean depletion of ten systems is three times the error of the mean.

Silicon depletion is stronger than expected: its mean value,  ≃ −0.27 dex, is not particularly weaker than that of iron,  ≃−0.42 dex. This is surprising because iron is much more depleted than silicon in the ISM, as shown in Table 3. While  ⟨ δ_Fe ⟩  in DLA systems is consistent with the lowest Galactic ISM depletions of iron (warm halo gas), the mean DLA value  ⟨ δ_Si ⟩  resembles the Galactic depletions of silicon in warm disk gas, which represents an ISM phase of somewhat higher level of depletion. The mean depletion of Mg is in between those of the warm halo gas and the warm disk gas.

The different behaviors of the silicon and iron depletions are also apparent from their dispersions: silicon depletions are less scattered (σ_Si = 0.16 dex) than iron depletions (σ_Fe = 0.28 dex). The existence of systematic differences between silicon and iron depletions is confirmed by the detailed analysis of the trend with metallicity.

6.2. Depletions versus metallicity

In Fig. 5, we plot the Mg, Si, and Fe depletions versus [Fe/H]. To establish a common scale of dust-free [Fe/H] for the systems of the (Zn+S), , and  samples, we used the [Fe/H] value predicted by the model⁹. The depletions obtained from the different samples (Sect. 3.3) are all indicated with a blue diamond. Proximate DLA systems (Sect. 3.2) are marked with a red triangle; their depletions are generally consistent with those of intervening DLA systems.

From a simple inspection of Fig. 5, one can see that δ_Fe increases with metallicity. To quantify the correlation between δ_Fe and [Fe/H], we used a Spearman’s rank correlation analysis. The Spearman correlation coefficient, r_S, is a non-parametric measure of the statistical dependence of two variables related by any monotonic function (not just a linear function as in the case of the Pearson analysis). By applying this analysis to δ_Fe versus  [Fe/H] , we find that r_S =  −0.74 for the merged sample of iron measurements (n = 105). The iron data appear to be correlated, albeit with a large scatter in δ_Fe at each value of [Fe/H].

The iron result is not surprising because a general rise in DLA depletions with metallicity was reported in previous investigations (e.g. Ledoux et al. 2003; Vladilo 2004). For this reason, we expected to find a similar rise for silicon. However, this is not the case: by applying the Spearman analysis to δ_Si versus  [Fe/H]  we find that r_S =  + 0.05 for the merged sample of silicon measurements (n = 74), i.e., that silicon depletions are not statistically correlated with metallicity. This explains the smaller dispersion in silicon depletions seen in Table 3 (σ_Si < σ_Fe): iron varies with [Fe/H] while silicon does not.

Despite the lack of statistical correlation, a careful inspection of Fig. 5 shows that the maximum  |δ_Si|  at a given [Fe/H] does increase with metallicity, put in other words the lower envelope of silicon data decreases with [Fe/H]. However, the upper envelope of silicon depletions stays roughly constant (actually showing a hint of an increase with [Fe/H]) and this explains why silicon depletion is uncorrelated with metallicity. By applying the same analysis to iron, we find a different behavior: both the minimum and the maximum  |δ_Fe|  at a given [Fe/H] increase with metallicity, or in simple words the lower and upper envelopes of iron depletions decrease with [Fe/H]. The minimum value of  |δ_Fe|  vanishes at [Fe/H]  ≲   −1.5.

Even if the magnesium sample is too small to derive firm conclusions, the data suggest similarities with silicon results. The Spearman analysis yields r_S =  + 0.19 for the merged sample of δ_Mg (n = 10): magnesium depletions are not statistically correlated with metallicity. The inspection of Fig. 5 suggests that the upper envelope of magnesium depletions is roughly constant, while the lower envelope seems to decrease with [Fe/H]; these results require more data to be confirmed.

In spite of the differences between iron depletions on the one hand, and silicon and magnesium depletions on the other, the comparison with Galactic ISM depletions suggests that these two sets of results have a common behavior. One can see from Fig. 5 that all the lower envelopes of δ_Mg, δ_Si, and δ_Fe, extrapolated to [Fe/H] = 0, tend to match the maximum values of Galactic ISM depletions with increasing metallicity. In contrast, all the upper envelopes of δ_Mg, δ_Si and δ_Fe tend to match the minimum values of Galactic ISM depletions (warm-halo gas) at [Fe/H] = 0. From these results, we tentatively propose that the depletions of different elements in DLA systems may follow a common pattern of metallicity evolution: for a given element X, the envelope of the highest values of DLA depletions evolves with metallicity in such a way as to come close to the highest Galactic depletion of X at [Fe/H] = 0; the envelope of the lowest DLA depletions approaches the lowest Galactic depletion of X at [Fe/H] = 0. Future investigations, based on large datasets for many chemical elements, will prove or disprove the validity of this general picture.

6.3. Depletions and physical state

The spread of Galactic depletions at [Fe/H] = 0 provides a means of interpreting the spread of DLA depletions at a given [Fe/H]. We know that the different depletion levels in the ISM reflect the variations in the physical conditions of Galactic clouds. By analogy, it is reasonable to assume that the same holds true in DLA systems, the depletions being weaker in low-density, high-temperature gas and stronger in high-density, low-temperature gas. We argue that the spread in DLA depletions at a given [Fe/H] reflects the variation in the physical conditions of clouds embedded in galaxies with the same level of chemical enrichment. To prove this assumption, we need to compare the depletions with measurements of physical parameters. However, we know in advance that this comparison might be biased: our measurements of depletions are probably underestimated for the lines of sight with the highest levels of depletions, owing to the possible depletion of sulfur and zinc. The highest levels of depletion are expected to occur in molecular systems that we now discuss.

6.4. Molecular DLA systems

Molecular systems are marked with a red circle in Fig. 5. As explained in Sect. 2.5.1, zinc and sulfur could be depleted in molecular systems, in which case the depletions of Mg, Si, and Fe would be underestimated by an amount equal to δ_Zn or δ_S (Eq. (5)). To have an approximate idea of how strong this effect might be, we estimated the maximum values expected for zinc and sulfur depletion in the most depleted regions of the Galactic ISM. Owing to the paucity of accurate measurements of sulfur depletions in high-column density Galactic lines of sight, we estimated δ_Zn and δ_S using Eq. (10) of Jenkins (2009) with F_∗ = 1.0. This value is representative of the highest level of Galactic depletion, generally found in molecular gas. From this indirect estimate, we find that δ_Zn ≃  −0.55 dex and¹⁰δ_S ≃  −0.72 dex. If we take these figures to indicate how much the depletions in molecular DLA systems may be underestimated, we can study the potential implications for our results. One can see from Fig. 5 that if we decreased the depletions of molecular systems (symbols inside red circles) by an amount in the range  ≃−0.55/−0.72 dex, these systems would end up more or less close to the lower envelope of DLA depletions in most cases. This would confirm our hypothesis that the spread of depletions at a given metallicity is due to variations in the physical conditions: molecular clouds would lie close to the lower envelope of maximum depletions for this reason. Therefore, the main conclusions of present discussion would be reinforced if sulfur and zinc were depleted as they are in the Galactic ISM.

6.5. Differential sulfur/zinc depletion

Six of seven molecular DLA systems lie below the reference model in Fig. 2. This systematic effect, albeit small as we discussed in Sect. 5.1.4, may indicate the existence of differential sulfur/zinc depletion if sulfur is more depleted than zinc in dust-rich, molecular lines of sight. To test this possibility, we used again the Galactic ISM as a reference. From the above-mentioned values of δ_Zn and δ_S along Galactic lines of sight with F_∗ = 1.0 (Jenkins 2009) we expect sulfur depletion to exceed zinc depletion by  ≃ 0.17 dex. The deviations that we find between molecular data and reference model in Fig. 2 are typically  ≃  − 0.05 dex ( − 0.10 dex in two cases), consistent with an excess of sulfur over zinc depletion of that magnitude. All together, these results may represent evidence of differential sulfur/zinc depletion in DLA systems, which remains to be confirmed by future investigations.

6.6. Depletions versus absorption redshift

In Fig. 6, we plot the depletions versus the absorption redshift of the DLA systems. In this figure, the results obtained from the (Zn+S), , and  samples are indicated with different symbols (black circles, red diamonds and blue squares, respectively). For the sake of clarity, molecular systems are not labeled in this case, even if they are present in the figure.

Figure 6 is useful to see how the Zn data (red diamonds) give a good coverage of low-redshift systems, while the S data (cyan squares) of high-redshift ones. The (Zn+S) sample for which we have measurements of both volatile elements (black circles) is limited between the redshift for detection of S ii lines from ground (z_abs ≳ 2) and the redshift at which Zn ii has mostly been detected so far (z_abs ≲ 3). The advantage of using the  and  samples, in addition to the (Zn+S) sample, is evident from this figure. The redshift range 2 ≲ z_abs ≲ 3 could be used to calibrate other reference elements in future studies of DLA abundances and dust evolution.

From a simple inspection of the figure, one can see that there is no trend between depletions and redshift. The Spearman correlation analysis confirms this result, yielding r_S =  −0.05 for δ_Si versus z_abs (n = 74). In the case of iron, we find that r_S =  + 0.27 for δ_Fe versus z_abs (n = 105). The very modest trend found for iron probably reflects the weak decrease in metallicity with redshift, found in previous investigations (e.g. Vladilo et al. 2000; Prochaska et al. 2001), combined with the trend between δ_Fe and metallicity discussed above. Evidently, the lack of a statistical trend between δ_Si and metallicity translates into the null result that we find for silicon.

The lack of dependence of depletion on absorption redshift suggests that the abundance of DLA dust and its potential effects on the extinction of background quasars should not vary significantly in the course of cosmic evolution. This result is consistent with the results from the CORALS survey, which found DLA statistics and metallicity to be largely unbiased by dust at both z ≳ 1.8 (Ellison et al. 2001; Akerman et al. 2005) and 0.7 ≲ z ≲ 1.6 (Ellison et al. 2004; Ellison & Lopez 2009).

The absence of any significant trend with redshift suggests that cosmic time is not a good clock for measuring the evolution of the dust content in DLA galaxies. The metallicity is a better clock than time, at least in the case of iron. These results are in line with what we expect from studies of galactic chemical evolution: the absolute abundance of metals is a better clock of evolution than the time elapsed since the beginning of star formation.

7. Summary and conclusions

We have presented a new method for measuring elemental depletions in DLA systems. The method was developed with the aim of measuring the depletion of silicon, a basic constituent of cosmic silicates, at the early stages of chemical evolution. In addition to silicon, we applied the method to iron, for comparison with previous studies of DLA depletion, and to magnesium, an important ingredient of silicates that had not been previously investigated in studies of DLA depletions.

The method is based on the comparison of abundance ratios measured in the gas and total abundance ratios (gas plus dust) calculated from models of galactic chemical evolution. We assume that the volatile elements sulfur and zinc can be used to trace the total abundances, while the refractory elements magnesium, silicon, and iron are partly depleted from the gas into dust grains. The diagram of measured ratios S/Zn versus Zn/H is used as a dust-free diagnostic tool to constrain models of galactic chemical evolution. The models are also constrained by means of an age test that compares the look back time of each DLA system, inferred from its redshift, with the time elapsed after the beginning of star formation, inferred from the model.

In applying the method, we used column density measurements accumulated from the literature for 293 DLA systems. After selecting the highest quality measurements and discarding DLA systems affected by ionization, we were left with 10, 131, 66, 167, and 94 column densities of Mg ii, Si ii, S ii, Fe ii, and Zn ii, respectively. These include a few measurements of S ii column densities presented here.

From a set of 956 models of galactic chemical evolution, representative of a large variety of putative DLA host galaxies (Sect. 4), we selected the model that best fits the observational diagram S/Zn versus Zn/H (Fig. 2); this model was later used as a reference to calculate the depletions. A total of 322 models, consistent with the S/Zn diagnostic tool according to the chi-square condition shown in Eq. (8), were also selected; these models were used to estimate the spread of the depletions, acknowledging that DLA systems are likely to originate in a variety of galaxies.

Our selection of galactic chemical evolution models yields conclusions consistent with previous work: most DLA systems appear to originate in a mixture of low-mass galaxies; a minority of them may originate in spiral disks. Since the population of DLA systems is probably inhomogeneous, the chi-square of the model that best fits the S/Zn versus Zn/H data, , is not particularly high. This suggests that, after all, the chemical histories of DLA systems do not differ widely, at least for the systems of the high-resolution spectroscopic sample considered here, and within the range of metallicity covered by our data (−2 ≲  [Fe/H]  ≲ 0).

After selecting the models using the abundances of volatile elements, we tested the ratios of refractory to volatile elements, X/V, predicted by these models. The calculated ratios, plotted versus V/H, should never lie below the measured ratios if our assumptions are correct. The comparison of predictions and measurements of the 6 ratios Mg/Zn, Mg/S, Si/Zn, Si/S, Fe/Zn, and Fe/S for a total of 189 data points gives a positive result in 99% of cases (Fig. 3). After this test, we calculated the depletions. With the available data, we were able to derive δ_Mg, δ_Si, and δ_Fe for 10, 74, and 105 DLA systems, respectively. The main results that we find can be summarized as follows.

Silicon and magnesium depletions were detected, despite the mild depletion of these elements in the Galactic ISM and the low dust content of DLA systems.

The mean depletion of silicon that we find,  ⟨ δ_Si ⟩  =  −0.27  ±  0.16 dex, is almost as high as that of iron,  ⟨ δ_Fe ⟩  =  −0.42  ±  0.28 dex. The similarity between these two values is surprising since iron is much more depleted than silicon in the Galactic ISM (Table 3). The typical iron depletion in DLA systems is comparable to that of the ISM warm-halo gas, while the typical silicon depletion in DLA systems is comparable to that of the ISM warm-disk gas.

The analysis of depletions versus metallicity reveals an important difference between silicon and iron: while δ_Fe correlates with [Fe/H] (Spearman correlation coefficient r_S =  −0.74), δ_Si is not correlated (r_S =  + 0.05). In spite of the lack of statistical correlation, the lower envelope of silicon depletions decreases with metallicity, approaching the Galactic ISM silicon depletion of cool-disk gas (Fig. 5). The upper envelope of silicon depletions plotted versus metallicity is instead roughly constant. For iron, both the lower and the upper envelope decrease with metallicity; at [Fe/H]  ≲   − 1.5 dex, the iron depletion tends to vanish.

Magnesium depletions seem to behave more in accordance with silicon than with iron. For this element, the conclusions are tentative because of the limited size of the sample.

The behavior of iron depletions is consistent with that found in previous studies of DLA systems, which already pointed out the resemblance of  ⟨ δ_Fe ⟩  with warm halo gas depletions, the rise in  | δ_Fe |  with metallicity and the tendency of δ_Fe to vanish at [Fe/H]  ≲   − 1.5 dex. The different behavior that we find for silicon, and possibly magnesium, is an unexpected result of the present work.

In spite of the different behavior of Fe depletions on the one hand, and of Si and Mg depletions on the other, the comparison with Galactic ISM data suggests that all DLA depletions may follow a common pattern of evolution with metallicity: for a given element X, the envelope of the highest values of DLA depletions evolves approaching the highest value of Galactic depletion of X at [Fe/H] = 0; the envelope of the lowest values of DLA depletions approaches the lowest value of Galactic depletion of X.

The Mg, Si, and Fe depletions generally display a significant spread at a given value of [Fe/H]. By analogy with the interpretation of the spread in the Galactic depletions at constant metallicity ([Fe/H] = 0), we argue that the spread in the DLA depletions reflects the variation in the physical conditions in clouds that have attained the same level of chemical enrichment. The distribution of molecular systems, expected to have the highest levels of depletion at a given [Fe/H], is found to be broadly consistent with this hypothesis, once we take into account that their depletions might be underestimated by up to  ≈ 0.5/0.7 dex due to S and Zn depletion in dust-rich gas. We find tentative evidence of differential S/Zn depletion of  ≲ 0.1 dex in molecular systems, to be confirmed by future investigations.

No trend between depletions and redshift is detected, apart from a weak tendency for iron depletion to decrease with redshift. The lack of trends with redshift and the presence of trends with [Fe/H], at least for iron, indicates that metallicity is a better clock than cosmic time for the purpose of tracing the history of dust evolution.

The different properties of silicon and iron depletions in DLA systems have several implications. From the practical point of view, one should be careful about using silicon as a dust-free indicator of DLA metallicity in the absence of abundance measurements of volatile elements. In fact, when [Fe/H]  ≲   − 1.5 dex, it is probably safer to use iron rather than silicon for this purpose.

From the point of view of the evolution of the dust-to-metal ratio in galaxies, the silicon results change the perspective of previous work based on iron. If a constant fraction of metals is condensed into dust in the course of chemical evolution, we expect the dust-to-metal ratio and the depletions to remain roughly constant with metallicity, apart from variations in the physical conditions of the clouds. The depletions of silicon in DLA systems, which are approximately constant with metallicity, are in line with the expectations for a constant dust-to-metal ratio. This is possibly also true for magnesium. At this point, the behavior which is more difficult to understand is that of iron. The decrease in δ_Fe at the lowest DLA metallicities, already known from previous studies, suggests the existence of some mechanism that inhibits the production of iron-rich dust at the early stages of chemical evolution ([Fe/H]  ≲   − 1.5 dex).

Taken together, our results indicate that the behavior of DLA depletions is more complex than expected and may vary from element to element. To study the variation of the total dust-to-metal ratio, we should first investigate the evolution of the depletions of all the cosmically abundant elements. In general, measuring depletions for a variety of elements in DLA systems is essential to constrain models of dust production in the course of chemical evolution (Dwek 1998; Calura et al. 2008; Zhukovska et al. 2008). We hope that our methodology and results may open new possibilities for future studies of this kind and, in particular, for studies of cosmic silicates at high redshift.

Acknowledgments

We thank Cristina Chiappini and Francesco Calura for helpful discussions at the early stages of preparation of this work; C.A. thanks Miriam Centurión for her support in the analysis of UVES spectra; J.Y. thanks the financial support from the National Science Foundation of China No. 10573028, the Key Project No. 10833005, the Group Innovation Project No. 10821302, 973 program No. 2007CB815402, and the Knowledge Innovation Program of the Chinese Academy of Sciences No. Y090761009. We thank an anonymous referee for her/his helpful comments and suggestions.

References

Online material

All Tables

All Figures

Fig. 1 Multi-component Voigt profile fits of S ii and Fe ii lines for the DLA systems at zabs = 2.2930 in Q0216+080 (left panels) and at zabs = 1.9200 in Q2206-199 (right panels). In each spectral region, we indicate the ion and rest-frame wavelength of the transition; the continuum and zero levels are indicated with solid and dashed lines, respectively. The radial velocity scale is obtained from the wavelength scale using the redshift of the strongest absorption component of each system as a reference. The velocity components used in the fits are marked with vertical ticks. The positions of the S ii velocity components (top panels) were constrained from the analysis of Fe ii lines outside the Ly α forest (bottom panels). The spectral region around the redshifted 2249 Å Fe ii lines in Q2206-199 (bottom right panel) is disturbed by a few cosmic events (emission spikes).

In the text

Fig. 2 Abundance ratios S/Zn versus Zn/H in DLA systems. Symbols with error bars: measurements for the samples of DLA systems listed in Table 4; curves: predictions of the models of galactic chemical evolution presented in Sect. 4. Blue diamonds: sample used to select the models of galactic chemical evolution (first part of Table 4); red circles: molecular DLA systems (second part of Table 4); black square: DLA system with peculiar nucleosynthetic abundances (last row in Table 4). Top panel: thin, cyan curves: models that pass the  selection condition (Eq. (8)); thick, magenta curve: best-fit reference model. Bottom panel: solid, red curves: dwarf models rejected by the  selection process; black, dotted curves: Milky Way models rejected by the  selection process.

In the text

Fig. 3 Abundance ratios of the refractory elements Mg, Si, and Fe over the volatile element Zn (left panels) and S (right panels), plotted versus the absolute abundance of the volatile element. Filled symbols with error bars: measurements in DLA systems obtained from the (Zn+S) sample (black circles), the  sample (red diamonds), and the  sample (blue squares); see Sect. 3.3. Curves: predictions obtained from the models of galactic chemical evolution selected by our procedure (same legenda as in Fig. 2, top panel).

In the text

	Fig. 4 Comparison of Mg, Si and depletions of the (Zn+S) sample of DLA systems obtained from different refractory/volatile abundance ratios. Abscissae: depletions obtained from ratios of refractories to Zn (see left panels in Fig. 3). Ordinates: depletions obtained from ratios of refractories to S (see right panels in Fig. 3).
In the text

Fig. 5 Depletions of Mg, Si, and Fe in DLA systems plotted on a common scale of [Fe/H] metallicity defined in Sect. 6.2. Blue diamonds: results obtained by merging all the observational samples (Tables 4 − 6); molecular DLA systems are marked with red circles; proximated DLA systems with red triangles; the peculiar system at zabs = 1.6265 in QSO 0142-100 is marked with a red square. The labels CD, WD, and WH indicate the typical values of Galactic ISM depletions in cool disk gas, warm disk gas, and warm halo gas, respectively (see Table 3).

In the text

	Fig. 6 Depletions of Mg, Si, and Fe versus absorption redshift in DLA systems. Symbols with error bars: data obtained from the (Zn+S) sample (black circles), the  sample (red diamonds), and the  sample (blue squares); see Sect. 3.3. The labels CD, WD, and WH indicate the typical values of Galactic ISM depletions in cool disk gas, warm disk gas, and warm halo gas, respectively.
In the text