The dust obscuration bias in Damped Lyman\,$\alpha$ systems % :\\ support from Milky Way interstellar studies % observed at high spectral resolution
Giovanni Vladilo \inst{1}
C\'eline P\'eroux \inst{2}
We present a new study of the effects of quasar obscuration on the statistics of Damped Ly\,$\alpha$ (DLA) systems. % % We show that the extinction of any Galactic or extragalactic \ion{H}{i} region, $A_\lambda$, increases % REVISION 3 linearly with the column density of zinc, % END REVISION 3 with a %characteristic turning point $\partial A_\lambda / \partial (\log N_\mathrm{Zn}) =1$, above which background sources are suddenly obscured. %%% [start revision2] %{% We derive a relation $A_\lambda=A_\lambda(N_\mathrm{H},Z,z)$ between the extinction of a DLA system and its \ion{H}{i} column density, $N_\mathrm{H}$, metallicity, $Z$, fraction of iron in dust, $f_\mathrm{Fe}(Z)$, and redshift, $z$. % From this relation we estimate the fraction of DLA systems missed as a consequence of their own extinction in magnitude-limited surveys. % of quasars/DLAs. % % By ignoring the role of multiple DLA absorptions in a quasar line of sight, We derive a method for recovering the true frequency distributions of $N_\mathrm{H}$ and $Z$ in DLAs, $f_{N_\mathrm{H}}$ and $f_Z$, using the biased distributions measured in the redshift range where the observations have sufficient statistics ($1.8 \leq z \leq 3$). % % revision3 By applying our method we find that the well-known empirical thresholds of DLA column densities, $N_\mathrm{Zn} \la 10^{13.1}$ atoms cm$^{-2}$ and $N_\ion{H}{i} \la 10^{22}$ atoms cm$^{-2}$, can be successfully explained in terms of the obscuration effect without tuning of the local dust parameters. %% %(1) %by approximating the true \ion{H}{i} column density %distribution with a power law $f_{N_\mathrm{H}} \sim N_\mathrm{H}^{-\beta}$ we derive %$\beta \simeq 1.5$, %a slope similar to that of quasar absorbers of lower column density; %% % The obscuration has a modest effect on the distribution of quasar apparent magnitudes, but plays an important role in shaping the statistical distributions of DLAs. % in particular the metallicity distribution. % The exact estimate of the bias is still limited by the paucity of the data ($\simeq$ 40 zinc measurements at $1.8 \leq z \leq 3$). % We find that the fraction of DLAs missed as a consequence of obscuration is $\sim$ 30\% to 50\%, consistent with the results of surveys of radio-selected quasars. % By modelling the metallicity distribution with a Schechter function we find that the mean metallicity can be $\sim 5$ to 6 times higher than the value commonly reported for DLAs at $z \sim 2.3$. % \keywords{ ISM: dust, extinction -- Galaxies: ISM -- Galaxies: high-redshift -- Quasars: absorption lines } } % \maketitle %\end{document} % %________________________________________________________________ \section{Introduction} Quasar absorption line systems probe the diffuse gas in the Universe over cosmological scales and at various stages of evolution. Among these absorbers, the Damped Ly$\alpha$ systems (hereafter DLAs) have the highest \ion{H}{i} column densities ($N_\ion{H}{i} \geq 2 \times 10^{20}$ atoms cm$^{-2}$) and are believed to originate in \ion{H}{i} regions of high-redshift galaxies (Wolfe et al. 1986). Also, they have higher metallicities\footnote{ We adopt the usual definition [X/H] = log (X/H) - log (X/H)$_{\sun}$.} than any other class of quasar absorbers ([M/H] $\approx -1.1$ dex; Pettini et al. 1994) and as such are expected to contain dust. % %Dust plays an important role in many physical processes %in galaxies and affects several types of astronomical observations. %In particular, If present, the dust in DLAs will absorb and scatter the radiation of the background quasar, dimming its apparent magnitude ("extinction" effect) and changing the slope of its spectral distribution ("reddening"). % In magnitude limited surveys, the extinction may obscure the quasar, leading to an observational selection effect known as obscuration bias (Ostriker \& Heisler, 1984; Fall \& Pei 1989, 1993). %Competing to this effect, is the possible gravitational magnification %of the background quasar by the absorber itself. %Many studies other the years have attempted to estimated %the magnitude of this phenomenon for both DLAs (Smette et al. 1997, Le %Brun et al. 2000, Murphy \& Liske 2004) and metal absorption lines (Vanden %Berk 1996, M\'enard \& P\'eroux 2003). The results show that for the DLAs %selected at optical wavelength, gravitational effects are small. The bias can lead to misleading conclusions on the nature of the galaxies associated with the DLA systems (herefater DLA galaxies), because the \ion{H}{i} regions most enriched by metals and dust would be systematically missed in the surveys. % In addition, some properties of the high-redshift Universe that can only be derived from studies of DLAs, would also be affected by the bias. One example is the comoving mass density of neutral gas, $\Omega_\mathrm{DLA}$, and its evolution with redshift, which is an indicator of gas consumption due to star formation (Wolfe et al. 1995). Another example is the mean cosmic metallicity inferred from abundance studies of DLAs (Pettini et al. 1997), which is related to the star formation rate %in the high-redshift Universe in the early Universe (Pei \& Fall 1995, hereafter PF95; Pei et al. 1999). Whether the obscuration effect is important, depends on the actual amount of dust in DLAs. % The first evidence of dust, in the form of reddening, was reported by Pei et al. (1991). The complex variability of the quasar continuum makes it impossible to determine the reddening in individual cases. However, from a statistical comparison of quasars with and without foreground DLAs one can search for a systematic change of the continuum slope. In this way, Pei et al. found a systematic difference, suggestive of reddening, using a simplified power-law fit to the continuum distribution. %% %This can be estimated a posteriori from studies %of the systems discovered in quasar samples. %% %One approach consists in estimating the reddening %of the continuum of the quasar. % The detection or reddening motivated detailed studies of quasar obscuration. % Fall \& Pei (1993; hereafter FP93) presented an analytical method of computing the obscuration from the observed luminosity function of quasars, the typical dust-to-gas ratio of DLAs and the empirical distribution of \ion{H}{i} column densities. By modelling the radial distribution of the neutral gas in DLA galaxies, they concluded that a large fraction of quasars may be obscured in optically selected samples. % %This bias would preclude the detection of DLAs with high dust content %in magnitude-limited samples. % The large uncertainty of the reddening measurements prevented reaching firm conclusions on the dust-to-gas ratios and on the magnitude of the bias. An alternative approach to measure the amount of the dust in DLAs consists in measuring the differential depletions between refractory and volatile elements (Pettini et al. 1994; Hou et al. 2001; Prochaska \& Wolfe 2002; Ledoux et al. 2003; Vladilo 2002, 2004). %Detailed studies of depletions can be used to %infer which fraction Vladilo 2002 % % The differential %depletion of these elements gives an indication of the amount of dust %There is now general agreement that dust depletion exists %() %and increases with metallicity (). %%The fraction of dust %of iron has values typical of Milky-Way warm interstellar gas %when the metallicity is $\ga 0.1$ the solar value, but becomes %negligible at very low metallicities (Vladilo 2004). % %%However, a recent study based on a larger and %%more homogeneous sample of quasars %%%selected by the Sloan survey (Richards et al) %%do not confirm the detection of such an effect %%(Murphy \& Liske, 2004). % The depletions can be converted into dust-to-gas ratios, yielding an approximate estimate of the quasar extinction %. %So far, estimates of this type %yielded somewhat different results, leading to %different conclusions on the importance of the obscuration bias (Vladilo et al. 2001b; Prochaska \& Wolfe 2002). This method, as well as the reddening measurements, only give information on the detected DLAs, leaving open the possibility that more dusty \ion{H}{i} regions remain undetected. %% %, % %Fall \& Pei (1993) subsequently investigated the potential effects %of DLA extinction %using a model of the radial distribution of the gas in the %intervening galaxies. %% %Comparing their predictions with the observed %frequency distribution of \ion{H}{i} column densities, they %concluded that %%This obscuration effect %%should lead to a selection bias against detection of DLAs with %%high dust content in optically selected quasar samples. %Nevertheless, the detection %of reddening by Pei et al. (1991) %has not been confirmed so far (Murphy \& Liske, 2004). A more direct probe of the obscuration effect is a comparison of DLA statistics in optically selected and radio selected quasar samples, given that the latter are unaffected by dust extinction. Ellison et al. (2001) compiled a homogeneous sample of radio selected quasars and searched for DLAs towards every target, irrespective of its optical magnitude (CORALS survey). They concluded that dust-induced bias in previous magnitude limited surveys may have led to underestimating the \ion{H}{i} mass density, $\Omega_\mathrm{DLA}$, and the number density per unit redshift interval, $n(z)$, of DLAs by at most a factor of two at $z \sim 2.3$. In addition, they found tentative evidence that $n(z)$ is greater in fainter quasars, as expected by the obscuration effect. % (but only at 1 $\sigma$ level). % REVISION 3 In a follow-up study of CORALS metallicities, Akerman et al. (2005) find $ [ < \mathrm{Zn/H} > ] = -0.88 \pm 0.21 $. This value is higher than in a control sample, % in the same redshift interval, %($1.9 < z < 3.5$), as expected by the obscuration, but only at $1 \sigma$ level. % In an extension of the CORALS survey at $0.6 < z < 1.7$ Ellison et al. (2004) find $n(z) = 0.16 ^{+0.08}_{-0.06}$ using MgII absorbers as DLA candidates. This value is consistent with magnitude-limited estimates at the same $z$, but the $1 \sigma$ error permits a factor of 2.5 difference in the sense predicted by the obscuration. % END REVISION 3 The results of the CORALS survey are not yet conclusive since the original sample of Ellison et al. had limited statistics, in particular at the high values of $N(\ion{H}{i})$ where the effect is expected to be more important. This weak evidence of obscuration, together with the non-detection of reddening in a large sample of quasars with and without candidate DLAs, recently reported by Murphy \& Liske (2004), are conveying the impression that the obscuration effect is not important. Yet, indirect evidence for the existence of the bias comes from other studies. % abundance studies of known DLAs. Indeed, The analysis of [Zn/H] versus $\log N(\ion{H}{i})$ by Boiss\'e et al. (1998) showed that none of the % dozen DLAs with zinc detection known at the time have a metallicity above the threshold [Zn/H] $+\log N(\ion{H}{i}) \simeq 20.5$, corresponding to $N(\ion{Zn}{ii}) \simeq 10^{13.1}$ atoms cm$^{-2}$. They proposed that this threshold could be attributed to obscuration effect, assuming that metal column density tracks the dust column density. While this assumption has not been proven by subsequent investigations, the threshold proposed by Boiss\'e et al. has been invoked % by different authors % revision 3 to reconcile several predictions of galactic evolution models (Prantzos \& Boissier 2000; Hou et al. 2001) and cosmological simulations (Cen et al. 2003) with the observed properties of DLAs. % end revision 3 Another potential evidence of the bias is the lack of DLAs with \ion{H}{i} column densities $N(\ion{H}{i}) \ga 10^{22}$ atoms cm$^{-2}$. In this case, the obscuration has been invoked to explain the significant fraction of model galaxy disks predicted to have higher \ion{H}{i} column densities (Churches et al. 2004). % As an alternative explanation, % to explain the column density thresholds observed in DLAs, it has been proposed that hydrogen may undergo a sudden transition from the atomic to the molecular form above a critical column density threshold (Schaye 2001), in which case the absorber would not be detected as a DLA system. %Competing to the obscuration effect, is the possible gravitational magnification %of the background quasar by the absorber itself. %Many studies other the years have attempted to estimated %the magnitude of this phenomenon for both DLAs (Smette et al. 1997, Le %Brun et al. 2000, Murphy \& Liske 2004) and metal absorption lines (Vanden %Berk 1996, M\'enard \& P\'eroux 2003). The results show that for the DLAs %selected at optical wavelength, gravitational effects are small. In the present work we address several open questions concerning the existence and importance of the obscuration bias. %% The paper is structured as follows. %%% TAKEN OUT IN REVISION In Section 2, % in order to understand if the bias is present, we perform a careful investigation of all the factors that determine the extinction of a Galactic or extragalactic \ion{H}{i} region. % revision 3 %We explain how the %extinction gives rise to a barrier that prevents %the detection of faint background sources. % In Section 3 we report some indirect evidence of obscuration effects. % in the sample of DLAs %with zinc measurements. % In Section 4 we present a %[start revision] mathematical formulation %[end revision] aimed at recovering the frequency distributions of \ion{H}{i} column densities and metallicities of DLAs from the %[start revision] observed distributions affected by obscuration bias. % The first implementation of this method is described in Section 5 and the results are discussed in Section 6. % In Section 7 we summarize the conclusions of the work. % %A critical point of this procedure is the %conversion between dust-to-gas ratios, derived from depletion %measurements, and extinction. In fact we know that, even for a fixed %depletion pattern, the extinction can vary widely by only changing the %dust grain size distribution. As a consequence, it is not guaranteed %that the extinction scales linearly with the fraction of metals in %dust form. For this reason, and because this approach is limited to %DLAs observed in abundance surveys, definitive conclusions on the %obscuration bias cannot be derived from this type of work. % %Furthermore, Fall \& Pei (1993) speculated that extinction, if it %exist, should lead to a selection bias against detection of DLAs with %high dust content in optically selected quasar samples. Understanding %whether dust obscuration is present or not has important implications %concerning the metallicity distribution and its evolution in %DLAs. This in turn is relevant both for studies of cosmic metal %production (Fall ... ???) and of galactic chemical evolution %(Prantzos \& Boissier 2000). % %At variance with previous works, the study presented here is motivated %by observational evidences, based on Milky Way interstellar studies, %which supports the suggestions from Boiss\'e et al. (1998). We start %by reassessing the existence of a cutoff of \ion{Zn}{ii} column %densities in an up-dated larger DLA samples (Section 2). We then show %results concerning the Galactic interstellar extinction which is shown %to increase with $N(\ion{Zn}{ii})$, with a sharp rise at the cutoff %observed in DLAs (Section 3). In section 4, we develop a method %allowing for the estimation of the obscuration of quasars by dusty %DLAs and impletment this analysis to real data in Section 5. Section 6 %provides predictions of the effects of such obscuration on HI column %density distribution of DLAs, the metallicity measurements and the %estimates of the zinc column densities of these systems. All these %results are summarized in Section 7. % %Among quasar absorption line systems, %Damped Ly $\alpha$ systems (DLAs) are %the most likely candidates to have sufficient dust content to %cause extinction of the background quasar given the fact that they have % the highest HI column density %($N(\ion{H}{i})\geq 2 \times 10^{20}$ atoms cm$^{-2}$; Wolfe et al. 1986) %and the highest metallicity\footnote{ %We adopt the usual definition %[X/H] = log (X/H) - log (X/H)$_{\sun}$.} %([M/H] $\approx -1.1$ dex; Pettini et al. 1994). %Evidence for reddening induced by dust in DLAs was provided %by Pei et al. (1991), who found a systematic difference %in the continuum slope of quasars with and without foreground %DLAs. %The possibility such extinction could % lead to a selection bias against detection of %DLAs with high dust content was first considered by Fall \& Pei (1993). %% %Indirect evidence for the existence of the dust obscuration bias was provided by %Boiss\'e et al. (1998) %from the analysis of the plot of [Zn/H] versus $\log N(\ion{H}{i})$. %These authors did not found % DLAs above the threshold %[Zn/H] $+\log N(\ion{H}{i}) \simeq 20.5$, %corresponding to %$N(\ion{Zn}{ii}) \simeq 10^{13.1}$ atoms cm$^{-2}$, % and suggested that this %threshold could be attributed to obscuration effect, %assuming that metal column density tracks the dust column density. %% %Understanding whether dust obscuration is present or %not has important implications concerning the metallicity %distribution and its evolution in DLAs. This in turn %is relevant both for studies of cosmic metal %production (Fall ... ???) %and of galactic chemical evolution % (Prantzos \& Boissier 2000). %Different approaches have been used in order to %assess the importance of the dust obscuration effect. %One approach consists in %estimating the quasar extinction due to DLAs by converting %differential the dust depletions measured in the DLAs %into dust-to-gas ratios. %Optical extinctions of %$\la 1$ mag (Vladilo et al. 2001) %and $\la 0.4 $ mag (Prochaska \& Wolfe 2002) %have been obtained in this way, leading to %different conclusions concerning the importance of %the dust obscuration effect. %A critical point of this procedure is the conversion between %dust-to-gas ratios, derived from depletion measurements, %and extinction. In fact we know that, even for a fixed depletion pattern, %the extinction can vary widely by only changing the dust grain %size distribution. As a consequence, it is not guaranteed that the extinction scales %linearly with the fraction of metals in dust form. For this reason, %and because this approach is limited to DLAs %observed in abundance surveys, % definitive conclusions on the % obscuration bias cannot be derived from this %type of work. % %A powerful approach to attack the problem is to %compare the statistics of %optically selected and radio selected DLA surveys, % the latter %unaffected by dust extinction, to search for evidence of the %dust obscuration bias. %Ellison et al. (2001) compiled a homogeneous sample %of radio selected quasars and searched for DLAs towards every %target, irrespective of its optical magnitude. %They conclude that dust-induced bias in previous magnitude limited surveys %may have lead to underestimate the neutral gas mass density, $\Omega_\mathrm{DLA}$, and the number %density per unit redshift interval, $n(z)$, of DLAs by at most a factor of two. %In addition, they find that $n(z)$ is greater in fainter quasars, %as expected by the dust obscuration effect (but only at 1 $\sigma$ level). %The results by Ellison et al. (2001) are %not yet conclusive since their sample is still limited, in particular %at the high values of $N(\ion{H}{i})$ where the effect is expected to be more important. % % %At variance with previous work, here % we consider empirical evidence in support %of Boiss\'e et al's suggestion obtained from Milky Way %interstellar studies. %% %First we reassess the existence of a cutoff of \ion{Zn}{ii} %column densities, not expected on the basis of %our measuring capabilities (Section 2). Then we show that %Galactic interstellar extinction increases with $N(\ion{Zn}{ii})$, %with a sharp rise at the cutoff observed in DLAs (Section 3). %These results are discussed in Section 4 and summarized in Section 5. % %\end{document} \section{ The relation between $N_\mathrm{Zn}$ and extinction } %We investigate the relation between $N(\ion{Zn}{ii})$ %and interstellar extinction to understand if there is a threshold for dust obscuration %at $N(\ion{Zn}{ii}) \simeq 10^{13.1}$ atoms cm$^{-2}$, as originally %suggested by Boiss\'e et al. (1998). % We adopt the simplified notation $N_\mathrm{Zn} = N(\ion{Zn}{ii})$ and $N_\mathrm{H} = N(\ion{H}{i})$, % and similar for other elements, ignoring contributions from ionization states other than the dominant one (see Vladilo et al. 2001a and references therein). We start by considering a refractory and a volatile\footnote{ Refractory elements easily condense into dust form, while volatile elements tend to remain in the gas phase. % revision 3 next sentence eleiminated %This different behaviour appears to be related to differences in the %condensation temperature characteristic of each element (see Field 1974). % end revision 3 } element, $\mathrm{E_r}$ and $\mathrm{E_v}$, with interstellar abundance ratio by number $( \mathrm{E_r} / \mathrm{E_v} )$. The ratio of the total column densities (gas plus dust) of these two elements along an interstellar line of sight will be $ N^\mathrm{tot}_\mathrm{r} / N^\mathrm{tot}_\mathrm{v} = ( \mathrm{E_r} / \mathrm{E_v} ). $ We call $N^\mathrm{d}_\mathrm{E}$ and $N_\mathrm{E}$ the column densities of an element E in the dust and in the gas, respectively. The fraction of atoms of E in dust form will be $f_{\mathrm{E}} \equiv N^\mathrm{d}_{\mathrm{E}}/N^\mathrm{tot}_{\mathrm{E}}$. % From these definitions we have $ N^\mathrm{tot}_{\mathrm{r}} % = N(\mathrm{X}_r) + N_d(\mathrm{X}_r) = N^\mathrm{d}_\mathrm{r} / f_{\mathrm{r}}$ and $N^\mathrm{tot}_\mathrm{v} % = N(\mathrm{X}_v) + N_d(\mathrm{X}_v) = N_\mathrm{v} / (1-f_\mathrm{v})$. %From (Eq. \ref{RoverV1}) Combining these relations we have \begin{equation} N^\mathrm{d}_{\mathrm{r}} = \, { f_\mathrm{r} \over (1-f_\mathrm{v})} \, \left( { {\mathrm{E_r} \over \mathrm{E_v} } } \right) \, N_\mathrm{v} ~. \label{RoverV} \end{equation} % We next consider the relation between $N^\mathrm{d}_\mathrm{r}$ and the column density of dust grains, $N_\mathrm{gr}$, i.e. the number of grains per cm$^{2}$ along the line of sight. For spherical grains of radius $r_\mathrm{gr}$ (cm) and internal mass density $\varrho_\mathrm{gr}$ (g cm$^{-3}$) it is easy to show that \begin{equation} N^\mathrm{d}_{\mathrm{r}} \, \mathrm{A_r} \, m_\mathrm{H} = {4 \over 3 } \, \pi ~ r_\mathrm{gr}^{3} ~ \varrho_\mathrm{gr} ~ N_\mathrm{gr} ~ X^\mathrm{d}_\mathrm{r} ~, \label{Rgrains} \end{equation} where $\mathrm{A_r}$ is the atomic mass of the refractory element and $X^\mathrm{d}_\mathrm{r}$ the mass fraction in the dust of the same element. Finally, we consider the relation between the interstellar extinction in magnitudes at the wavelength $\lambda$, $A_\lambda$, and the column density of dust grains. For spherical grains with extinction efficiency factor $Q_{e,\lambda}$ (see Spitzer 1978, Chapter 7) this relation is \begin{equation} A_\lambda = 1.086 ~ % N_d(\mathrm{grains}) \, \sigma_{gr} \, Q_e ~, N_\mathrm{gr} \,\, Q_{e,\lambda} \,\, \pi \, r_\mathrm{gr}^2 ~. \label{AVgrains} \end{equation} By combining Eqs. (\ref{RoverV}), (\ref{Rgrains}) and (\ref{AVgrains}) we derive \begin{equation} A_\lambda = a_\lambda ~ N_{\mathrm{v}} ~, \label{BigA_Nv} \end{equation} where we define \begin{equation} a_\lambda = 1.817 \times 10^{-24} ~ \mathrm{A_r} ~ g_\lambda ~ { f_\mathrm{r} \over (1-f_\mathrm{v})} \, \left( { {\mathrm{E_r} \over \mathrm{E_v} } } \right) ~, \label{SmallA_Nv} \end{equation} and we isolate in the term \begin{equation} g_\lambda = s ~ { Q_{e,\lambda} \over r_\mathrm{gr} \, \varrho_\mathrm{gr} \, X^\mathrm{d}_\mathrm{r} } ~ \label{glambda} \end{equation} all the properties of the grains (shape, extinction efficiency, size, density, and composition). For spherical grains $s = 3/4$. The factor ${ f_\mathrm{r} / (1-f_\mathrm{v})}$ is approximately constant by definition of volatile and refractory elements. In a medium of constant composition, such as the local ISM, also the abundance $\left( { {\mathrm{E_r} / \mathrm{E_v} } } \right)$ is constant. Therefore, for a given type of dust, the term $a_\lambda$ is constant and the expression (\ref{BigA_Nv}) implies that {\em the interstellar extinction increases linearly with the gas-phase column density of any volatile element measured in the same line of sight}. %The dispersion in the relation will be dominated by the dispersion %of $g_\lambda$, which characterizes the properties of the dust grains. % %Eqs. (\ref{Rgrains}) and (\ref{AVgrains}) are based on the assumptions %that all grains are identical. In the actual case, Differences in the shape, size, density, composition and extinction efficiency of the grains will induce a scatter in the relation. %However, %it is important to note that The dependence of $g_\lambda$ on the grain size $r_\mathrm{gr}$ is linear, %in the final Eq. (\ref{Alambda_Nv}) while in Eqs. (\ref{Rgrains}) and (\ref{AVgrains}) it was cubic and quadratic. %This milder dependence % on $r_\mathrm{gr}$ %suggests that the relation between extinction and %column density of volatile elements may not undergo dramatic variations %even if the grain size distribution changes. % Since the grain size distribution is a key factor in determining the wavelength dependence of the interstellar extinction (see e.g. Draine 2003), the mild dependence of $g_\lambda$ on $r_\mathrm{gr}$ suggests that {\em the relation between $A_\lambda$ and $N_\mathrm{v}$ may be similar even in interstellar regions with different types of extinction curves}. \begin{figure} \centering \includegraphics[width=7.5cm,angle=0]{1570f1.eps} \caption{Extinction $A_V=R_V \, E_{B-V}$ (mag) versus \ion{Zn}{ii} column density (atoms cm$^{-2}$) in Galactic interstellar lines of sight. Filled diamonds: HST data from Roth \& Blades (1995). Empty diamonds: IUE data from Van Steenberg \& Shull (1988). %Error bars are not quoted in the original work. Curve: extinction law (\ref{AV_NZn}) with $ a_V = 0.29 \times 10^{-13}$ mag cm$^2$ derived empirically as explained in the text. } \label{figAV_NZn} \end{figure} By adopting zinc as the volatile element and considering the extinction in the $V$ band ($\lambda_V=0.55 \mu$m), we obtain from (\ref{BigA_Nv}) the relation \begin{equation} A_V \simeq a_V % \times \, N_\mathrm{Zn} ~, \label{AV_NZn} \end{equation} where the parameter $a_V$ (mag cm$^2$) % = a( Z_d(\mathrm{X}_r), %\varrho_{d}, r_{d}, % Q_e, ( {\mathrm{X}_r / \mathrm{X}_v } )_m)$ can be determined by comparing measurements of $A_V$ and $N_\mathrm{Zn}$ in individual lines of sight. In this way it is possible to measure the scatter of $a_V$ due to variations of the grain properties. % in different interstellar regions. %% = a( r_{d}, \varrho_{d}, Z_d(\mathrm{X}_r), %% Q_{e,V}, ( {\mathrm{X}_r / \mathrm{X}_v } )_m)$ %in lines of sight characterized by different types of dust. To estimate $a_V$ we gathered %$A_V$ and $N_\mathrm{Zn}$ %data from the literature. %from Eq. (\ref{EBVlaw}) %%estimated by comparing measurements of %% $E_{B-V}$ and $N_\mathrm{Zn}$. %%% %%To do so %We collected the $N_\mathrm{Zn}$ measurements obtained from HST (Roth \& Blades 1995) and IUE (Van Steenberg \& Shull 1988) high-resolution spectra. The IUE data are more numerous, but have lower resolution and signal-to-noise ratio. %than the HST data. To reduce the impact of these differences we selected the IUE measurements with total error $< 0.3$ dex in $\log N_\mathrm{Zn}$. %(last column of Table 5 in Van Steenberg \& Shull 1988). % much more numerous but of lower quality than the HST data. % but provide us with a much larger sample. To minimize the contamination by stellar absorptions on top of the IUE \ion{Zn}{ii} interstellar lines (Van Steenberg \& Shull 1988), we selected only IUE spectra of fast-rotating stars ($V \sin i \geq$ 200 km s$^{-1}$). %To combine data of comparable quality We then scaled the IUE column densities to match the Bergeson \& Lawler (1993) \ion{Zn}{ii} oscillator strengths adopted in the HST data and in the current research. As far as the extinctions are concerned, we used the $A_V$ values given by Guarinos (1991). As a result, we obtained a sample of 11 lines of sight. The mean value of extinction per unit zinc column density that we derive is $<\!\! a_V \!\!> \, = 0.27 (\pm 0.06) \, 10^{-13}$ mag cm$^2$ and $0.38 (\pm 0.20) \, 10^{-13}$ mag cm$^2$ for the HST and IUE sub-samples (4 and 7 lines of sight, respectively). %The two results are consistent with each other %within their standard error of the mean %($\pm 0.03$ and $\pm 0.07$ mag cm$^2$, respectively). In order to enlarge the sample we also considered the color excess $E_{B-V}$, which is more commonly measured than $A_V$. %from the comparison of reddened and unreddened stars %of same spectral type. %The standard color excess is usually measured by comparing the fluxes in %the B and V photometric bands. The ratio of general-to-selective extinction, $ R_V = { A_V / E_{B-V}} $, is approximately constant, with a mean value $<\!\! R_V \!\!> \, \simeq 3.1$ in the diffuse ISM of the Milky Way (see e.g. Draine 2003). We used this property to derive a mean value $<\!\! a_V \!\!> \, = \,<\!\! R_V \!\!> \, <\!\! E_{B-V}/N_\mathrm{Zn} \!\!>$ from reddening and zinc column density measurements. % The $E_{B-V}$ data %for both the HST and IUE sub-samples, were taken from the extinction catalog of Savage et al. (1985). % The resulting sample includes 24 lines of sight, shown in Fig. \ref{figAV_NZn}. The mean value %of $e_V=E_{B-V}/ N_\mathrm{Zn}$ of the HST sub-sample (9 lines of sight) is $<\!\! a_V \!\!> \, = 3.1 \times 0.089 \, (\pm 0.030) \, 10^{-13}$ mag cm$^2$, and of the IUE sub-sample (15 lines of sight), %$<{a_{\circ} \over R_V }>= $3.1 \times 0.123 \, (\pm 0.071) \, 10^{-13}$ mag cm$^2$. %Again, the results from the two sub-samples are consistent %with each other. % within their standard error of the mean. % We did not find significant correlations between $<\!\! E_{B-V}/N_\mathrm{Zn} \!\!>$ and the parameters of UV extinction provided by Savage et al. (1985), such as the strength of the emission bump %(parameters $E_\mathrm{Bump}$ and $\Delta_\mathrm{Bump}$) or the steepness of the UV rise. % (parameter $\Delta_{15-V}$). The lack of these correlations gives us a justification for averaging the results obtained from lines of sight with different types of dust. The results obtained from the $E_{B-V}$ analysis are identical to those obtained from the more limited sample of direct measurements of $A_V$. The larger dispersion of the IUE sub-sample is probably due to the lower quality of the data. % REVISION 3 next sentence eliminated % %The slight offset might be due to saturation effects %hard to disentangle in the IUE data. The saturation %would lead to underestimating the zinc column densities %and, as a consequence, to overestimating $a_V$. % % end revision 3 We adopted the weighted average of the IUE and HST sub-samples, which yields $<\!\! a_V \!\!> \, = 0.29 (\pm 0.07) \, 10^{-13}$ mag cm$^2$, a value almost identical to that of the HST sub-sample. % REVISION 3 next sentence eliminiated %The error %is obtained from the propagation of the standard errors of the mean. % end revision 3 % The relation (\ref{AV_NZn}) obtained from this estimate of $a_V$ % $E_{B-V} = e_V \, N_\mathrm{Zn}$ is displayed in Fig. \ref{figAV_NZn} (solid curve). The exponential rise of the extinction is due to the fact that the \ion{Zn}{ii} column density is plotted on a logarithmic scale. % In this scale the rise of the extinction is mild at low values of $\log N_\mathrm{Zn}$, but very fast at high values. % of $\log N(\ion{Zn}{ii})$. The turning point %$y_{\circ}$ marking the transition between these two regimes can be estimated from the condition $\partial A_V / \partial (\log N_\mathrm{Zn}) =1$. This criterion yields $\log N_\mathrm{Zn} = - \log (<\!\! a_V \!\!> \, \ln 10) = 13.18$. {\em Above this turning point the extinction undergoes a dramatic rise, acting as a barrier against detection of faint background sources. } This barrier %may prevent the detection of faint %quasars lying beyond DLA systems with $\log N_\mathrm{Zn}$ above the %turning point, which is remarkably similar to the detection threshold proposed by Boiss\'e et al. (1998). This motivated us to investigate the relation between extinction and metal column density in DLA systems. % \subsection{Extinction versus $N_\mathrm{Zn}$ in DLA systems } The relations (\ref{BigA_Nv}), (\ref{SmallA_Nv}) and (\ref{glambda}) are valid in the ISM of any galaxy, including DLA systems, which are \ion{H}{i} regions of high-redshift galaxies. % Applying these relations to a DLA system and to the local ISM, and adopting $\mathrm{E_v}$=Zn, $\mathrm{E_r}$=Fe and $\lambda=\lambda_V$, we obtain % \begin{equation} A_{V} = a_{V,i} \, { g_V \over g_{V,i} } ~ { (1-f_{\mathrm{Zn},i}) \over f_{\mathrm{Fe},i} } ~ { f_{\mathrm{Fe}} \over (1-f_{\mathrm{Zn}}) } ~ { (\mathrm{Fe/Zn}) ~ \over (\mathrm{Fe/Zn})_i } ~ N_{\mathrm{Zn}} ~, \label{AVdla1} \end{equation} % where the quantities in the local ISM are indicated with the index $i$ and those in the DLA system are not labelled. To take into account the dependence on metallicity, which varies among DLAs, we introduce the metallicity $Z \equiv N^\mathrm{tot}_{\mathrm{Zn}}/N_\mathrm{H}$, where $N^\mathrm{tot}_{\mathrm{Zn}} = N_{\mathrm{Zn}} /(1-f_{\mathrm{Zn}})$ is the total zinc column density (gas plus dust). In this way we obtain the expression % \begin{equation} A_{V} \simeq A_\circ ~ { G } ~ f_{\mathrm{Fe}} ~ { (\mathrm{Fe/Zn}) ~ \, \over (\mathrm{Fe/Zn})_{\sun} } ~ Z ~ N_{\mathrm{H}} ~, %N^\mathrm{tot}_{\mathrm{Zn},j} ~. \label{AVdla2} \end{equation} % where $A_\circ \equiv a_{V,i} \, (1-f_{\mathrm{Zn},i})/ f_{\mathrm{Fe},i}$ and $G=g_V / g_{V,i}$. The constant $A_\circ $ is estimated using the value $<\!\! a_V \!\!>$ derived in the previous section and the mean values of dust fractions of the local ISM sample ($f_{\mathrm{Zn},i}=0.40$ and $f_{\mathrm{Fe},i}=0.94$). % We obtain $A_\circ \simeq 1.85 \times 10^{-14}$ mag cm$^2$, with a relative error of $\simeq \pm 0.15$ dex including the uncertainties of the dust fractions. % The DLA extinction will vary according to the Fe/Zn abundance in the medium, the iron depletion and the properties of the grains. %% We briefly discuss each one of these factors. %The dependence on the composition of the medium enters %through the term %s $Z$ and %${ (\mathrm{Fe/Zn}) / (\mathrm{Fe/Zn})_{\sun} }$. %$ \left( { {\mathrm{X}_r \over \mathrm{Zn} } } \right)_{m} $, which represents the %abundance ratio of the refractory element X$_r$ relative to zinc. %The metallicity level $Z$ can be estimated quite well from %the measured column densities of zinc and hydrogen since %the dust fraction of zinc is almost negligible in DLAs and so %$Z = N_{\mathrm{Zn}}/N_\mathrm{H} %\simeq N^\mathrm{tot}_{\mathrm{Zn}}/N_\mathrm{H} $ %with a good approximation. Relative abundances of metals do not show strong variations in DLAs, despite significant changes of the absolute metallicity $Z$. The Fe/Zn ratio %, one of the most abundant refractory elements, is very close to solar in metal-poor stars with metallicities typical of DLA systems (Mishenina et al. 2002; Gratton et al. 2003; Nissen et al. 2004; see discussion in Vladilo 2004). Deviations from the solar ratio are measured in stars with extremely low metallicities. Such deviations do not affect the present work in any case, because the dust fraction and the extinction vanish at very low metallicity, as we now discuss. We adopt therefore ${ (\mathrm{Fe/Zn}) / (\mathrm{Fe/Zn})_{\sun} } = 1$. %This choice is conservative since it might lead to underestimate the extinction %if the Zn/Fe ratio in the DLAs is overabundant, %as found in metal-poor stars of extremely low metallicity. %For these reasons we do not expect significant variations of %$ \left( { {\mathrm{X}_r \over \mathrm{Zn} } } \right)_{m} $ to occur %in DLA systems, relative to %the Miky Way value implicitely adopted in deriving $a_V$. %The dependence on the depletion enters through the term %$ f_{\mathrm{Fe}}$. % The depletion of DLAs increases with metallicity (Ledoux et al. 2003). In particular, the dust fraction of iron undergoes a fast rise between between metallicities [Zn/H] $\sim -2$ dex and $-1$ dex, from values close to zero up to values typical of the Milky-Way warm interstellar gas (Vladilo 2004). Here we model this trend with the analytical expression $$ f_{\mathrm{Fe}}(Z) = %({f_{\mathrm{Fe},\circ} / \pi} ) { 1 \over \pi} \, \left[ \arctan \left({\mathrm{[Zn/H]} - \mathrm{[Zn/H]}_\circ \over \Delta(\mathrm{[Zn/H]})} \right)+{\pi \over 2} \right] $$ % $f_{\mathrm{Fe}}=f_{\mathrm{Fe}}(Z)$ designed to smoothly fit the ISM and DLA data, providing a gradual decrease to zero % $ f_{\mathrm{Fe}}=0 $ at very low metallicity. %with [Zn/H]$= \log_{10} (Z/Z_{\sun})$, %%$f_{\mathrm{Fe},\circ} = 1$, % $\mathrm{[Zn/H]}_\circ=-1.25$ dex and %$\Delta(\mathrm{[Zn/H]})=0.4$ dex. % This expression also fits % the $ f_{\mathrm{Fe}} $ measured in the local ISM %%value $ f_{\mathrm{Fe},i} \simeq 0.94$ [{check}] when the metallicity is solar. %and %provides a smooth decrease to $ f_{\mathrm{Fe}}=0 $ %at very low metallicity. %% and a smooth increase to $ f_{\mathrm{Fe}}=1$ %%at very high metallicity. The dependence on the changes of the grain properties enters through the factor $G$, which scales as $s \, Q_{e,\lambda} \, (r_\mathrm{gr} \, \varrho_\mathrm{gr} \, X^{d}_\mathrm{Fe}) ^ {-1}$ (Eq. \ref{glambda}). This term is mostly determined by the physical conditions of the medium, rather than by its chemical enrichment, with the possible exception of the abundance by mass of iron in the grains, $X^{d}_\mathrm{Fe}$. % At the very early stages of chemical evolution we may expect variations of $X^{d}_\mathrm{Fe}$ as a consequence of non-solar relative abundances of metals. However, these effects do not affect our estimate of the extinction since, at the very low metallicities typical of these early stages ($Z \ll Z_{\sun}$), the factor $f_{\mathrm{Fe}}(Z)$ vanishes in any case, yielding a null extinction. The variations of $G$ due to changes of the physical conditions can be assessed by sampling different regions in a medium of constant composition, such as the local ISM. % The low scatter of $a_V$ derived in the previous section indicates that the scatter of $G$ in the local ISM is low, even if the MW sample includes lines of sight with different physical conditions. % To explain this qualitatively, we can imagine that variations of the grain size $r_\mathrm{gr}$ may compensate variations of the dust density $\varrho_\mathrm{gr}$. For instance, in a harsh interstellar environment, volatile elements tend to leave the grains and we may expect an increase of $\varrho_\mathrm{gr}$ because carbon, the element potentially most abundant in the grains, is volatile and has low atomic mass. At the same time, we may expect that larger grains will be more easily destroyed in a harsh environment and the mean $r_\mathrm{gr}$ will decrease, countering the increase of $\varrho_\mathrm{gr}$ in Eq. (\ref{glambda}). %% revision 3 next sentence eliminated %% %As far as the abundance by mass of iron is concerned, %we can imagine that variations of $X^{d}_\mathrm{Fe}$ %are not very large because the bulk of the grain mass %comes from the most refractory and heavy elements, %and iron itself is one of these elements. % These general considerations apply to any type of interstellar environment, suggesting that $G$ may not vary dramatically in DLAs with moderately low abundances ($Z \ga 0.1 \, Z_{\sun}$). To test this hypothesis we computed $G$ in the Small Magellanic Cloud (SMC) from the expression \begin{equation} { G_{\mathrm{SMC}} } \simeq { ( { A_V \over N_\mathrm{H} } )_\mathrm{SMC} \over ( { A_V \over N_\mathrm{H} } )_\mathrm{MW} } \, % \left[ { Z_\mathrm{SMC} \over Z_{\sun} } \, { f_{\mathrm{Fe},\mathrm{SMC}} \over f_{\mathrm{Fe},\mathrm{MW}} } \right]^{-1} \, % \end{equation} obtained by applying Eq. (\ref{AVdla2}) to the SMC with $ (\mathrm{Fe/Zn})_\mathrm{SMC} = (\mathrm{Fe/Zn})_{\sun}$. % Inserting $( { A_V \over N_\mathrm{H} } )_\mathrm{SMC} = 8.7 \times 10^{-23}$ mag cm$^{2}$ (Gordon 2003; SMC bar), $( { A_V \over N_\mathrm{H} } )_\mathrm{MW} = 2.1 \times 10^{-22}$ mag cm$^{2}$ (Bohlin et al. 1978), $ Z_\mathrm{SMC} \simeq 0.25 \, Z_{\sun} $ (Russel \& Dopita 1992), and $ f_{\mathrm{Fe},\mathrm{SMC}} (Z_\mathrm{SMC}) \simeq 0.91 \, f_{\mathrm{Fe},\mathrm{MW}}$ from %% REVISION 3 our analytical relation $f_{\mathrm{Fe}}=f_{\mathrm{Fe}}(Z)$, %% relation (\ref{fFeZeta}), % we obtain ${ G_{\mathrm{SMC}} } \simeq 0.6 $. % Considering the uncertainties in this derivation, this result is consistent with the Milky-Way value, % ($G=1 \pm 0.3$), suggesting that, indeed, the term $G$ may not vary dramatically in galaxies of moderately low abundances %%% revision 3 even if the extinction curve is different. %%% NEXT SENTENCE ELIMINATED %This result is strengthened by %the fact that SMC-type dust is commonly considered to be quite different %from Milky-Way dust. % %%% end revision 3 Eq. (\ref{AVdla2}) is valid in the rest frame of the \ion{H}{i} region. % To derive the extinction in the observer's frame we need to consider the wavelength dependence of the extinction. % In the rest frame of the DLA system the extinction % of the background quasar at the wavelength $\lambda^\prime$ will be %% %\begin{equation} $ A_{\lambda^\prime} \simeq \xi(\lambda^\prime) \,\, A_{V} $, %\end{equation} where $\xi(\lambda^\prime)=A(\lambda^\prime)/A(\lambda_V)$ is, by definition, the normalized extinction curve. In the observer's frame the same extinction % at $\lambda = \lambda^\prime$ %in the rest frame of the DLA system will appear at $\lambda = \lambda^\prime \, (1+z)$, where $z$ is the absorption redshift. \begin{figure} \centering \includegraphics[width=7.5cm,angle=0]{1570f2.eps} \caption{ Extinction of a quasar at a fixed wavelength in the observer's frame, plotted versus the redshift of an intervening DLA system with $N^\mathrm{tot}_\mathrm{Zn} = 10^{13.50}$ cm$^{-2}$. %The extinction is calculated with Eqs. (\ref{ext_zNZn}) or %(\ref{ext_zNH}) for different extinction % laws and photometric bands. Solid and dashed curves represent MW-type and SMC-type extinction curves, respectively (Cardelli et al. 1998; Gordon et al. 2003). Curves left to right: photometric bands $u^\prime$, $g^\prime$, and $r^\prime$, respectively. } \label{ext_z} \end{figure} In conclusion, the observer's frame extinction of a quasar at wavelength $\lambda$ due to an intervening DLA system at redshift $z$ is %with hydrogen column density $N_\mathrm{H}$, %metallicity $Z$ \begin{equation} A_{\lambda} (N_\mathrm{H},Z; z) \simeq A_\circ ~ G ~\, %2.0 \times 10^{-12}\, { g \over g_i } \, % a_{V,i} \, (1-f_{\mathrm{Zn},i}) \, \xi \! \left( \! { \lambda \over 1 \!+ \! z } \! \right) ~ % { p_{V,j} \over p_{V,i} } \, %{ f_{\mathrm{Fe}}(Z) ~\, %\over f_{\mathrm{Fe},i} } \, N_{\mathrm{H}} ~ Z ~ , %N^\mathrm{tot}_{\mathrm{Zn},j} ~. \label{AlamHzeta} \end{equation} % with %${ g / g_i } = 1 $ and $0.6$ $A_\circ \simeq 1.85 \times 10^{-14}$ mag cm$^2$, and $G=1$ and $0.6$ for Galactic-type and SMC-type dust, respectively. For each type of dust we adopt the corresponding extinction curve representative of the Milky Way and of the SMC. %% %Concerning the choice of the extinction curve, %Our knowledge of the extinction curve of DLAs is very poor %with the exception of the work by Pei et al. (1991), which suggests %that 217.5\,nm extinction bump is weak or absent, as %observed in SMC extinction curves. %Considering the uncertainty of this result and the fact that a Milky-Way type %extinction curve has been found at redshift $z=0.83$ (Motta et al. 2002), a safe %approach is to consider both Galactic-type and SMC-type extinction %curves as possible proxies for the DLAs. Since $\xi(\lambda^\prime)>1$ for $\lambda^\prime < \lambda_V$, the net effect of the cosmological redshift is an amplification of the rest-frame extinction at $\lambda < \lambda_V ~ (1+z)$. %a condition certainly satisfied by UV and optical bands. At $z \ga 2$ this amplification compensates for the reduction of $f_{\mathrm{Fe}}(Z)$ in systems of moderately low metallicity and for the mild reduction of $G$ in SMC-type dust. Only at very low metallicity the extinction becomes negligible since $f_{\mathrm{Fe}}(Z)$ tends to vanish. Overall, {\em the extinction per unit column density of zinc atoms predicted for most DLAs % } by Eq. (\ref{AlamHzeta}) {\em is similar, or even larger, than that typical of the Milky-Way ISM}. % % revision 3 NEXT SENTENCE ELIMINATED %% %We expect therefore that the zinc detection threshold found %for the Milky-Way can also apply to DLAs. % % end revision 3 Examples of quasar extinction estimated with Eq. (\ref{AlamHzeta}) %for different effective wavelengths $\lambda$ and extinction curves $\xi(\lambda)$ are plotted in Fig. \ref{ext_z} as a function of $z$, keeping fixed the other terms. This type of representation puts in evidence the amplification of the extinction with increasing redshift. Each curve was computed at a constant $\lambda$ equal to the effective wavelength of the SLOAN Digital Sky Survey (SDSS) photometric bands $u^\prime$, $g^\prime$ and $r^\prime$ (Fukugita et al. 1996). % plus the $V$ band of the Johnson system. We adopted the extinction law by Cardelli et al. (1998; hereafter CCM law) with $R_V=3.1$ %(Cardelli et al. 1998) for the Milky-Way extinction curve and the mean SMC bar data by Gordon et al. (2003) for the SMC curve. The term $f_\mathrm{Fe}(Z)$ was estimated at [Zn/H]\,$= \log (Z/Z_{\sun}) = -1$ dex. The total zinc column density was fixed at $N^\mathrm{tot}_{\mathrm{Zn}} = N_\mathrm{H} \, Z =10^{13.50}$ atoms cm$^{-2}$, roughly two times the zinc column density threshold proposed by Boiss\'e et al. (1998). %of the turning point $\log N_\mathrm{Zn} = 13.18$ dex %found in the previous section, %that separates the regimes of slow and fast rise of the extinction. For the redshift interval typical of most DLAs ($z \approx 2/3$) the predicted extinction % at $\log N_\mathrm{Zn} = 13.15$ dex is $\gsim 1$ mag, a value sufficient to obscure quasars in spectroscopic surveys. % for all the photometric bands and extinction curves considered, %with an amplification factor of $\approx 3$ of the rest-frame extinction. %% %The fast rise of $A_b$ at $\log N_\mathrm{Zn}$ predicted %above the turning point % $\log N_\mathrm{Zn} = 13.15$ dex %will give extinctions well in excess of 1 magnitude, \begin{figure} \centering \includegraphics[width=7.5cm,angle=0]{1570f3.eps} \caption{Diamonds: column densities $N_\mathrm{H}$ versus $N_\mathrm{Zn}$ in DLAs. % the sample of DLAs with \ion{Zn}{ii} measurements. Horizontal lines: DLA definition threshold (bottom) and empirical upper boundary (top) of $N_\mathrm{H}$. Vertical lines: detection limit (left) and obscuration threshold (right) of $N_\mathrm{Zn}$. Dashed lines: lines of constant metallicity. % REVISION 3 Circles: Milky Way data of Fig. 1. % END REVISION 3 Star: GRB-DLA from Savaglio et al. (2003). % See Sections ... for explanations. } \label{sample} \end{figure} \section{Evidence for extinction from DLA data} To search for some extinction effect related to the zinc column density %On the basis of the above discussion we expect %the zinc column density to be an indicator of quasar extinction. % we collected from the literature all DLAs with %$N_\mathrm{H} \geq 10^{20.3}$ atoms cm$^{-2}$ and available $N_\mathrm{Zn}$ measurements. % The resulting sample, shown in Fig. \ref{sample}, %and called hereafter the zinc sample, includes 41 measurements, most of which derived from Hires/Keck and UVES/VLT spectra. References to these data can be found in Vladilo (2004; table 1), with the exception of 4 systems (without \ion{Fe}{ii} lines) % with \ion{Zn}{ii}, but , published by Pettini et al. (1994, 1997) and Boiss\'e et al. (1998). % % From the analysis of Fig. \ref{sample} we can derive the % following considerations. % %First, The updated sample includes more than twice the measurements of the one discussed by Boiss\'e et al. (1998) but, %, which included 17 measurements and several limits. in spite of this increase, column densities above the original threshold % $N_\mathrm{Zn} > 10^{13.1}$ atoms cm$^{-2}$ (line AB in the figure) are not found. % %This limit is remarkably similar to the %one found by Boiss\`e et al. with a half number of measurements. % % Second, The lack of DLAs with $N_\mathrm{Zn} > 10^{13.1}$ atoms cm$^{-2}$ is surprising, since Milky Way lines of sight with \ion{Zn}{ii} measurements do show values above this limit in a significant fraction of cases % REVISION 3 (Fig. \ref{sample}). % % Studies of gamma-ray bursts (GRBs) observed inmediately after the explosion demonstrate that zinc column densities above the threshold exist and can be detected in extragalactic absorbers\footnote{ % Absorbers detected in front of GRBs, however, may have different properties from classical DLAs; %% FINAL ADDITION a claim for gray extinction in GRB/DLAs has been made by Savaglio et al. (2003). % } % when the background source is sufficiently bright (Savaglio et al. 2003; the only case with measured $N_\mathrm{H}$ is indicated with a star in Fig. \ref{sample}). % These results confirm that some selection effect is preventing the detection of values of $N_\mathrm{Zn}$ above the threshold when the background source is faint. %The bias is not of instrumental origin, since we are able to measure %column densities above the threshold. % % %An intrinsic cutoff at high metallicity cannot explain the lack of systems %% in Fig. \ref{sample}, but not %in the triangle ABC, since the lines of constant metallicity to the left of the line AC %intersect observational data points. %% where such systems would have %%metallicities inside the observed range. % The remarkable similarity between the DLA cutoff at $N_\mathrm{Zn} = 10^{13.1}$ atoms cm$^{-2}$ and the ISM turning point % of the derivative $\partial A_V / \partial (\log N_\mathrm{Zn}) $ at $N_\mathrm{Zn} \simeq 10^{13.2}$ suggests that the rapid rise of the extinction above the turning point can be responsible for the cutoff. % This is consistent with the conclusion of the previous section that the extinction per unit zinc column density of many DLAs is similar to that of the local ISM. %% %{\em The sharpness of the cutoff suggests that the %extinction per unit zinc column density %is approximately constant among DLAs. %% %All together, these arguments support %the notion that the parameter $G$, which %describes the properties of the dust grains, does not vary %dramatically among different galaxies. } %Given the faintness of quasars, compared to the brightness %of the Galactic stars observed in interstellar studies, a rapid rise of the %extinction would be critical. % %The lack of a clear cutoff with %increasing $N_\mathrm{H}$ or metallicity %in Fig. \ref{sample} % the ($\log N_\mathrm{Zn},\log N_\mathrm{H}$) plane % is consistent with our previous conclusion, based on Eq. (\ref{ext_zNH}), % that it is hard to disentangle the dependence of the extinction % on $N_\mathrm{H}$ and [Zn/H]. \begin{figure} \centering \includegraphics[width=7.5cm,angle=0]{1570f4.eps} % \hspace{1.5cm} %\includegraphics[width=6.cm,angle=0]{Vmaghist.eps} \caption{Normalized frequency distribution of quasar apparent magnitudes for the %DLA sample of Fig. \ref{sample}. % Bottom panel: full sample. Top panel: sub-sample of DLAs with $\log N_\mathrm{Zn} > 12.53$ (shaded histogram) and $\log N_\mathrm{Zn} \leq 12.53$ (light histogram).} \label{Vmaghist} \end{figure} If the extinction increases with $N_\mathrm{Zn}$, we expect to find evidence of this effect also below the threshold, in the range $\log N_\mathrm{Zn} \la 13.1$ %i.e. in the interval % of $ N_\mathrm{Zn}$ where we do detect DLAs. To search for this evidence we analysed the frequency distribution of quasar apparent magnitudes of the zinc sample. We used the $V$ magnitude, which is measured in most cases. % REVISION 3 In 4 quasars without $V$ measurements we adopted the magnitude of nearby optical bands. % END REVISION 3 In practice, we compared the behaviour of the two sub-samples with $\log N_\mathrm{Zn}$ below and above the median value, $(\log N_\mathrm{Zn})_\mathrm{median}= 12.53$. %The resulting distributions are represented %as a light and a shaded histogram, respectively, in %the top panel of %Fig. \ref{Vmaghist}. % The differences between the two distributions, shown in Fig. \ref{Vmaghist}, are consistent with a rise of the quasar extinction with increasing $N_\mathrm{Zn}$. % The analysis of the three bins with more statistics % between $V$\,=\,16 and 19 mag, indicates that the maximum of the sub-sample with high $N_\mathrm{Zn}$ (shaded histogram) is shifted by $\approx 1$ bin relative to the maximum of the other sub-sample (light histogram). A Kolmogorov-Smirnov test shows that there is only a 9.7\% level of probability that the two distributions are drawn from the same parent population. % The shift to fainter magnitudes suggests that even at $\log N_\mathrm{Zn} \lsim 13. 1$ dust extinction is already present, affecting more the high-$N_\mathrm{Zn}$ sub-sample. % The magnitude of this effect can only be explained by invoking a redshift amplification of the extinction, % of a factor of $\gsim 3$, as predicted by relation (\ref{AlamHzeta}), since the extinction expected at rest-frame is only $\simeq 0.35$ mag at $\log N_\mathrm{Zn} = 13. 1$ (Fig. \ref{figAV_NZn}). % \section{ Mathematical formulation of the bias } In Eq. (\ref{AlamHzeta}) the variation of the depletion in DLAs is accounted for by the term $f_\mathrm{Fe}(Z)$, which is determined by the metallicity. The term $G$ is approximately constant, at least in galaxies with metallicity similar to those of the SMC and Milky Way. Possible variations of $G$ in galaxies of very low metallicity do not affect the prediction of the extinction since, in any case, $f_\mathrm{Fe}(Z)$ vanishes when $Z \ll Z_{\sun}$. For these reasons, relation (\ref{AlamHzeta}) allows us to estimate the extinction of a DLA system using, in practice, only its \ion{H}{i} column density and metallicity. %In fact, % %For the term $G$, we assume that it is similar to % %convert the frequency distribution %of hydrogen column densities and metallicities into a frequency distribution %of extinctions. %$f_{N_\mathrm{Zn}} $ into the frequency distribution of the quasar extinctions %due to intervening DLAs. We use this property to derive a mathematical formulation of the obscuration bias based on the study of the distribution functions of $N_\mathrm{H}$ and $Z$ in DLA systems. % % [start revision1] %In practice, we consider the extinction due to the bulk of the population %of observed DLAs, which is peaked at redshift $z \sim 2.3$. %We neglect the extinction due to DLAs with Ly\,$\alpha$ absorption below %the atmospheric cutoff ($z \lsim 1.8$), for which it is practically %impossible to determine $N_\mathrm{H}$ and $Z$. %Luckily, there are two factors that make less important the extinction %contribution of low-redshift DLAs: %(1) the redshift dependence of the extinction, shown in relation (\ref{AlamHzeta}); %(2) the intrinsic reduction of the number density of DLAs at %low redshift (Rao \& Turnshek 2000; Ellison et al. 2004). %The fraction in dust should not significantly increase with cosmic time below $z \lsim 1.8$ %because $f_\mathrm{Fe}$ already approaches a value close to unity well before % $z \sim 2$ (Vladilo 2004). Instead, %DLAs at very high $z$ probably do not yield a strong contribution %to the extinction since there is evidence for a drop %of $f_\mathrm{Fe}$ at the earliest stages of chemical evolution (see Eq. 10 above). %In summary, by considering only the extinction of DLAs in the interval %$1.8 \lsim z \lsim 3.0$ %we can obtain a reasonable estimate of the overall effect of quasar obscuration %and, at the same time, make the best use of available observations %without modelling the extinction properties in redshift intervals %that cannot be tested observationally. %% START REVISION 2 %{In practice, we determine the fraction of DLAs that are missed as a consequence of their own extinction. %, where %the extinction is computed from (\ref{AlamHzeta}) %given $N_\mathrm{H}$, $Z$ and $z$. We then assume that multiple DLA absorbers in a given line of sight play a negligible role in the estimate of the obscuration bias. %With these empirical considerations in mind, we have derived a mathematical %formulation of the extinction bias which considers %DLAs in a given redshift interval as the only %sources of quasar extinction. % In this way, as we show in the Appendix, we derive a mathematical formulation of the bias which only makes use of observable statistical distributions of DLAs and quasars. % together with the empirical-analytical relation (\ref{AlamHzeta}). % To quantify the obscuration effect we need a good statistics of the observed distributions and this requirement limits the redshift range where our method can be presently applied, since the bulk of DLAs data are currently concentrated in $1.8 \lsim z \lsim 3.0$. An advantage of our formulation is that it does not require a knowledge of % (but does not make predictions on) the intrinsic luminosity function of quasars or the geometrical distribution of the neutral gas in DLA galaxies. % We refer to FP93 for a mathematical treatment of the effect of quasar obscuration % REVISION 3 fottnote eliminated %\footnote %{ %%This is equivalent to ignore multiple DLA absorbers in a given line of sight. %% %The way multiple absorptions are treated in the formulation by FP93 can be understood %from the derivation of their Eqs. (5) and (6); this derivation can be found in Eqs. (18)-(21) of %Fall \& Pei (1989), where the probability of finding more than one absorber in the %interval $(z, z+\delta z)$ is an %infinitesimal of order $(\delta z)^2$. %} % which describes the relation with the quasar luminosity function and the geometrical distribution of the gas in the intervening galaxies. %} %%%%%%%%% END REVISION 2 % As an advantage over previous work, %In our formulation the extinction is computed making %use of statistical distributions that can be tested %observationally. In particular, the metallicity distribution can be %probed more easily than the distribution of dust-to gas ratios %employed in the formulation by Fall \& Pei (1993). %%%% % [end revision1] \subsection{ True and biased distributions } %%% REVISION 3 % We consider the DLAs in the redshift interval $(z_\mathrm{min},z_\mathrm{max})$ that are detectable in a survey with limiting magnitude $m_\ell$. % We call $f_{N_\mathrm{H}} d N_\mathrm{H}$ the number of such DLAs with column densities between $N_\mathrm{H}$ and $N_\mathrm{H}+dN_\mathrm{H}$ and $f_{Z} \, dZ$ the number of those with metallicity between $Z$ and $Z+d \,Z$. % The distributions $f_{N_\mathrm{H}}$ and $f_{Z}$ are defined in absence of obscuration bias and we call them the "true" distributions. % When we consider the effect of the extinction generated by these DLAs, the number of detectable systems will be $f^\mathrm{b}_{N_\mathrm{H}} d N_\mathrm{H}$ and $f^\mathrm{b}_{Z} \, dZ$. % We call $f^\mathrm{b}_{N_\mathrm{H}}$ and $f^\mathrm{b}_{Z}$ the "biased" distributions. % In the Appendix we derive the following relations between true and biased distributions [see Eqs. (\ref{fb_H}) and (\ref{fb_zeta})], % \begin{equation} f^\mathrm{b}_{N_\mathrm{H}} \simeq { \int_{0}^{\infty}%_{Z_\mathrm{min} }^{Z_\mathrm{max} } \mathcal{B}_{m_\ell} ( N_\mathrm{H}, Z) \, f_{Z} \, d\, Z \over \int_{0}^{\infty}%_{Z_\mathrm{min} }^{Z_\mathrm{max} } f_{Z} \, d\, Z } ~ f_{N_\mathrm{H}} ~~ \label{fb_H_paper} \end{equation} % and % %In a similar way, by integrating Eq. (\ref{HZdistr}) in column density, %it is easy to obtain a relation for the biased distribution %of metallicities, i.e. % \begin{equation} f^\mathrm{b}_{Z} \simeq { \int_{N_\mathrm{DLA}}^{\infty} \mathcal{B}_{m_\ell} ( N_\mathrm{H}, Z) \, f_{N_\mathrm{H}} \, d\, N_\mathrm{H} \over \int_{N_\mathrm{DLA}}^{\infty} f_{N_\mathrm{H} } \, d\, N_\mathrm{H} } ~ f_{Z} ~~. \label{fb_zeta_paper} \end{equation} %where $N_\mathrm{DLA}=10^{20.3}$ atoms cm$^{-2}$ %is the definition threshold of DLA systems. For simplicity we omit the dependence of the equations % of the functions on the integration limits on $z_\mathrm{min}$ and $z_\mathrm{max}$. We assume that the statistical distributions of interest smoothly vary inside the redshift interval, so that we can approximate them with their value at mean redshift $\overline{z}=(z_\mathrm{max}-z_\mathrm{min})/2$, % where % $(\overline{z}- {\Delta z\over 2}, \overline{z}+ {\Delta z\over 2})$ $\overline{z}$ is close to the peak of the observed distribution. The "bias function" $ \mathcal{B}_{m_\ell} (N_\mathrm{H}, Z) \equiv B_{m_\ell} (N_\mathrm{H}, Z, \overline{z}) %\label{BigCalB_paper} $ that transforms the true distributions into the biased ones is given by the relation [see Eq. (\ref{BigBHZz})] \begin{equation}%\left[ B_{m_\ell} ( N_\mathrm{H}, Z,z) \equiv { \int_\circ^{m_{\ell}-A_\lambda(N_\mathrm{H},Z, z)} \, % n(m; z) \, d m \, % \over \int_\circ^{m_\ell} \, n(m; z) \, dm \, , \label{BigBHZz_paper} } \end{equation} where $A_\lambda(N_\mathrm{H},Z, z)$ is the DLA extinction and $n(m; z)$ the quasar magnitude distribution. In the next section we show how to estimate $n(m; z)$, while in Section 4.3 we present a procedure for deriving the true distributions from Eqs. (\ref{fb_H_paper}) and (\ref{fb_zeta_paper}), starting from the observed distributions. % In deriving these equations %Eqs. (\ref{fb_H_paper}) and (\ref{fb_zeta_paper}) we made the assumption that the distributions of $N_\mathrm{H}$ and $Z$ are statistically independent. We discuss this assumption in Section 4.4. % revision 3 next sentence eliminated % In the Appendix we test the consistency of the above mathematical % formulation with the equations derived by FP93 for the ideal case % of a constant dust-to-gas ratio and power-law quasar luminosity function % (Section A.1.2). % \subsubsection{ The quasar magnitude distribution } We call $n(m; z) \, dm$ the number of quasars with apparent magnitude $m \in (m,m+dm)$ observable beyond redshift $z$ all over the sky. This number is defined in absence of quasar obscuration and we call $n(m; z) \, dm$ the "true" distribution. % %To use Eqs. (\ref{fb_H}) and (\ref{fb_zeta}) we need to estimate %the unbiased frequency distribution of quasar apparent magnitudes %$n(m; \overline{z})$ %which enters in the definition of the term %$\mathcal{B}_{m_\ell} ( N_\mathrm{H}, Z,)$. %%[see Eq. (\ref{BigBHZz})]. % Our goal is to infer the true distribution from the observed one, % REVISION 3 which must equal the distribution biased by the obscuration effect, $n^\mathrm{b}(m; z)$. % The relation between true and biased distribution is discussed in the Appendix (Section A.2), where we derive the equations Eq. (\ref{nb_m}), i.e. \begin{equation} n^\mathrm{b}(m; z) \simeq \left[ 1-\overline{\mathcal{F}}_1 \right] \, n(m; z) + \, \overline{\mathcal{F}}_1 \, n^\mathrm{b}_1(m;z) ~, \label{nb_m_paper} \end{equation} and Eq. (\ref{p_m}), i.e. \begin{equation} n^\mathrm{b}_1(m; z) = K \int_{-\infty}^{\infty} n(m^\prime; z) ~ f_{A_\lambda}\!(m-m^\prime; \overline{z}_e) ~ dm^\prime ~. \label{p_m_paper} \end{equation} % In these relations $\overline{\mathcal{F}}_1$ is the fraction of quasars with one foreground DLA, $n^\mathrm{b}_1(m; z) $ the magnitude distribution of quasars with one foreground DLA, $K$ is a normalization factor and $f_{A_\lambda}(A_\lambda; \overline{z}_e)$ %\, dA_\lambda$ the frequency distribution of the extinctions % $A_\lambda = m-m^\prime$ of the DLAs in the redshift range $z \in (0,\overline{z}_e)$ ($\overline{z}_e$ is the mean redshift of the quasars in the survey). % In Section 4.3, we show how to use Eqs. (\ref{nb_m_paper}) and (\ref{p_m_paper}) to infer $n(m; z)$ (see also Appendix A.2). %% %To solve these equations we estimate $\overline{\mathcal{F}}_1$, %$f_{A_\lambda}(A_\lambda; \overline{z}_e)$ %and $K$ as we explain below. %%%% END REVISION 3 \subsection{Obscuration fraction} % We call {\em obscuration fraction} the fraction of DLA systems % of a survey with limiting magnitude $m_\ell$ % lost as a consequence of the extinction bias. % carried on . We define the obscuration fraction using the mathematical relations between the true and biased distributions. We call {\em total obscuration fraction} the quantity \begin{equation} \Phi_{m_\ell} = 1 - { \int^\infty_{N_\mathrm{DLA}} f^\mathrm{b}_{N_\mathrm{H}} \, d N_\mathrm{H} \over \int^\infty_{N_\mathrm{DLA}} f_{N_\mathrm{H}} \, d N_\mathrm{H} } %= 1 - %{ %\int f^\mathrm{b}_{Z} \, d\, Z %\over %\int f_{Z} \, d\, Z %} %= 1 - %{ %\int \mathcal{B}_\ell (N_\mathrm{Zn}) \, f_{N_\mathrm{Zn}} \, d N_\mathrm{Zn} %\over %\int f_{N_\mathrm{Zn}} \, d N_\mathrm{Zn} %} ~. \label{BigPhi} \end{equation} The total obscuration fraction represents the number ratio of DLAs missed due to quasar obscuration in a survey with limiting magnitude $m_\ell$. The dependence on the limiting magnitude enters in $f^\mathrm{b}_{N_\mathrm{H}} $ through Eqs. (\ref{BigBHZz}) and (\ref{fb_H}). The total obscuration fraction can also be derived using the true and biased distributions of metallicity. % % The {\em cumulative obscuration fraction}, i.e. % the total fraction of DLA systems that {\em cannot} be detected % owing to the obscuration effect can instead be derived % by comparing the areas subtended by an intrinsic distribution % and by the corresponding biased distribution. % For instance, %the total number of {\em detectable} systems %can be estimated from the integral $\int f^\mathrm{b}_{N_\mathrm{Zn}} \, d N_\mathrm{Zn}$, %while the total number of systems % from $\int f_{N_\mathrm{Zn}} \, d N_\mathrm{Zn}$. %In this case the total obscuration fraction is given by %\ %Similar equations to derive $\Phi_\ell$ %can be obtained by integrating the $N_\mathrm{H}$ and %$Z$ distributions. Obviously, the same value of %$\Phi_\ell$ must be obtained independently of the particular distribution function %chosen to perform the calculation. We call {\em obscuration fraction}, $\phi_{m_\ell}$, the fraction by number of systems obscured at a particular value of column density or metallicity. % of the distribution functions $f_x$ discussed above % ($f_{N_\mathrm{H}}$, $f_{Z}$ and $f_{N_\mathrm{Zn}}$). % To measure $\phi(x)$ we use % the "biased-over-unbiased" ratio of the distribution function estimated % at a particular value of $x$. For instance, the fraction of systems obscured at a specific value of $N_\mathrm{H}$ is \begin{equation} \phi_{m_\ell} (N_\mathrm{H}) = 1 - {f^\mathrm{b}_{N_\mathrm{H}} \over f_{N_\mathrm{H}}} % = % 1 - \mathcal{B}_\ell (N_\mathrm{Zn}) ~~. \label{phi} \end{equation} %The meaning of this expression is clear from Eq. (\ref{BigBias_Zn}). % The function $\mathcal{B}_\ell (N_\mathrm{Zn})$ represents %the fraction of DLA/quasar pairs that {\em can} be detected at a given value %of $N_\mathrm{Zn}$. Therefore, its complement to unit represents %the fraction of pairs that {\em cannot} be detected. %Because $\mathcal{B}_\ell (N_\mathrm{Zn})$ is a function of the limiting magnitude, %also $\phi_\ell (N_\mathrm{Zn})$ depends on $m_\ell$, as indicated by the index $\ell$. Similar expressions can be derived % from Eqs. (\ref{fb_H}) and (\ref{fb_met}) to define the obscuration fractions $ \phi_{m_\ell} (Z)$ and $ \phi_\ell (N^\mathrm{t}_\mathrm{Zn})$, i.e. the fraction of systems missed at a particular value of $Z$ and $N^\mathrm{t}_\mathrm{Zn}$. \subsection{ Solution of the equations } %The biased distribution functions %$f^\mathrm{b}_{N_\mathrm{H}}$ and $f^\mathrm{b}_{Z}$ %can be determined from observational surveys %{%within the same limits of validity of the approximation %(\ref{approximation}). %} % The true functions $f_{N_\mathrm{H}}$ and $f_{Z}$ are the unknown quantities in Eqs. (\ref{fb_H_paper}) and (\ref{fb_zeta_paper}). % The procedure that we follow to solve these equations consists in (i) adopting trial functions $f_{N_\mathrm{H}}$ and $f_{Z}$, (ii) estimating $n_m \equiv n(m; \overline{z})$, (iii) computing $f^\mathrm{b}_{N_\mathrm{H}}$ and $f^\mathrm{b}_{Z}$ from the equations %Eqs. (\ref{fb_H}), (\ref{fb_zeta}) and (\ref{nb_m}) and (iv) comparing these biased distributions with the observed ones. % These steps are repeated by changing the trial functions until the predicted biased functions match the observed ones. % %% START REVISION 3 next sentence eliminated %{%This procedure is correct within the limits of validity of the approximation %(\ref{approximation}). %} % END REVISION 3 % We briefly describe each step of this procedure. In the first step, we adopt analytical functions in parametrical form as the trial functions $f_{N_\mathrm{H}}$ and $f_{Z}$. %of the true %distributions % $f_{N_\mathrm{H}}$ and $f_{Z}$ % revision 3 next sentence eliminated % %The choice of the shape of the functions is driven %by basic physical arguments and by the shape of %the empirical distributions %in the range of $N_\mathrm{H}$ and $Z$ %where the extinction is negligible. % % (low column densities and low metallicities). % %We choose functions with the lowest number of parameters %which satisfy the above criteria. % Once we choose the shape, the problem of determining the true functions $f_{N_\mathrm{H}}$ and $f_{Z}$ is equivalent to that of determining the values of their parameters. %In order to generate trial functions $f_{N_\mathrm{H}}$ and $f_{Z}$ %we therefore generate a set of trial values of their parameters. % %%% START REVISION 2 %{In the second step we use the trial functions $f_{N_\mathrm{H}}$ and $f_{Z}$ to derive the distribution of extinctions %$f_{A_\lambda}(A_\lambda; \overline{z}_e)$ from Eq. (\ref{AlamHzeta}). %All these distributions are defined %to include the DLAs with redshift %$z \in (z_\mathrm{min}, z_\mathrm{max})$. We use this extinction distribution % as an approximation of $f_{A_\lambda}(A_\lambda; \overline{z}_e)$ in Eq. (\ref{p_m_paper}). We then iterate Eqs. (\ref{nb_m_paper}) and (\ref{p_m_paper}), starting with a trial function\footnote{ The observed distribution $n^\mathrm{b}(m; z)$ can be used as the starting trial function of $n(m; z)$.} % $n(m; z)$, until we find a $n(m;z)$ that yields a perfect agreement between $n^\mathrm{b}(m; z)$ and the distribution $n^\mathrm{obs}(m; z)$ measured from the surveys. % revision 3 next part eliminated %and make a preliminary estimate of %$n^\mathrm{b}_1(m; z)$ from Eq. (\ref{p_m_paper}). %We insert this %$n^\mathrm{b}_1(m; z)$ % in Eq. (\ref{nb_m_paper}), together with %the estimates of $\overline{\mathcal{F}}_1$ and %$n^\mathrm{b}(m; z)$. %This last distribution is estimated from %In this way we obtain from Eq. (\ref{nb_m_paper}) an improved guess of $n(m; z)$. %% %Proceeding by iterations we obtain a function $n(m; z)$ %such that the calculated distribution $n^\mathrm{b}(m; z)$ %can match $n^\mathrm{obs}(m; z)$ with any degree of accuracy desired. % end revision 3 % %} %%%%% END REVISION 2 %With this choice of $n_m$ In the third step we use $n(m; \overline{z})$ to derive $\mathcal{B}_{m_\ell} ( N_\mathrm{H}, Z)$ from Eq. (\ref{BigBHZz_paper}) % and (\ref{BigCalB_paper}) and, finally, the biased distributions %$f^\mathrm{b}_{N_\mathrm{Zn}}$, $f^\mathrm{b}_{N_\mathrm{H}}$ and $f^\mathrm{b}_{Z}$ from Eqs. (\ref{fb_H_paper}) and (\ref{fb_zeta_paper}). %{In the last step we compute the $\chi^2$ deviation between the predicted distributions $f^\mathrm{b}_{N_\mathrm{H}}$ and $f^\mathrm{b}_{Z}$ and the corresponding observed distributions. %} By iterating the steps we determine the parameters of $f_{N_\mathrm{H}}$ and $f_{Z}$ which allow $f^\mathrm{b}_{N_\mathrm{H}}$ and $f^\mathrm{b}_{Z}$ to best fit the empirical distributions. The search is done via $\chi^2$ minimization. Since we are not guaranteed a priori of the unicity of the solution, we carefully investigate the variation of $\chi^2$ in the parameter space to make sure the minimum is unique. % \subsection{Are $N_\mathrm{H}$ and $Z$ statistically independent ?} The mathematical formulation presented above %derivation of Eqs. \ref{convolution}, \ref{fb_H}, and \ref{fb_met} relies on the assumption that $N_\mathrm{H}$ and $Z$ are statistically independent of each other in the true population of DLAs. % The local ISM offers an example in support of this independence, since a broad range of \ion{H}{i} column densities, including the high values typical of DLAs, are found at constant metallicity, $Z \simeq Z_{\sun}$. Similarly, a DLA system of given metallicity $Z \simeq Z_\circ$ will display a distribution of \ion{H}{i} column densities, if sampled along random lines of sight. This distribution will reflect the mass spectrum of interstellar clouds, which is likely to be determined by some physical mechanism, such as interstellar turbulence (see discussion in Khersonsky \& Turnshek 1996 and refs. therein), rather than by the metallicity. In this sense, it is reasonable to assume that $f_{N_\mathrm{H}}$ and $f_{Z}$ are independent distributions. Independent support of these arguments comes from the hydrodynamic simulations of DLAs evolving in $\Lambda$CDM cosmology (Cen et al. 2003), which indicate that metallicity and hydrogen column density are almost completely uncorrelated. %{Similar results are found from the cosmological SPH simulations performed by Nagamine et al. (2004). %} Notwithstanding, a trend is expected because the metallicity correlates with the star formation rate, which in turn increases with the gas density in the host galaxy. Therefore, lines of sight with high $N_\mathrm{H}$ may have a tendency to be associated with higher metallicities. A weak evidence for this trend was found by Cen et al. (2003), but the effect is particularly small at redshift $z \simeq 2-3$. % % If the metallicity is correlated with $N_\mathrm{H}$, the effect of the obscuration is more important than in the ideal case considered above. In this case the application of our mathematical formulation gives a conservative estimate of the bias. % \subsection{Gravitational magnification} Competing with the obscuration effect, is the possible gravitational magnification of the background quasar by the absorber itself. Many studies over the years have attempted to estimate the magnitude of this phenomenon for both DLAs (Smette et al. 1997, Le Brun et al. 2000) and metal absorption lines (Vanden Berk et al. 1996, M\'enard \& P\'eroux 2003). The results show that for the DLAs selected at optical wavelengths, gravitational effects are small. % %In principle, the gravitational lensing could be formally %included in our equations by adding a magnification term %in the bias function (\ref{BiasFun}). %In practice, this would require introducing a number %of extra parameters not constrained by the empirical %distributions considered here. % In the present application of our method, based on DLAs selected at optical wavelength, we therefore neglect the magnification effect. % revision 3 next sentence elminiated %We note that the magnification {\em due to other absorbers}, if present, %is automatically accounted for by the method. %In fact such an effect would change the observed distribution %of apparent magnitudes, which is the input required by our method %to derive the magnitude distribution not affected by extinction, $n_m$. % \section{Implementation of the procedure} We describe the statistical distribution functions adopted here as a first example of implementation of the procedure. % For the intrinsic distributions we only need to adopt a functional dependence (i.e. a shape), leaving to the procedure the task of determining the parameter values. \subsection{ The empirical distributions} We consider the sub-sample of DLA systems with available measurements of \ion{Zn}{ii} column densities, for which it is possible to derive metallicities corrected for depletion effects (Vladilo 2004). %The observed frequency distribution of %metallicities was estimated from the zinc sample, %taking into account dust depletion effects (Vladilo 2004). %where the effects of dust depletions are taken into account. The differences between the corrected metallicities and the [Zn/H] values taken at face value are small, with a mean value of $+0.12$ dex. %The total redshift interval considered corresponds to an interval %of cosmic time of xxx Gyr, equivalent to about ...\% of the age %of the universe. ... % Given the size of the zinc sub-sample ($\sim 40$ systems; see list of references in Table 1 of Vladilo 2004) we can estimate the frequency distribution of metallicities with a Poisson statistical error $\la 10\%$ in only a few bins. %of the empirical frequency distributions. To make the best use of the available data we selected the systems in the redshift range $1.8 \leq z \leq 3$, where most of the $N_\mathrm{Zn}$ measurements have been obtained so far. The mean redshift of the sample (28 systems) is $z \simeq 2.3$. % corresponding %to an age of the Universe of $\simeq 2.9$ Gyrs %for current cosmological parameters. % The redshift interval corresponds %to a cosmic time interval of $\simeq$ 1.5 Gyr, starting at about %$\simeq 2.2$ Gyr after the Big Bang. % (look-back time between $\simeq 10.0$ and %$\simeq 11.5$ Gyrs before present) %By excluding systems at $ z > 3$ we should be safe from % dramatic changes in the metallicity distribution expected %at very high redshift, where we get closer to the onset %of metal production in the first protogalaxies. % % To derive the empirical distribution of $N_\mathrm{H}$ we used the surveys by Storrie-Lombardi \& Wolfe (2000) and P\'eroux et al. (2003), which have more statistics than the zinc sub-sample. Selecting the DLAs in the range $1.8 \leq z \leq 3$ we obtained a sample of $59$ systems. Then $\log N(\ion{H}{i})$ and [Zn/H]=$\log (Z/Z_{\sun})$, we transformed the derived distributions in linear space, $N(\ion{H}{i})$ and $Z$, for later comparison with the distributions $f_\mathrm{H}$ and $f_Z$. % In computing the biased distributions with our procedure, we took into account the fact that the typical limiting magnitude of the $N_\mathrm{H}$ surveys ($m_\ell \sim 19.5$) is somewhat larger than that of the metallicity surveys ($m_\ell \sim 19$), owing to the different requirements in spectral resolution. In fact, Eqs. (\ref{fb_H}) and (\ref{fb_zeta}) can be solved independently of each other, inserting in each case the appropriate value of $m_\ell$. \subsubsection{The frequency distribution of quasar magnitudes} To derive the empirical distribution of quasar magnitudes we used the data of the Sloan Digital Sky Survey (SDSS) (Schneider et al. 2003). The distribution was determined for two photometric bands well representative of the visual part of the spectrum, namely the $g^\prime$ and $r^\prime$ bands, with effective wavelengths $\lambda=0.48 \mu$m and $0.62 \mu$m, respectively (Fukugita et al. 1996). In each band we binned the data at steps of 0.5 mag, counting all the quasars located beyond the typical absorption redshift of our sample (in practice, we adopted $ 2 \leq z_e \leq 5 $). % % revision 3 An example of empirical distribution computed for the $g^\prime$ band is shown in Fig. \ref{plotQSOmag} (diamonds with error bars). % After binning the data, we performed a polynomial fit to the observed $\log n$ - $\log m$ distribution, up to the magnitude $21.5$, for which the statistics and the completeness of the SDSS sample are still good. % % revision 3 This fit was adopted as a smooth model of the observed distribution in the magnitude range $ 17 \leq m \leq 21.5$. An example of fit is shown in Fig. \ref{plotQSOmag} (solid line). % The limit $m \sim 21.5 $ is sufficient for studying the effects of quasar obscuration on the statistics on DLAs since the typical limiting magnitude of DLA surveys is currently $m_\ell \lsim 21$. \begin{figure} \centering \includegraphics[width=7.cm,angle=0]{1570f5.eps} \caption{ %%% REVISION 3 Diamonds: distribution of the SDSS $g^\prime$ apparent magnitudes for quasars in the interval $2 \leq z_e \leq 5$. Smooth lines: example of a pair of model distributions in which the "true" model (dashed line) is determined from the biased model that fits the observations (solid line), % by means Eqs. (\ref{nb_m_paper}) and (\ref{p_m_paper}), once the distribution of DLA extinctions and $\overline{\mathcal{F}}_1$ are specified (see Section 4.1.1). } \label{plotQSOmag} \end{figure} \begin{center} \begin{table*} \caption{ Results. } \begin{tabular}{cccccccccc} \hline \hline Ext.curve & Band & $\beta$ & $\alpha$ & $\log ( {Z_\ast \over Z_{\sun} }) $ & $\Phi_{22} (m_\ell) ^a $ & $\Phi_{23} (m_\ell) ^b $ & $\log ( {<Z> \over Z_{\sun}}) $& $\Omega_{22}^a$ & $ \Omega_{23}^b$ \\ & & & & & $19.0 / 19.5$ & $19.0 / 19.5$ & & $(10^{-3})$ & $(10^{-3})$ \\ \hline MW & $r^\prime$ & 1.56 & $-0.44$ & $-0.13$ & $0.41 / 0.34$ & $0.44 / 0.38$ & $-0.37$ & $2.3$ & $~7.1$ \\ MW & $g^\prime$ & 1.51 & $-0.40$ & $-0.10$ & $0.48 / 0.42$ & $0.51 / 0.46$ & $-0.31$ & $2.8$ & $~9.5$\\ SMC & $r^\prime$ & 1.61 & $-0.46$ & $-0.19$ & $0.33 / 0.27$ & $0.37 / 0.30$ & $-0.44$ & $1.9$ & $~5.5$ \\ SMC & $g^\prime$ & 1.48 & $-0.38$ & $-0.07$ & $0.52 / 0.46$ & $0.56 / 0.51$ & $-0.27$ & $3.1$ & $11.3$\\ \hline \end{tabular} \\ $^a$ Upper limit of integration in Eq. (\ref{BigPhi}) or (\ref{Omega}) truncated at $\log N_\ion{H}{i}=22$. \\ $^b$ Upper limit of integration in Eq. (\ref{BigPhi}) or (\ref{Omega}) truncated at $\log N_\ion{H}{i}=23$. \\ \end{table*} \end{center} \subsection{ The shape of the true $N_\mathrm{H}$ distribution } One way to approximate the true shape of $f_{N_\mathrm{H}}$ is to assume a geometrical distribution of the gas in DLA galaxies. % For planar disks with radial exponential profiles $N_{\bot }(r) = N_{\bot 0} \exp (-\alpha \, r)$, the expected frequency distribution is $f_{N_\mathrm{H}} \propto N_\mathrm{H}^{-1} ~ \ln N_\mathrm{H}$ for $N_\mathrm{H}\ll N_{\bot 0}$ and $f_{N_\mathrm{H}} \propto N_\mathrm{H}^{-3} $ for $N_\mathrm{H}\gg N_{\bot 0}$ (FP93). For central column densities $N_{\bot 0} \ga 3 \times 10^{21}$ cm$^{-2}$ or much larger, as inferred by FP93, this gives $f_{N_\mathrm{H}} \propto N_\mathrm{H}^{-1} ~ \ln N_\mathrm{H}$ for most of the column-density range typical of DLAs. A realistic approximation of $f_{N_\mathrm{H}}$ should also take into account the fact that, in a given galaxy, the interstellar clouds show a spectrum of masses, $m_c$, and sizes, $r_c$, which is not accounted for by a smooth radial profile. % A large number of studies indicate that the mass spectrum of Milky-Way interstellar clouds follows a power law $f(m_c) \sim m_c^\gamma$, with $\gamma = -1.5 \pm 0.2$ over a 6 orders of magnitude range of masses (Scalo \& Lazarian 1996 and refs. therein). % From this mass spectrum, together with a relation between the internal density of the cloud, $n$, and the cloud size we can infer a column-density distribution. Adopting $n \sim r_c^p$ with $-1.2 < p \leq 0$ (Scalo \& Lazarian 1996 and refs. therein) we find\footnote{ With the adopted relation we find $m_c \sim N_\mathrm{H}^{p+3 \over p+1}$ and then derive the lower limit of $\beta$ assuming $p=0$. } that the column-density distribution should follow a power law $f_{N_\mathrm{H}} \sim N_\mathrm{H}^{-\beta}$ with $\beta \geq 2.5$. If this column-density spectrum applies to DLAs, we would expect a decline of column densities faster than that predicted by the exponential profile model. %% revision 3 next sentence eliminated %Modelling the real distribution $f_{N_\mathrm{H}} $ in DLAs %while taking into account both a smooth radial profile and %a column-density spectrum for different types %of galaxies, %would yield %a complex analytical expression with a large number %of unconstrained parameters. %On the other hand, %% end revision 3 % Both the exponential profile model and the cloud mass-spectrum hypothesis suggest that the true distribution may be approximated with a power law. % at least over the range of column densities %typical of DLAs, which span less than two orders of magnitude. The hydrodynamic computations of DLAs by Cen et al. (2003) yield a simulated true distribution of $N_\mathrm{H}$ consistent with this conclusion. % %% %Independent support to this conclusion comes from the fact that % the frequency distribution of \ion{H}{i} column densities %%$N_\mathrm{H}$ %in quasar absorbers follows a power-law %%\begin{equation} %%$f_{N_\ion{H}{i}} \propto N_\ion{H}{i}^{-\beta} $ %%%\label{f_H} %%%\end{equation} %%%with $\beta \simeq -1.46$ % over %more than 9 orders of magnitude in column density %$N_\mathrm{H}$, %%from the Ly\,$\alpha$ forest % , with $\beta \simeq 1.1 [?]$, %%to % Lyman Limit systems and %%DLA systems %(Tytler 1987; Storrie-Lombardi \& Wolfe 2000). %The typical value of the spectral index %is $\beta \simeq 1.5$, %even if optically thick absorbers are better represented by %$\beta \simeq 1.3$ (Petitjean et al. 1993). %%For the %%A simultaneous fit with a single power law over the whole range %%yields $\beta \simeq 1.5$, but the data are better represented by %%two power laws with a break %%at $N_\ion{H}{i} \simeq 10^{16}$ atoms cm$^{-2}$ %%and %$\beta \simeq 1.83$ and %%in the high column-density regime %Since these values of $\beta$ apply also to the column density regime %inmediately below the DLA threshold, where the extinction %is negligible, %this empirical evidence suggests that a power law can be %a good approximation of the true distribution. %%$f_{N_\mathrm{H}}$ in DLAs. %% % %However, detailed studies of the very high %column-density interval indicate that the %frequency of DLAs decreases faster than a power law % with $\beta \simeq 1.32$ %and is matched by a function %% %The parameter $N_{\star}$ represents the turning point of %the $N_\mathrm{H}$ distribution. % %with %$f_\star=0.03$, %$\log N_{\mathrm{H},\star} = 21.39$, and $\beta=1.52$ %Here we adopt the working hypothesis that %the true distribution of DLAs follows a simple power law, %as observed in quasar absorbers with negligible extinction, %and that the fast decline %is due to the onset of the quasar extinction when %$N_\mathrm{H}$ is sufficiently high. %%The hypothesis that the true %%distribution is a power law is %%consistent with the observations of quasar absorbers of lower %%$N_\ion{H}{i}$, for which the extinction is negligibile. %In the application of our procedure we do not make %any assumption on the value of $\beta$, which is instead %determined by the procedure itself. %%By adopting the power law (\ref{f_H}) %%we only need to determine the parameter $\beta$ because %%the normalization factor is not required in the solution of our %%problem. In light of these considerations, we adopt the simple law $f_{N_\mathrm{H}} \sim N_\mathrm{H}^{-\beta}$ for the true distribution, with the parameter $\beta$ to be determined by the method itself. % If this is a good approximation, % of $f_{N_\mathrm{H}}$, %we should find %$\beta \approx 1.3/1.5$ for consistency for the value typical %of quasar absorbers with negligible extinction. %In addition, the resulting biased distribution $f^\mathrm{b}_{N_\mathrm{H}}$ % obtained starting from a simple power law %(\ref{f_H}) % in Eq. (\ref{fb_H}) must reproduce the observed fast decline of the number of DLAs with column density, which %, at variance with low-$N_\mathrm{H}$ absorbers, is commonly fitted by a Schechter-type function \begin{equation} f_{N_\mathrm{H}} \propto % = {\left( f_\star \over N_{\mathrm{H},\star} \right) } {\left( N_\mathrm{H} \over N_\ast \right)}^{-\beta} \, e^{-\left( N_\mathrm{H} \over N_{\ast} \right)} ,~ \label{schechter} \end{equation} % characterized by a fast decline at $N_\mathrm{H} > N_{\ast}$ (Pei \& Fall 1995; Storrie-Lombardi \& Wolfe 2000; P\'eroux et al. 2003). \subsection{ The shape of the true metallicity distribution } %\subsubsection{The metallicity distribution function} The arguments leading to the choice of $f_{Z}$ can be summarized as follows. The observed distribution of [Zn/H] measurements peaks at [Zn/H] $\simeq -1.1$ dex and does not show systems below $\approx -2$ dex or above solar metallicity (Pettini et al. 1997). Although this distribution is likely biased at both ends, we do expect a genuine %The absence of systems at very low metallicity %is probably due to the $N_\mathrm{Zn}$ %observational cutoff below the detection limit of the \ion{Zn}{ii} resonant %lines. % most often used in this class of investigation. %On the other hand, the lack of systems at high metallicity %may result from the obscuration bias. %In spite of these observational biases, we expect to exist %an intrinsic decline in the number of DLAs below and above some critical values of metallicity. At [Zn/H] $\lsim -1.5$ dex the zinc sample shows a decrease of the number of DLAs in a region of the ($N_\mathrm{Zn}$, $N_\mathrm{H}$) plane not affected by any bias (the triangle DEF in Fig. \ref{sample}). At high metallicity it is reasonable to predict a decrease since we expect a natural decline of the number of systems with higher and higher star formation rates. % These arguments indicate that the true distribution of [Zn/H] starts from a negligible value at low metallicity, shows a rise around $\approx -1.5$ dex and declines after reaching a maximum. % % REVISION 3 We have been particularly careful in modelling this trend with an analytical function because the results, and in particular the mean metallicity, depend on the adopted functional form. % To choose the function we started by making an educated guess of the shape of the metallicity distribution based on our knowledge of the statistics of galaxies and DLAs. % We then made sure that the adopted function satisfies two requirements: % (1) that it is a "conservative" one, i.e. a function that does not overpredict the effects of the obscuration; %% %We also made sure %that (2) that the predicted extension of the function at high metallicities, where the obscuration is important, is not affected by the poor knowledge of the low end of the distribution. %does not affect our estimate of the high end, %where the obscuration is important. % %On the basis of these requirements, After considering several possibilities, % among which a lognormal distribution, we adopted the function % %For the same reasons, we prefer functional form with %a mild rise in the range of low metallicities. In this way we minimize the number %of systems lost due to observational bias other than %quasar obscuration. % % \begin{equation} f_Z ~ \propto ~ (Z/Z_\ast)^\alpha ~ e^{-Z/Z_\ast} ~, \label{MetalDistFunction} \end{equation} which satisfies very well the above requirements, as we explain below. This function is % similar to the one proposed by Schechter (1976) for describing the luminosity function of galaxies. To make an educated guess for the shape of the distribution %(first criterion) we started from the metallicity-luminosity relation in galaxies. % Evidence is building up that this relation is valid not only in the local Universe (e.g. Lamareille et al. 2004), but also in DLA systems (Ledoux et al. 2005). The empirical relation between the logarithmic metallicity and the absolute magnitude implies that the linear metallicity $Z$ increases linearly with the luminosity. The luminosity in turn follows a Schechter distribution both in the local Universe (e.g. Cuesta-Bolao \& Serna 2003), at redshift $z \simeq 0.1$ (e.g. Blanton et al. 2003) and, apparently, up to $z \simeq 3.5$ (Poli et al. 2003). Combining these different pieces of evidence, it is reasonable to assume that $Z$ may follow a Schechter distribution in DLA systems. % The requirement (1) of a "conservative" function is equivalent to make sure that decline at high metallicity is fast. In fact, the faster the decline of the "true" distribution, the lower the estimated number of missed DLAs. % and the mean metallicity. % Luckily, the Schechter satisfies this requirement since it provides a very fast decline at $Z \!\!>\!\! Z_\ast$. In fact, the decline is faster than in any polynomial in $Z$ and in a lognormal distribution. The requirement (2) implies that we should avoid functions for which a single parameter specifies the behaviour at both ends of the distribution. % %the shape at high metallicities, %where we expect the obscuration, %and the shape at low metallicities %are determined by the same parameter(s). %% %With the third criterion one can model the high-metallicity tail %of the distribution independently of the observed behaviour at %low metallicities. %%This criterion is important because %%at very low metallicities we have a poor statistics. %Thanks to this criterion, the statistical error due to the fit %at very low metallicities, where the statistics is poor, will not propagate at high metallicities, %where the obscuration can be important. The Schechter function satisfies this requirement since % for the Schechter function (\ref{MetalDistFunction}), the parameter $\alpha$, which in practice controls the extension of the distribution at low metallicities, has a negligible effect on the exponential decrease at $Z > Z_\star$. % %Therefore the poor statistics of the measurements at %low metallicity does not affect the %determination of $Z_\star$ and hence of $<\!\! Z \!\!>$. % For a lognormal distribution, instead, the width parameter % (and the extension %of the distribution at high metallicities) is affected by the fit of the distribution at the low-metallicity end, which is poorly constrained by the observations. The resulting error on the width parameter will affect the extension of the distribution at high metallicities and therefore the estimate of the obscuration effect. % The same considerations apply to any distributions that, like the lognormal one, are specified by a centroid and a width. % For these types of functions, the uncertain behaviour of the low end of the metallicity distribution affects the determination of the obscuration effect. % %When $\alpha \!\!>\!\! -1 $, %the Schechter function provides %the rise at low metallicities of the logarithmic distribution %$f_{(\log Z)}= (\log_e 10) ~ 10^{(\log Z)} ~ %f_Z (10^{(\log Z)}) $, % expected from the measurements of %[Zn/H] = $\log (Z/Z_{\sun})$. %% % end revision 3 % %%With the choice of the function (\ref{MetalDistFunction}) %%we need only two %The parameters $\alpha$ and $Z_\ast$ %%to specify the metallicity distribution. %%We do not make any assumption on the values %%of these parameters, which %are determined by %the application of the procedure. \section{Results and discussion} In Table 1 we summarize the results obtained from the application of our procedure. The quasar extinctions were estimated in the photometric bands $r^\prime$ and $g^\prime$ for an absorption redshift $z=2.3$. % The differences between the results obtained for the two %$g^\prime$ and $r^\prime$ bands give an estimate of the uncertainty % in the determination of the bias effect due to the lack of a homogeneous set of visual magnitudes % in a single visual band for all the quasars of the adopted surveys. % The obscuration fraction in the $r^\prime$ band is more conservative than that in the $g^\prime$ band, given the increase of the extinction with decreasing effective wavelength. The results for the $g^\prime$ band ($\lambda=0.44 \mu$m) are more comparable to those derived in previous studies of the obscuration bias, which considered the $B$ band ($\lambda=0.44 \mu$m; Ellison et al. 2001). Two types of dust extinction curves were considered, namely the MW curve by Cardelli et al. (1988) and the SMC curve by Gordon et al. (2003). The presence or absence of the 2175\AA\ extinction bump in DLAs is irrelevant for most of our results, because at $z=2.3$ the extinction curves in the $r^\prime$ and $g^\prime$ bands do not sample the bump (Fig. \ref{ext_z}). The only marginal exception is the combination of the $r^\prime$ band with the MW-type extinction. If the bump is completely absent in DLAs, the MW results become closer to the SMC results for the $r^\prime$ band. For each combination of extinction type and photometric band we derived the best-fit values of the parameters % $\beta$, $\alpha$ and $Z_\ast$ via $\chi^2$ minimization in the intervals $1.0 \leq \beta \leq 2.0$, $-1.0 \leq \alpha \leq 0.0$, and $-1.0 \leq \log (Z_\ast/Z_{\sun}) \leq +0.5$. % The best-fit parameters are listed in Table 1. % We made sure the minimum is unique by careful inspection of the variation of $\chi^2$ in the specified parameter space. The typical fit errors of $\beta$, $\alpha$ and $\log (Z_\ast/Z_{\sun})$ are $\simeq \pm 0.12$, $\sim \pm 0.25$ and $\simeq \pm 0.15$ dex, respectively. % revision 3 % {% The total error % on $\log (Z_\ast/Z_{\sun})$, including the uncertainties of the % empirical distributions, % is % %$\simeq \pm 0.2$, $\sim \pm 0.4$ and % $\simeq \pm 0.3$ dex. % % respectively. % } % In Fig. \ref{fits_mw} % and \ref{fits_smc} we show the true and biased distributions (dashed and solid lines) of the best-fit solution for the case of MW-type extinction and $g^\prime$ band. The empirical distributions (circles with error bars) are well reproduced by the computed biased distributions. Similar results are found for all the cases considered in Table 1. %In panel (c) we display the $f_{N_\mathrm{Zn}}$ and %$f^\mathrm{b}_{N_\mathrm{Zn}}$ distributions resulting from the application %of the procedure, without no attempt to match %the corresponding empirical $N_\mathrm{Zn}$ distribution. %The comparison between adopted and observed distributions %for the best-fit solution is shown in Figs. \ref{fits_mw} %and \ref{fits_smc} , %respectively. \begin{figure*} \centering \includegraphics[width=7.5cm,angle=0]{1570f6a.eps} \hspace{0.5cm} \includegraphics[width=7.5cm,angle=0]{1570f6b.eps} % \hspace{0.2cm} % \includegraphics[width=5.7cm,angle=0]{plotfitZn_mw.eps} \caption{ Frequency distributions of of \ion{H}{i} column densities (left panel) and metallicities (right panel) in DLAs. %Input parameters %for the intrinsic distributions % are indicated in the labels. % $\beta = 1.5$; [Zn/H]$_\circ = -0.7$ dex; $\sigma(\mathrm{[Zn/H]}) = 0.5$ dex. Dashed lines: true distributions. Solid lines: distributions biased by quasar obscuration. % computed for %the limiting magnitude $m_\ell$ of the corresponding surveys. Circles: empirical distributions obtained from DLA surveys. % {a)} $N_\mathrm{H}$ distribution; $m_\ell=19.5$. % {b)} metallicity distribution; $m_\ell=19.0$. % {c)} $N_\mathrm{Zn}$ distribution; $m_\ell=19.0$. By imposing that the biased distributions best fit the observed ones, we infer the true distributions (see Section 4.4). The solution shown in the figure was derived for DLAs with a MW-type extinction and quasars observed in the $g^\prime$ band. } \label{fits_mw}% \end{figure*} The effect of DLA extinction on the shape of distribution function of quasar apparent magnitudes %is generally modest, at least up $m \sim 21$ in the visual. % % after the \ion{H}{i} column density %and metallicity distributions are specified. %% can be seen in Fig. \ref{plotQSOmag}, where we show %{the adopted distribution $n_m \equiv n(m;z)$ (dashed line) and the predicted biased distribution $n^\mathrm{b}_m \equiv n^\mathrm{b}(m;z)$ (solid line) %} for the $g^\prime$ band and MW-type extinction. % The agreement between $n^\mathrm{b}_m$ and the empirical distribution (diamonds) demonstrates the capability of the procedure to account for the bias in a self-consistent way. %%%%%% START REVISION 2 %{The results are very stable for variations of $\pm 0.1$ of the parameter $\overline{\mathcal{F}}_1$ around the adopted value $\overline{\mathcal{F}}_1 = 0.5$. % Fig. \ref{plotQSOmag} shows that the obscuration effect does not change substantially the shape of the distribution of apparent magnitude of the quasars, at least within the current values of limiting magnitude of the surveys. % This is due to the fact that a large fraction of DLAs has relatively low \ion{H}{i} column density, as shown in Fig. \ref{fits_mw} and, as a consequence, low extinction. % The modest bias of the quasar statistics does not imply that the bias is unimportant for the DLAs statistics, as we discuss below. %} %%%%%% END REVISION 2 % % Fig. \ref{fits_mw}{a} is reminiscent of Fig. 8 by FP93 where these authors compare the model and empirical distributions of $N_\mathrm{H}$ and $k$. In fact, the dust-to-gas ratios $k$ is defined by FP93 as an extinction per unit \ion{H}{i} column density and is therefore almost equivalent to a metallicity, as one can see in our Eq. (\ref{AlamHzeta}). The fact that the metallicity distribution is much better constrained by the observations than the $k$ distribution is an advantage of our approach. In FP93 the model distribution of $N_\mathrm{H}$ is derived using an exponential \ion{H}{i} radial profile; in our work the model % the conversion from adopted to biased distributions of $N_\mathrm{H}$ and %metallicity $Z$ is done %taking into account self-consistently the adopted distributions. is meant to be the simplest analytical approximation of the true distribution in the range probed by the observations. % with the lowest number of free parameters. %%%%%%%%%%%%%%%%
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\begin{document}
%
   \title{  The dust obscuration bias in  Damped Lyman\,$\alpha$ systems
%   :\\ support from Milky Way interstellar studies
%   observed at high spectral resolution 
}
 
   \author{Giovanni Vladilo 
          \inst{1}
          \and
          C\'eline P\'eroux 
          \inst{2} 
          }

   \offprints{G. Vladilo}

   \institute{
              Osservatorio Astronomico di Trieste - Istituto Nazionale di Astrofisica, 
              Via Tiepolo 11, 34131 Trieste, Italy. \\
              \email{vladilo@ts.astro.it}
  \and
             European Southern Observatory, Karl-Schwarzschild-Str. 2,
	85748 Garching-bei-M\"unchen, Germany. \\
             \email{cperoux@eso.org}
                }

   \date{Received ...; accepted ...}
   
 \titlerunning{Dust obscuration bias in  Damped Ly\,$\alpha$ systems}
 \authorrunning{G. Vladilo and C. P\'eroux }


 \abstract{  
 We present a new study of the effects 
of quasar obscuration  on the
statistics of Damped Ly\,$\alpha$ (DLA)  systems. 
%
%
We show that the extinction
of   any Galactic or extragalactic \ion{H}{i} region,
 $A_\lambda$,    increases
 % REVISION 3
  linearly with the column density of zinc, 
% END REVISION 3
with a %characteristic 
turning point 
$\partial A_\lambda / \partial (\log N_\mathrm{Zn}) =1$, 
above which background sources are suddenly obscured.  
%%% [start revision2]
%{\bf 
%
We derive a relation 
$A_\lambda=A_\lambda(N_\mathrm{H},Z,z)$
between the extinction of a DLA system and its     
\ion{H}{i} column density, $N_\mathrm{H}$, metallicity, $Z$, 
fraction of iron in dust, $f_\mathrm{Fe}(Z)$, and redshift, $z$. 
%
From this relation we estimate the fraction of  DLA  systems
    missed as a consequence of their own   extinction
in magnitude-limited surveys. % of quasars/DLAs.  
%
% By ignoring the role of multiple DLA absorptions in a quasar line of sight,  
We derive a method for recovering the true
frequency distributions of $N_\mathrm{H}$ and $Z$ in DLAs, 
$f_{N_\mathrm{H}}$ and $f_Z$,  
using the biased distributions measured in the redshift range
where the observations have sufficient statistics ($1.8 \leq z \leq 3$). 
%  
% revision3 
By applying our method   we find that the
   well-known empirical thresholds  of   DLA  column densities,
$N_\mathrm{Zn} \la 10^{13.1}$ atoms cm$^{-2}$ and
$N_\ion{H}{i} \la 10^{22}$ atoms cm$^{-2}$, can be 
successfully explained in terms of the obscuration effect
without tuning of the local dust parameters. 
%% 
%(1)   
%by approximating the true \ion{H}{i} column density
%distribution with a power law $f_{N_\mathrm{H}} \sim N_\mathrm{H}^{-\beta}$ we derive
%$\beta \simeq 1.5$, 
%a slope similar to that of quasar absorbers of lower column density;
%%
%
The obscuration has a modest effect on the distribution of quasar apparent magnitudes,
but plays an important role in shaping the statistical distributions of DLAs.
%  in particular the metallicity distribution.  
%
The exact estimate of the bias  is still
limited by the  paucity of the data  
 ($\simeq$ 40  zinc measurements at $1.8 \leq z \leq 3$). 
%
We find that the fraction of DLAs missed  
as a consequence of   obscuration is $\sim$ 30\% to 50\%, 
   consistent  with the results of  
 surveys of  radio-selected quasars.
% 
By modelling the   
metallicity distribution with a Schechter function
we find that the mean metallicity  can be
$\sim 5$ to 6 times higher than the  value  
commonly reported for DLAs at $z \sim 2.3$.
%
 \keywords{ ISM: dust, extinction -- Galaxies: ISM -- Galaxies: high-redshift
 -- Quasars: absorption lines  }
  }
% 
 
\maketitle

%\end{document}   
   
%
%________________________________________________________________
 
\section{Introduction}

Quasar absorption line systems probe the diffuse gas in the Universe
over cosmological scales and at various stages of evolution. 
Among these absorbers, the Damped Ly$\alpha$ systems (hereafter DLAs) 
have the highest \ion{H}{i} column
densities ($N_\ion{H}{i} \geq 2 \times 10^{20}$ atoms cm$^{-2}$)
and  are believed to originate in \ion{H}{i} regions of high-redshift galaxies  
(Wolfe et al. 1986). 
Also, they have   higher metallicities\footnote{
We adopt
the usual definition [X/H] = log (X/H) - log (X/H)$_{\sun}$.}  than
any other class of quasar absorbers   ([M/H] $\approx -1.1$
dex; Pettini et al. 1994) and as such are expected to contain  dust.
%
%Dust plays an important role in many physical processes
%in galaxies and affects several types of astronomical observations. 
%In particular,   
If present, the  dust in DLAs  will absorb and scatter the radiation
of the background quasar, dimming   its apparent magnitude
("extinction" effect) and changing the slope of its spectral distribution
("reddening"). 
%
In magnitude limited surveys, the extinction may obscure the quasar,
leading to an observational selection effect known  as
obscuration bias (Ostriker \& Heisler, 1984; Fall \& Pei 1989, 1993). 

%Competing to this effect, is the possible gravitational magnification 
%of the background quasar by the absorber itself. 
%Many studies other the years have attempted to estimated
%the magnitude of this phenomenon for both DLAs (Smette et al. 1997, Le
%Brun et al. 2000, Murphy \& Liske 2004) and metal absorption lines (Vanden
%Berk 1996, M\'enard \& P\'eroux 2003). The results show that for the DLAs
%selected at optical wavelength, gravitational effects are small.

The bias can lead to misleading conclusions on the nature
of the galaxies associated with the DLA systems (herefater DLA galaxies),
because the \ion{H}{i} regions most enriched by metals and dust would
be systematically missed in the surveys.  
%
In addition,  some properties
of the high-redshift Universe that can only be derived from studies of DLAs,
would also be affected by the bias. 
One example is
the  comoving mass density of neutral gas,
$\Omega_\mathrm{DLA}$, and its evolution with redshift,
which is an indicator of gas consumption due to
 star formation (Wolfe et al. 1995). 
Another example is the mean cosmic metallicity
inferred from abundance studies of DLAs (Pettini et al. 1997),
which is related to  the star formation rate %in the high-redshift Universe
in the early Universe
(Pei \& Fall 1995, hereafter PF95; Pei et al. 1999).
 
Whether the obscuration effect is important, depends 
on the actual amount of dust in DLAs. 
%
The first evidence of dust, in the form of reddening,
was reported by Pei et al. (1991). 
The complex variability 
of the  quasar continuum  makes it impossible to determine the
reddening in individual cases.  However, 
from a statistical
comparison of quasars with and without foreground DLAs
one can search for a systematic change of the continuum slope.
In this way,   Pei et al. found a systematic difference,  
suggestive of reddening,
using a simplified power-law fit to the continuum distribution. 


%%
%This  can be estimated a posteriori from studies 
%of the systems discovered in quasar samples. 
%%  
%One approach consists in estimating the reddening 
%of the continuum of the quasar.  
%


The detection or reddening motivated detailed studies    
of quasar obscuration. 
%
Fall \& Pei (1993;  hereafter FP93)  presented an analytical method of computing
the obscuration   from the observed luminosity function
of quasars, the typical dust-to-gas ratio of DLAs and the empirical
distribution of \ion{H}{i} column densities. 
By modelling the radial distribution of the neutral gas in DLA galaxies, 
they concluded  that a large fraction of quasars may be obscured
in optically selected samples.   
%
%This bias would preclude the detection of DLAs with high dust content
%in magnitude-limited samples.  
%
The large uncertainty   of the reddening measurements prevented
  reaching firm conclusions on the dust-to-gas ratios and
on the magnitude of the bias. 

An alternative approach to measure the amount of the  dust  in DLAs   
consists in measuring the differential depletions between
refractory and volatile elements 
(Pettini et al. 1994;  Hou et al. 2001; Prochaska \& Wolfe 2002;
Ledoux et al. 2003; Vladilo 2002, 2004).
%Detailed studies of depletions can be used to
%infer which fraction  Vladilo 2002
%   
%    The differential
%depletion of these elements gives an indication of the amount of dust  
%There is now general agreement that   dust depletion exists  
%()
%and increases with metallicity (). 
%%The fraction of dust
%of iron has values typical of Milky-Way warm interstellar gas
%when the metallicity is $\ga 0.1$ the solar value, but becomes
%negligible at very low metallicities (Vladilo 2004). 
%
%%However, a recent study based on a larger and
%%more homogeneous sample of quasars 
%%%selected by the Sloan survey (Richards et al) 
%%do not confirm the detection of such an effect
%%(Murphy \& Liske, 2004). 
%
The  depletions can be converted into dust-to-gas ratios, yielding
an approximate estimate of  the quasar extinction  
%.   
%So far, estimates of this type 
%yielded somewhat different results, leading to
%different conclusions on the importance of the  obscuration bias 
(Vladilo et al. 2001b; Prochaska \& Wolfe 2002).   
This method, as well as the reddening measurements,
only   give information on the detected DLAs, leaving
open the possibility that more dusty \ion{H}{i} regions
remain undetected. 

%%
%, 
% 
%Fall \& Pei (1993) subsequently investigated the potential effects 
%of DLA extinction 
%using a model of the radial distribution of the   gas in the
%intervening galaxies. 
%%
%Comparing their predictions with the observed
%frequency distribution of \ion{H}{i} column densities, they    
%concluded that 
%%This obscuration effect
%%should lead to a selection bias against detection of DLAs with
%%high dust content in optically selected quasar samples. 


%Nevertheless, the  detection
%of reddening   by Pei et al. (1991)
%has not been confirmed so far (Murphy \& Liske, 2004). 
 
A more direct probe of the obscuration effect is a comparison
of DLA statistics in optically selected and radio selected quasar
samples, given that the latter are unaffected by dust extinction.
Ellison et al. (2001) compiled a homogeneous sample of radio selected
quasars and searched for DLAs towards every target, irrespective of
its optical magnitude (CORALS survey). They concluded that dust-induced bias in
previous magnitude limited surveys may have led to underestimating the
\ion{H}{i} mass density, $\Omega_\mathrm{DLA}$, and the number
density per unit redshift interval, $n(z)$, of DLAs by at most a
factor of two at $z \sim 2.3$. 
In addition, they found tentative evidence that $n(z)$ is greater in
fainter quasars, as expected by the obscuration effect.
% (but only at 1 $\sigma$ level). 
%   REVISION 3
In a follow-up  study of CORALS metallicities,
Akerman et al. (2005) find 
$ [ < \mathrm{Zn/H} > ] = -0.88 \pm 0.21 $. This value is  
   higher  than in a 
control sample, % in the same redshift interval, %($1.9 < z < 3.5$), 
as expected by the obscuration, but only at $1 \sigma$ level. 
%
In an extension of the CORALS survey at $0.6 < z < 1.7$
 Ellison et al. (2004)
find  $n(z) = 0.16 ^{+0.08}_{-0.06}$
using MgII absorbers as DLA candidates.
 This value is  consistent with magnitude-limited estimates
at the same $z$, but the $1 \sigma$ error permits a factor of 2.5
difference in the sense predicted by the obscuration. 
% END REVISION 3

The results of the CORALS survey are not yet conclusive since the
original sample of Ellison et al. had limited statistics, in particular at the
high values of $N(\ion{H}{i})$ where the effect is expected to be more
important. This weak evidence of obscuration, 
together with the non-detection of reddening in a large sample
of quasars with and without candidate DLAs, 
recently reported by Murphy \& Liske (2004), 
are conveying the impression that the obscuration effect
is not  important.

Yet, indirect evidence for the existence of the bias  comes  
from other studies.
%  abundance studies of known DLAs. Indeed, 
 The analysis of
[Zn/H] versus $\log N(\ion{H}{i})$ by Boiss\'e et al. (1998) showed
that none of the % dozen 
DLAs with zinc detection known at the time have
a metallicity above the threshold [Zn/H] $+\log N(\ion{H}{i}) \simeq
20.5$, corresponding to $N(\ion{Zn}{ii}) \simeq 10^{13.1}$ atoms
cm$^{-2}$. They proposed that this threshold could be attributed to
obscuration effect, assuming that metal column density tracks the dust
column density. 

While this assumption has not been proven by subsequent investigations,
the  threshold proposed by Boiss\'e et al.  
has been invoked
% by  different authors 
% revision 3 
to reconcile  several predictions of galactic evolution models 
(Prantzos \& Boissier 2000; Hou et al. 2001)
and cosmological simulations 
(Cen et al. 2003) with the observed properties of DLAs. 
% end revision 3

Another potential evidence of the bias is the lack of DLAs with \ion{H}{i} column densities $N(\ion{H}{i}) \ga 10^{22}$ atoms cm$^{-2}$. 
In this case, the obscuration   has been invoked to
explain  the significant fraction of model galaxy disks 
predicted to have higher \ion{H}{i} column densities  (Churches et al. 2004). 
%
As an alternative explanation,
% to explain the column density thresholds observed in DLAs,
it has been proposed that  hydrogen may undergo a sudden transition from 
the atomic
to the molecular form  above a critical
column density threshold  (Schaye 2001), in which
case the absorber would not be detected as a DLA system. 
 
%Competing to the obscuration effect, is the possible gravitational magnification 
%of the background quasar by the absorber itself. 
%Many studies other the years have attempted to estimated
%the magnitude of this phenomenon for both DLAs (Smette et al. 1997, Le
%Brun et al. 2000, Murphy \& Liske 2004) and metal absorption lines (Vanden
%Berk 1996, M\'enard \& P\'eroux 2003). The results show that for the DLAs
%selected at optical wavelength, gravitational effects are small.

In the present work  we address several open questions
concerning the existence and   importance of the obscuration bias. 
%%   The paper is structured as follows.   %%% TAKEN OUT IN REVISION
In Section 2, % in order to understand if the bias is present, 
we   perform a careful investigation of all the factors
that determine the extinction of a Galactic or extragalactic \ion{H}{i}
region. 
% revision 3
%We  explain how the
%extinction  gives rise to a barrier that prevents
%the detection of faint background sources.   
%
In     Section 3 we   report some indirect evidence 
of obscuration effects.
%  in the sample of DLAs
%with zinc measurements. 
%  
In Section 4 we present a  
%[start revision]
  mathematical formulation
%[end revision]
 aimed at  recovering  the frequency distributions
of  \ion{H}{i} column densities and metallicities of DLAs
 from the    
 %[start revision] 
observed  distributions affected by obscuration bias. 
%
The first implementation of this method is described
in Section 5 and the results are   discussed in Section 6. 
% In Section 7 we summarize the conclusions of the work. 

   
%
%A critical point of this procedure is the
%conversion between dust-to-gas ratios, derived from depletion
%measurements, and extinction. In fact we know that, even for a fixed
%depletion pattern, the extinction can vary widely by only changing the
%dust grain size distribution. As a consequence, it is not guaranteed
%that the extinction scales linearly with the fraction of metals in
%dust form. For this reason, and because this approach is limited to
%DLAs observed in abundance surveys, definitive conclusions on the
%obscuration bias cannot be derived from this type of work.
%

%Furthermore, Fall \& Pei (1993) speculated that extinction, if it
%exist, should lead to a selection bias against detection of DLAs with
%high dust content in optically selected quasar samples. Understanding
%whether dust obscuration is present or not has important implications
%concerning the metallicity distribution and its evolution in
%DLAs. This in turn is relevant both for studies of cosmic metal
%production (Fall ... ???)  and of galactic chemical evolution
%(Prantzos \& Boissier 2000). 
 
%
%At variance with previous works, the study presented here is motivated
%by observational evidences, based on Milky Way interstellar studies,
%which supports the suggestions from Boiss\'e et al. (1998). We start
%by reassessing the existence of a cutoff of \ion{Zn}{ii} column
%densities in an up-dated larger DLA samples (Section 2). We then show
%results concerning the Galactic interstellar extinction which is shown
%to increase with $N(\ion{Zn}{ii})$, with a sharp rise at the cutoff
%observed in DLAs (Section 3). In section 4, we develop a method
%allowing for the estimation of the obscuration of quasars by dusty
%DLAs and impletment this analysis to real data in Section 5. Section 6
%provides predictions of the effects of such obscuration on HI column
%density distribution of DLAs, the metallicity measurements and the
%estimates of the zinc column densities of these systems. All these
%results are summarized in Section 7.

%
 

%Among quasar absorption line systems,
%Damped Ly $\alpha$ systems (DLAs) are
%the most likely candidates to have sufficient dust content to
%cause extinction of the background quasar given the fact that they have
% the highest HI column density
%($N(\ion{H}{i})\geq 2 \times 10^{20}$ atoms cm$^{-2}$; Wolfe et al. 1986)
%and the highest metallicity\footnote{
%We adopt the usual definition
%[X/H] = log (X/H) - log (X/H)$_{\sun}$.}
%([M/H] $\approx -1.1$ dex; Pettini et al. 1994).
%Evidence for reddening  induced by dust in DLAs was provided
%by Pei et al. (1991), who found a systematic difference
%in the continuum slope of quasars with and without foreground
%DLAs. 
%The possibility such extinction     could 
% lead to a selection bias   against detection of
%DLAs with high dust content  was first considered by Fall \& Pei (1993). 
%%
%Indirect evidence for the existence of the dust obscuration bias was provided by
%Boiss\'e et al. (1998)
%from the analysis of the plot of [Zn/H] versus $\log N(\ion{H}{i})$.
%These authors did not found   
% DLAs above the threshold
%[Zn/H] $+\log N(\ion{H}{i}) \simeq 20.5$,
%corresponding to 
%$N(\ion{Zn}{ii}) \simeq 10^{13.1}$ atoms cm$^{-2}$,
% and suggested that this
%threshold could be attributed to obscuration effect,
%assuming  that metal column density tracks the dust column density.
%%
%Understanding whether dust obscuration is present or
%not has important implications concerning the metallicity
%distribution and its evolution in DLAs. This in turn
%is relevant both for studies of   cosmic  metal
%production (Fall ... ???)
%and of galactic chemical evolution
% (Prantzos \& Boissier 2000). 

%Different approaches have been used in order to
%assess the importance of the dust obscuration effect.  
%One approach consists in
%estimating   the quasar extinction due to DLAs by converting
%differential the dust depletions measured in the DLAs
%into dust-to-gas ratios.
%Optical extinctions  of 
%$\la 1$ mag (Vladilo et al. 2001)
%and $\la 0.4 $ mag (Prochaska \& Wolfe 2002)
%have been obtained in this way, leading to
%different conclusions concerning the importance of
%the dust obscuration effect. 
%A critical point of this procedure is the conversion between 
%dust-to-gas ratios, derived from depletion measurements,
%and extinction. In fact we know that, even for a fixed depletion pattern,
%the extinction can vary widely by only changing the dust grain
%size distribution. As a consequence, it is not guaranteed that the extinction scales
%linearly with the fraction of metals in dust form. For this reason,
%and because this approach is limited to DLAs
%observed in abundance surveys, 
% definitive conclusions on the  
%    obscuration bias  cannot be derived from  this
%type of work.
% 
%A powerful   approach to attack the problem is to
%compare the statistics of
%optically selected  and radio selected DLA surveys, 
% the latter  
%unaffected by dust extinction, to search for evidence of the
%dust obscuration bias. 
%Ellison et al. (2001) compiled a homogeneous sample
%of radio selected quasars and searched for DLAs towards every
%target, irrespective of its optical magnitude.
%They conclude that dust-induced bias in previous magnitude limited surveys
%may have lead to  underestimate the neutral gas mass density, $\Omega_\mathrm{DLA}$, and the number
%density per unit redshift interval, $n(z)$, of DLAs by at most a factor of two.
%In addition, they find that $n(z)$ is greater in fainter quasars,
%as expected by the dust obscuration effect (but only at 1 $\sigma$ level).
%The results by Ellison et al. (2001) are  
%not yet conclusive since their sample is still limited, in particular
%at the high values of $N(\ion{H}{i})$ where the effect is expected to be more important. 
% 

% 
%At variance with previous work, here
%  we consider empirical evidence  in support
%of   Boiss\'e et al's suggestion obtained from Milky Way
%interstellar studies.  
%%
%First we reassess the existence of a cutoff of \ion{Zn}{ii}
%column densities, not expected on the basis of 
%our measuring capabilities (Section 2). Then we show that 
%Galactic interstellar extinction increases with $N(\ion{Zn}{ii})$,
%with a sharp rise at the cutoff observed in DLAs (Section 3).   
%These results are discussed in Section 4 and summarized in Section 5.

%

%\end{document}

\section{ The relation between $N_\mathrm{Zn}$ and extinction   }

%We   investigate the relation between $N(\ion{Zn}{ii})$
%and  interstellar extinction to understand if there is a threshold for dust obscuration
%at $N(\ion{Zn}{ii}) \simeq 10^{13.1}$ atoms cm$^{-2}$, as originally
%suggested by Boiss\'e et al. (1998).  
%
We adopt  
the simplified notation $N_\mathrm{Zn} = N(\ion{Zn}{ii})$ and
$N_\mathrm{H} = N(\ion{H}{i})$, % and similar for other elements,
ignoring contributions from ionization states other than the
dominant one (see Vladilo et al. 2001a and references therein). 
 
We start by considering  a refractory and a volatile\footnote{
Refractory elements easily condense
into dust form, while volatile elements tend to remain in the gas phase.
% revision 3 next sentence eleiminated
%This different behaviour appears to be related to differences in the
%condensation temperature characteristic of each element  (see Field 1974).  
% end revision 3
}    
element, $\mathrm{E_r}$ and $\mathrm{E_v}$,  
with   interstellar abundance ratio by number 
  $(  \mathrm{E_r} /  \mathrm{E_v}   )$.  
The ratio of the total column densities (gas plus dust) 
of these two elements along an interstellar line of sight will be 
$
 N^\mathrm{tot}_\mathrm{r} / N^\mathrm{tot}_\mathrm{v}  = 
 ( \mathrm{E_r} /  \mathrm{E_v}  ).
$
We call $N^\mathrm{d}_\mathrm{E}$ 
and $N_\mathrm{E}$ the column densities of an element E in the dust  
and  in the gas, respectively.  The fraction of atoms of E in dust form will be
$f_{\mathrm{E}} 
\equiv N^\mathrm{d}_{\mathrm{E}}/N^\mathrm{tot}_{\mathrm{E}}$. 
%
From these definitions we have 
$ N^\mathrm{tot}_{\mathrm{r}} % = N(\mathrm{X}_r) + N_d(\mathrm{X}_r) 
= N^\mathrm{d}_\mathrm{r} / f_{\mathrm{r}}$ and 
$N^\mathrm{tot}_\mathrm{v} % = N(\mathrm{X}_v) + N_d(\mathrm{X}_v) 
= N_\mathrm{v} / (1-f_\mathrm{v})$.
%From (Eq. \ref{RoverV1}) 
Combining these relations we have 
\begin{equation}
N^\mathrm{d}_{\mathrm{r}} 
=  \, { f_\mathrm{r} \over (1-f_\mathrm{v})} \,
 \left( { {\mathrm{E_r} \over \mathrm{E_v} } } \right)  \,  N_\mathrm{v} 
 ~.
\label{RoverV}
\end{equation} 
%
 

We next consider the relation between  $N^\mathrm{d}_\mathrm{r}$ and
the column density of dust grains,
$N_\mathrm{gr}$, i.e. the 
number of grains per cm$^{2}$ along the line of sight. 
For spherical grains of radius $r_\mathrm{gr}$ (cm)
and internal mass density $\varrho_\mathrm{gr}$ (g cm$^{-3}$)
it is easy to show that 
\begin{equation}
N^\mathrm{d}_{\mathrm{r}} \, \mathrm{A_r} \, m_\mathrm{H} =
{4 \over 3 } \, \pi 
 ~ r_\mathrm{gr}^{3} ~ \varrho_\mathrm{gr}  ~   N_\mathrm{gr} ~ X^\mathrm{d}_\mathrm{r}
  ~,
\label{Rgrains}
\end{equation}
where    $\mathrm{A_r}$ is the atomic mass of the refractory element and
$X^\mathrm{d}_\mathrm{r}$ the mass fraction
in the dust of the same element. 
 
Finally, we consider the relation between  the interstellar extinction in magnitudes
 at the wavelength $\lambda$, $A_\lambda$,   
 and the column density of dust grains. 
For spherical grains  with extinction efficiency factor $Q_{e,\lambda}$ 
(see Spitzer 1978, Chapter 7) this relation is  
\begin{equation}
A_\lambda  = 1.086 ~ % N_d(\mathrm{grains}) \, \sigma_{gr} \, Q_e   ~,
  N_\mathrm{gr} \,\,  Q_{e,\lambda} \,\, \pi \, r_\mathrm{gr}^2 ~.
\label{AVgrains}
\end{equation} 
 
By combining Eqs. (\ref{RoverV}), (\ref{Rgrains}) and (\ref{AVgrains})
we derive 
\begin{equation}
A_\lambda  = a_\lambda ~  N_{\mathrm{v}} ~,
\label{BigA_Nv}
\end{equation}
where we define
\begin{equation}
a_\lambda =  1.817 \times 10^{-24} ~ \mathrm{A_r} ~ g_\lambda  ~
 { f_\mathrm{r} \over (1-f_\mathrm{v})}  \,
 \left( { {\mathrm{E_r} \over \mathrm{E_v} } } \right) 
   ~,
\label{SmallA_Nv}
\end{equation}
and we isolate in the term
\begin{equation}
g_\lambda = s ~
{ Q_{e,\lambda}  \over 
r_\mathrm{gr}  \, \varrho_\mathrm{gr} \,   
X^\mathrm{d}_\mathrm{r}  } ~
\label{glambda}
\end{equation}
all the   properties of the grains (shape, extinction efficiency,
size, density, and composition). For spherical grains $s = 3/4$. 



The factor 
${ f_\mathrm{r} / (1-f_\mathrm{v})}$
is approximately constant  by definition of volatile and refractory elements. 
 In a medium of constant composition, such as the local ISM, also the abundance $\left( { {\mathrm{E_r} / \mathrm{E_v} } } \right)$
is constant. 
Therefore, for a given type of dust,   the term $a_\lambda$ is constant  
and the expression (\ref{BigA_Nv}) implies   that   {\em the interstellar
extinction   increases linearly with the gas-phase column density of any volatile element
measured in the same line of sight}. 
%The dispersion in the relation will be dominated by the dispersion
%of $g_\lambda$, which characterizes the properties of the dust grains. 
%
%Eqs. (\ref{Rgrains}) and (\ref{AVgrains}) are based on the assumptions
%that all grains are identical. In the actual case, 

Differences in the shape, size, density, composition and extinction efficiency
of the grains will induce a scatter in the relation.  %However, 
%it is important to note that 
The dependence of $g_\lambda$ on the grain size  $r_\mathrm{gr}$ is linear,  
%in the final Eq. (\ref{Alambda_Nv})  
while in Eqs. (\ref{Rgrains}) and (\ref{AVgrains}) it was cubic and quadratic.
%This milder dependence % on $r_\mathrm{gr}$
%suggests that  the relation between extinction and
%column density of volatile elements may not undergo dramatic variations 
%even if the grain size distribution changes. 
%
Since the grain size distribution
is   a key factor in determining the wavelength dependence  of the 
interstellar extinction (see e.g. Draine 2003), the mild dependence of
$g_\lambda$ on $r_\mathrm{gr}$ suggests   that
{\em  the
relation between $A_\lambda$ and $N_\mathrm{v}$ may be similar
even in interstellar regions with different types of
extinction curves}. 




\begin{figure}
   \centering
 \includegraphics[width=7.5cm,angle=0]{1570f1.eps}
\caption{Extinction $A_V=R_V \,  E_{B-V}$ (mag)
versus \ion{Zn}{ii} column density (atoms cm$^{-2}$)
in Galactic interstellar lines of sight. Filled diamonds: 
HST data 
from  Roth \& Blades (1995). Empty diamonds: 
IUE data from Van Steenberg \& Shull (1988). 
%Error bars are not quoted in the original work. 
Curve: extinction law (\ref{AV_NZn}) with    
$ a_V = 0.29 \times 10^{-13}$ mag cm$^2$ derived empirically as
explained in the text. }
\label{figAV_NZn}
\end{figure}




By adopting zinc as the volatile element       and considering the   extinction in    
 the $V$ band ($\lambda_V=0.55 \mu$m), we obtain from (\ref{BigA_Nv})
  the relation
\begin{equation}
A_V  \simeq a_V %  \times
\,   N_\mathrm{Zn} ~,
\label{AV_NZn}
\end{equation}
where the parameter $a_V$ (mag cm$^2$)
% = a( Z_d(\mathrm{X}_r),
%\varrho_{d}, r_{d},
%  Q_e, ( {\mathrm{X}_r / \mathrm{X}_v } )_m)$
can be determined  by comparing measurements of $A_V$ and  $N_\mathrm{Zn}$  
in individual lines of sight.   
In this way it is possible to  measure the scatter of $a_V$  
due to variations of the grain properties. % in different interstellar regions.    
%% = a( r_{d}, \varrho_{d},   Z_d(\mathrm{X}_r),
%%  Q_{e,V}, ( {\mathrm{X}_r / \mathrm{X}_v } )_m)$
%in   lines of sight characterized by different types of dust.  

To estimate $a_V$ we gathered 
%$A_V$ and  $N_\mathrm{Zn}$
%data from the literature. %from Eq. (\ref{EBVlaw})
%%estimated by   comparing measurements of
%%  $E_{B-V}$ and $N_\mathrm{Zn}$. 
%%%
%%To do so 
%We  collected 
the $N_\mathrm{Zn}$ measurements 
obtained from    HST    (Roth \& Blades 1995) and  
    IUE (Van Steenberg \& Shull 1988) high-resolution spectra.  
The IUE data are more numerous, but
have lower  resolution and signal-to-noise ratio.  
%than the HST data. 
To reduce the impact of these differences we selected the IUE
measurements  with total error $< 0.3$ dex 
  in $\log N_\mathrm{Zn}$.
%(last column of Table 5 in Van Steenberg \& Shull 1988).  
% much more numerous but of lower quality than the HST data.  
% but provide us with a much larger sample.
To minimize the contamination 
by stellar absorptions on top of the IUE \ion{Zn}{ii} interstellar lines
(Van Steenberg \& Shull 1988), we selected
only IUE spectra of fast-rotating stars ($V \sin i \geq$ 200 km s$^{-1}$). 
%To combine data of comparable quality 
We then scaled the IUE  column densities to match the 
Bergeson \& Lawler (1993)
\ion{Zn}{ii} oscillator strengths adopted in the HST data 
and in the current research.  
As far as the extinctions are concerned, we used the   $A_V$  
values given by Guarinos (1991). 
As a result, we obtained a sample of 11 lines of sight.
The mean value of extinction per unit zinc column density that we derive
is $<\!\! a_V \!\!> \, = 0.27 (\pm 0.06) \, 10^{-13}$ mag cm$^2$ and
$0.38 (\pm 0.20) \, 10^{-13}$ mag cm$^2$ for the HST and IUE sub-samples
(4 and 7 lines of sight, respectively).  
%The two results are consistent with each other
%within their standard error of the mean
%($\pm 0.03$ and $\pm 0.07$ mag cm$^2$, respectively).  

In order to enlarge the sample we also considered the color excess
$E_{B-V}$,  which is more commonly  measured than $A_V$.
%from the comparison of reddened and unreddened stars
%of same spectral type.
%The standard color excess is usually measured by comparing the fluxes in
%the B and V photometric bands. 
The ratio of general-to-selective extinction,
$
R_V = { A_V / E_{B-V}}
$,
is  approximately constant,  with a mean value
$<\!\! R_V \!\!> \, \simeq 3.1$ in the diffuse ISM of the Milky Way
(see e.g. Draine 2003).
We used this property to derive a mean value 
$<\!\! a_V \!\!> \, = \,<\!\! R_V \!\!> \, <\!\! E_{B-V}/N_\mathrm{Zn} \!\!>$ 
from reddening and zinc column density measurements. 
% 
The $E_{B-V}$  data %for both the HST and IUE sub-samples,
were taken from the
extinction catalog of Savage et al. (1985).   
%
The resulting sample includes 24 lines of sight, shown in Fig. \ref{figAV_NZn}. 
The mean value 
%of $e_V=E_{B-V}/ N_\mathrm{Zn}$ 
of the HST sub-sample (9 lines of sight) is
$<\!\! a_V \!\!> \, = 3.1 \times 0.089 \, (\pm 0.030) \, 10^{-13}$ mag cm$^2$,
and of the IUE sub-sample (15 lines of sight),
%$<{a_{\circ} \over R_V }>=
$3.1 \times 0.123 \, (\pm 0.071) \, 10^{-13}$ mag cm$^2$.
%Again, the results from the two sub-samples are consistent 
%with each other. % within their standard error of the mean. 
%
We did not find significant correlations  between
$<\!\! E_{B-V}/N_\mathrm{Zn} \!\!>$ and the parameters of
UV extinction provided by   Savage et al. (1985),
such as  the strength of the emission bump
%(parameters $E_\mathrm{Bump}$ and $\Delta_\mathrm{Bump}$)
or the steepness of the UV rise. % (parameter $\Delta_{15-V}$).  
The lack of these correlations gives us a justification for averaging
the results obtained from   lines of sight with
  different types of dust. 
 
The results obtained from the $E_{B-V}$ analysis  
are identical to those obtained from the more limited
sample of direct measurements of $A_V$. 
The larger dispersion of the IUE
sub-sample is probably due to the lower quality of the data.
% REVISION 3 next sentence eliminated
%
%The slight offset might be due to saturation effects 
%hard to disentangle in the IUE data. The saturation
%would lead to underestimating the  zinc column densities 
%and, as a consequence, to overestimating $a_V$.  
%
% end revision 3
We adopted the weighted average of the
IUE and HST sub-samples,   which yields 
$<\!\! a_V \!\!> \, = 0.29 (\pm 0.07) \, 10^{-13}$ mag cm$^2$,
a value almost identical to that of the HST sub-sample. 
% REVISION 3 next sentence eliminiated
%The   error
%is obtained  from the propagation of the standard errors of the mean. 
% end revision 3
%
The   relation   (\ref{AV_NZn}) obtained from this estimate of  $a_V$
% $E_{B-V} = e_V \, N_\mathrm{Zn}$ 
is displayed in Fig. \ref{figAV_NZn} (solid curve). The exponential rise
of the extinction is due to the fact that the \ion{Zn}{ii} column density
is plotted on a logarithmic scale. 
%
In this scale the rise of the extinction is mild 
at low values of  $\log N_\mathrm{Zn}$, but very fast at high values. %  of  $\log  N(\ion{Zn}{ii})$.
The turning point %$y_{\circ}$ 
marking the transition between these two regimes
can be estimated from the condition        
$\partial A_V / \partial (\log N_\mathrm{Zn}) =1$.
This criterion yields  $\log N_\mathrm{Zn} 
= - \log (<\!\! a_V \!\!> \, \ln 10) = 13.18$. 
{\em Above this turning point 
 the extinction undergoes a dramatic rise, acting as a barrier against
 detection of faint background sources.  }
This barrier 
%may prevent the detection of faint
%quasars lying beyond DLA systems with $\log N_\mathrm{Zn}$ above the
%turning point, which 
is remarkably similar to the detection threshold
proposed by Boiss\'e et al. (1998). 
This motivated us to investigate 
the  relation between extinction and metal
column density   in DLA systems. 
%
 
 
\subsection{Extinction versus $N_\mathrm{Zn}$ in DLA systems }
 

The relations (\ref{BigA_Nv}),  (\ref{SmallA_Nv}) and (\ref{glambda})
are valid in the ISM of any galaxy, including   DLA systems,
which are \ion{H}{i} regions of high-redshift galaxies. 
%
Applying these relations  to a DLA system and to the local ISM,  and
adopting $\mathrm{E_v}$=Zn, 
$\mathrm{E_r}$=Fe and $\lambda=\lambda_V$, we obtain  
%
\begin{equation}
A_{V} = a_{V,i} \, { g_V \over g_{V,i} } ~  
{ (1-f_{\mathrm{Zn},i}) \over f_{\mathrm{Fe},i} } ~
{ f_{\mathrm{Fe}} \over   (1-f_{\mathrm{Zn}}) } ~
{ (\mathrm{Fe/Zn}) ~ \over (\mathrm{Fe/Zn})_i  } ~
 N_{\mathrm{Zn}}  ~, 
 \label{AVdla1}
\end{equation}
%
where the quantities in the local ISM are indicated with the index $i$
and those in the DLA system are not labelled. 
To take into account the dependence on metallicity, which varies among DLAs,
we introduce the  metallicity
$Z \equiv N^\mathrm{tot}_{\mathrm{Zn}}/N_\mathrm{H}$,  
where 
$N^\mathrm{tot}_{\mathrm{Zn}} = N_{\mathrm{Zn}} /(1-f_{\mathrm{Zn}})$
is the total zinc column density (gas plus dust). In this way we obtain the expression
%
\begin{equation}
A_{V} \simeq A_\circ ~
{ G } ~
 f_{\mathrm{Fe}}  ~
{ (\mathrm{Fe/Zn}) ~ \, \over (\mathrm{Fe/Zn})_{\sun} } ~
Z ~ N_{\mathrm{H}} ~,
%N^\mathrm{tot}_{\mathrm{Zn},j}  ~.
\label{AVdla2}
\end{equation}
%
where $A_\circ  \equiv
a_{V,i} \, (1-f_{\mathrm{Zn},i})/ f_{\mathrm{Fe},i}$ and
$G=g_V / g_{V,i}$. 
The constant $A_\circ $ is estimated  using
the value $<\!\! a_V \!\!>$ derived in the previous section 
and the mean values of dust fractions of the local ISM sample
($f_{\mathrm{Zn},i}=0.40$ and $f_{\mathrm{Fe},i}=0.94$). 
%
We obtain $A_\circ \simeq 1.85 \times 10^{-14}$ mag cm$^2$,
with a relative error of $\simeq \pm 0.15$ dex
including the uncertainties of the dust fractions.
% 
The DLA extinction  
will vary according to the Fe/Zn abundance in the medium,
the iron depletion and the properties of the grains.  
%%
We briefly discuss each one of these  factors.      

%The   dependence on the composition of the medium enters  
%through the term %s $Z$ and
%${ (\mathrm{Fe/Zn})  / (\mathrm{Fe/Zn})_{\sun}  }$.
%$ \left( { {\mathrm{X}_r \over \mathrm{Zn} } } \right)_{m} $, which represents the
%abundance ratio of the refractory element X$_r$ relative to zinc. 
%The metallicity level $Z$ can be estimated  quite well from
%the measured column densities of zinc and hydrogen since
%the dust fraction  of zinc is almost negligible in DLAs and so
%$Z = N_{\mathrm{Zn}}/N_\mathrm{H} 
%\simeq N^\mathrm{tot}_{\mathrm{Zn}}/N_\mathrm{H} $ 
%with a good approximation.  
Relative abundances of metals   do not
show strong variations in DLAs, 
despite significant changes of the absolute metallicity  $Z$. 
The Fe/Zn ratio  
%, one of the most abundant refractory elements,  
is  very close to solar in metal-poor stars
with  metallicities typical of DLA systems
(Mishenina et al. 2002; Gratton et al. 2003; Nissen et al. 2004;
see discussion in Vladilo 2004).  
Deviations from the solar ratio are measured in stars with 
extremely low metallicities. Such deviations  do not affect
the present work in any case, because 
the dust fraction and the extinction vanish
 at very low metallicity,   as we now discuss. 
 We adopt therefore
${ (\mathrm{Fe/Zn}) / (\mathrm{Fe/Zn})_{\sun}  } = 1$.
%This choice is conservative since it might lead to underestimate the extinction
%if  the Zn/Fe ratio in the DLAs is overabundant, 
%as found in metal-poor stars of extremely low metallicity.
%For these reasons we do not expect significant variations of  
%$ \left( { {\mathrm{X}_r \over \mathrm{Zn} } } \right)_{m} $ to occur
%in DLA systems, relative to
%the Miky Way value implicitely adopted in deriving $a_V$. 

 
%The   dependence on the depletion   enters through the term
%$   f_{\mathrm{Fe}}$. %
The depletion of DLAs increases with metallicity (Ledoux et al. 2003). 
In particular,  the dust fraction of iron  undergoes a fast rise between 
between metallicities [Zn/H] $\sim -2$ dex  and $-1$ dex, from 
values close to zero up to  values typical 
of the Milky-Way warm interstellar gas
 (Vladilo 2004). 
Here we model this trend with  the analytical expression
$$
f_{\mathrm{Fe}}(Z) = 
%({f_{\mathrm{Fe},\circ} / \pi} )
{ 1 \over \pi}  \,
\left[ 
\arctan
\left({\mathrm{[Zn/H]} - \mathrm{[Zn/H]}_\circ \over 
\Delta(\mathrm{[Zn/H]})} \right)+{\pi \over 2} 
\right]   
$$
% $f_{\mathrm{Fe}}=f_{\mathrm{Fe}}(Z)$
designed to smoothly fit   the ISM  
and   DLA data, providing  a gradual decrease to zero % $ f_{\mathrm{Fe}}=0 $  
at very low metallicity. 
%with [Zn/H]$= \log_{10} (Z/Z_{\sun})$,
%%$f_{\mathrm{Fe},\circ} = 1$, 
%  $\mathrm{[Zn/H]}_\circ=-1.25$ dex and
%$\Delta(\mathrm{[Zn/H]})=0.4$ dex. 
% This expression also fits  
%  the $ f_{\mathrm{Fe}} $ measured in the local ISM
%%value $ f_{\mathrm{Fe},i} \simeq 0.94$ [{\bf check}] when the metallicity is solar.   
%and 
%provides a smooth decrease to  $ f_{\mathrm{Fe}}=0 $  
%at very low metallicity.
%% and a smooth increase to $ f_{\mathrm{Fe}}=1$ 
%%at very high metallicity. 

The dependence on the changes of the grain properties enters through
the factor $G$, which scales  as
$s \, Q_{e,\lambda} \, (r_\mathrm{gr} \, \varrho_\mathrm{gr} \, X^{d}_\mathrm{Fe}) ^ {-1}$
(Eq. \ref{glambda}). 
This term is mostly determined by the physical
conditions of the medium, rather than by its chemical enrichment,
with the possible 
exception of the abundance by mass of iron in the grains, $X^{d}_\mathrm{Fe}$. 
%
At the very early stages of chemical evolution we may expect variations of
$X^{d}_\mathrm{Fe}$ as a consequence of non-solar relative abundances
of metals. However, these effects do not affect our estimate of the
extinction since, at the very low metallicities typical of these early stages
($Z \ll Z_{\sun}$), the factor $f_{\mathrm{Fe}}(Z)$
vanishes in any case, yielding a null extinction. 

The variations of $G$ due to changes of the physical conditions can be
assessed by sampling different regions     in a medium of constant
composition, such as the local ISM. 
%
The low scatter of $a_V$ derived in the previous section
indicates that the scatter of $G$ in the local ISM   is  low,
even if the MW sample   includes   lines of sight 
with   different physical conditions.   
%
To explain this qualitatively, we can imagine that variations of the  
grain size $r_\mathrm{gr}$ 
may compensate variations of   the dust density $\varrho_\mathrm{gr}$.
For instance,  in a harsh interstellar environment, volatile elements
tend to leave the grains and we may expect an increase of
$\varrho_\mathrm{gr}$ because carbon, the element potentially    
most abundant in the grains,  is volatile and has  
low atomic mass. At the same time,  
we may expect that larger grains will be more easily destroyed
in a   harsh environment
and the mean      $r_\mathrm{gr}$ will decrease,
countering the increase of $\varrho_\mathrm{gr}$ in Eq. (\ref{glambda}).
%%  revision 3 next sentence eliminated
%%
%As far as  the abundance by mass of iron is concerned, 
%we can imagine that variations of $X^{d}_\mathrm{Fe}$
%are not very large because  the bulk of the grain mass  
%comes from the most refractory and heavy elements, 
%and iron itself is one of these elements.  
%

These general considerations apply to any type of interstellar environment, suggesting that $G$ may not vary dramatically in DLAs with moderately
low abundances ($Z \ga 0.1 \, Z_{\sun}$). To   test
this hypothesis we computed $G$ in the Small Magellanic Cloud (SMC)
 from the expression
\begin{equation}
{ G_{\mathrm{SMC}}  } \simeq
{ ( { A_V \over N_\mathrm{H} } )_\mathrm{SMC} \over 
   ( { A_V \over N_\mathrm{H} } )_\mathrm{MW}   } \, 
 %
\left[ 
{ Z_\mathrm{SMC}  \over Z_{\sun} } \,
{ f_{\mathrm{Fe},\mathrm{SMC}} \over f_{\mathrm{Fe},\mathrm{MW}} } 
\right]^{-1} \,
%
\end{equation}
obtained by applying Eq. (\ref{AVdla2}) to the SMC with
$ (\mathrm{Fe/Zn})_\mathrm{SMC} = (\mathrm{Fe/Zn})_{\sun}$.
%
Inserting
$( { A_V \over N_\mathrm{H} } )_\mathrm{SMC} = 8.7 \times 10^{-23}$ mag cm$^{2}$
(Gordon 2003; SMC bar), 
$( { A_V \over N_\mathrm{H} } )_\mathrm{MW} = 2.1 \times 10^{-22}$ mag cm$^{2}$
(Bohlin et al. 1978),
$ Z_\mathrm{SMC}  \simeq 0.25 \, Z_{\sun} $
(Russel \& Dopita 1992), and
$ f_{\mathrm{Fe},\mathrm{SMC}} (Z_\mathrm{SMC})
\simeq 0.91 \, f_{\mathrm{Fe},\mathrm{MW}}$
 from 
 %% REVISION 3
 our analytical relation $f_{\mathrm{Fe}}=f_{\mathrm{Fe}}(Z)$, 
%% relation (\ref{fFeZeta}), 
 %
 we obtain
${ G_{\mathrm{SMC}} } \simeq 0.6 $. 
%
Considering the uncertainties in this derivation, this result is
  consistent with the Milky-Way value, % ($G=1 \pm 0.3$),
suggesting that, indeed, the term $G$ may not vary
dramatically in galaxies of moderately low abundances
%%% revision 3
even if the extinction curve is different.  
%%%   NEXT SENTENCE ELIMINATED
%This result is strengthened by  
%the fact that SMC-type dust is commonly considered to be quite different
%from Milky-Way dust.  
%
%%% end revision 3

Eq. (\ref{AVdla2}) is valid in the rest frame of the \ion{H}{i} region.
%
To derive the extinction in the observer's frame
we   need to consider  the wavelength dependence of
the extinction.  
 %
In the rest frame of the DLA system
the extinction % of the background quasar  
at the wavelength $\lambda^\prime$  will be 
%%
 %\begin{equation}
$ A_{\lambda^\prime} \simeq \xi(\lambda^\prime)  \,\, A_{V} $,
%\end{equation}
where $\xi(\lambda^\prime)=A(\lambda^\prime)/A(\lambda_V)$
is, by definition,  the normalized extinction curve. 
In the observer's frame
the same extinction % at   $\lambda = \lambda^\prime$
%in the rest frame of the DLA system 
will appear  at   $\lambda = \lambda^\prime \, (1+z)$, 
where $z$ is the  absorption redshift. 
  


 
\begin{figure}
   \centering 
    \includegraphics[width=7.5cm,angle=0]{1570f2.eps} 
\caption{ 
Extinction of a quasar at a fixed wavelength
in the observer's frame, plotted versus
the redshift of an intervening  DLA system with
$N^\mathrm{tot}_\mathrm{Zn} = 10^{13.50}$ cm$^{-2}$. 
%The extinction  is calculated with Eqs. (\ref{ext_zNZn}) or
%(\ref{ext_zNH}) for different extinction
% laws and photometric bands. 
Solid and dashed curves represent MW-type and SMC-type
extinction curves, respectively (Cardelli et al. 1998; Gordon et al. 2003). 
Curves left to right: photometric bands $u^\prime$, $g^\prime$, and $r^\prime$, respectively. 
 }
\label{ext_z}
\end{figure}
 
 

In conclusion, the observer's frame extinction of a quasar 
at wavelength $\lambda$ due to an intervening DLA system 
at redshift $z$ is
%with hydrogen column density $N_\mathrm{H}$,
%metallicity $Z$   
\begin{equation}
A_{\lambda} (N_\mathrm{H},Z; z) 
\simeq A_\circ ~ G ~\,
%2.0 \times 10^{-12}\, { g \over g_i } \, 
% a_{V,i} \, (1-f_{\mathrm{Zn},i}) \, 
\xi \! \left(  \! { \lambda \over 1 \!+ \! z } \! \right)  ~
% { p_{V,j} \over p_{V,i} } \, 
%{ 
f_{\mathrm{Fe}}(Z) ~\,
%\over f_{\mathrm{Fe},i} } \,
 N_{\mathrm{H}} ~ Z ~ ,
%N^\mathrm{tot}_{\mathrm{Zn},j}  ~.
\label{AlamHzeta}
\end{equation}
% 
with  %${ g / g_i } = 1 $ and $0.6$ 
$A_\circ \simeq 1.85 \times 10^{-14}$ mag cm$^2$, 
and $G=1$ and $0.6$ 
for Galactic-type and SMC-type dust, respectively. 
For each type of dust we adopt the corresponding extinction curve
representative of the Milky Way and of the SMC. 

%%
%Concerning the choice of the extinction curve, 
%Our knowledge of the extinction curve of DLAs is very poor
%with the exception of the work by Pei et al. (1991), which suggests
%that 217.5\,nm  extinction bump is weak or absent, as
%observed in SMC extinction curves.  
%Considering   the uncertainty of this result and the fact that a Milky-Way type  
%extinction curve has been found at redshift $z=0.83$ (Motta et al. 2002), a safe
%approach  is to consider both Galactic-type and SMC-type  extinction
%curves as possible proxies for the DLAs.  

 
Since $\xi(\lambda^\prime)>1$ for $\lambda^\prime < \lambda_V$,
the net effect of the cosmological redshift is an
amplification of  the rest-frame extinction
at $\lambda <  \lambda_V ~ (1+z)$.
%a condition certainly satisfied   by  UV and optical   bands.   
At $z \ga 2$ this amplification compensates for the reduction of  
$f_{\mathrm{Fe}}(Z)$ in systems of moderately
low metallicity and for the mild reduction of $G$ in SMC-type dust. 
Only at very low metallicity the extinction becomes negligible
 since $f_{\mathrm{Fe}}(Z)$ tends to vanish. 
Overall, 
{\em the extinction per unit column density of zinc atoms
predicted for most DLAs
% } by Eq. (\ref{AlamHzeta}) {\em
 is similar, or even larger,
than that typical of the Milky-Way ISM}. 
%
% revision 3  NEXT SENTENCE ELIMINATED
%%
%We expect therefore that the zinc detection threshold found
%for the Milky-Way can also apply to DLAs. 
%
% end revision 3


Examples of quasar extinction   
estimated with Eq. (\ref{AlamHzeta})  
%for different effective wavelengths $\lambda$ and extinction curves  $\xi(\lambda)$ 
are plotted in Fig. \ref{ext_z} as a function of $z$, keeping
fixed the other terms. This type of representation puts in evidence
the amplification of the extinction with increasing redshift. 
Each curve was computed at a constant $\lambda$  equal to the effective
wavelength of  the SLOAN Digital Sky Survey (SDSS)
photometric bands $u^\prime$, $g^\prime$ and $r^\prime$  
(Fukugita et al. 1996). % plus the $V$ band of the Johnson system.
We adopted the extinction law by Cardelli et al. (1998;
hereafter CCM law) with $R_V=3.1$
%(Cardelli et al. 1998) 
for the Milky-Way extinction curve 
and the mean SMC bar data  by Gordon et al. (2003) for the  SMC curve.
The term $f_\mathrm{Fe}(Z)$ was estimated at  
 [Zn/H]\,$= \log (Z/Z_{\sun}) = -1$ dex.
The   total zinc column density was fixed
at $N^\mathrm{tot}_{\mathrm{Zn}} = N_\mathrm{H} \, Z
=10^{13.50}$ atoms cm$^{-2}$,
 roughly two times the zinc column density 
threshold proposed by Boiss\'e et al. (1998).
%of the turning point   $\log N_\mathrm{Zn} = 13.18$ dex 
%found in the previous section, 
%that separates the regimes of slow and fast rise of the extinction.  
For the   redshift interval typical of most DLAs ($z \approx 2/3$)
the predicted extinction % at $\log N_\mathrm{Zn} = 13.15$ dex
is $\gsim 1$ mag, a value sufficient to obscure quasars 
in  spectroscopic surveys. 
% for all the photometric bands and extinction curves considered,
%with an amplification   factor of $\approx 3$      of the rest-frame extinction.
%%
%The fast rise of $A_b$ at $\log N_\mathrm{Zn}$ predicted
%above the turning point % $\log N_\mathrm{Zn} = 13.15$ dex
%will   give  extinctions well in excess of 1 magnitude,  

 
 
 

 \begin{figure}
   \centering
 \includegraphics[width=7.5cm,angle=0]{1570f3.eps}
\caption{Diamonds: 
column densities $N_\mathrm{H}$  versus $N_\mathrm{Zn}$
in DLAs. % the sample of DLAs with \ion{Zn}{ii} measurements. 
Horizontal lines:  DLA definition threshold  (bottom) and empirical upper
boundary (top) of $N_\mathrm{H}$. 
Vertical lines: detection limit (left) and obscuration  threshold (right) of $N_\mathrm{Zn}$. 
Dashed lines: lines of constant metallicity.
% REVISION 3
Circles: Milky Way data of Fig. 1. 
% END REVISION 3
Star: GRB-DLA from Savaglio et al. (2003). 
% See Sections ...  for explanations. 
}
\label{sample}
\end{figure} 
 
 


\section{Evidence for extinction from DLA data}

To search for some extinction effect related to the zinc column density
%On the basis of the above discussion we expect    
%the zinc column density to be an indicator of  quasar extinction. 
%
we  collected  from the literature all   DLAs  with 
%$N_\mathrm{H} \geq 10^{20.3}$  atoms cm$^{-2}$ and  
available $N_\mathrm{Zn}$ measurements. 
%
The resulting sample, shown in Fig. \ref{sample},
%and called hereafter the zinc sample, 
 includes 41 measurements, most of which derived from
 Hires/Keck and UVES/VLT  spectra. 
 References to these data can be found in Vladilo (2004; table 1), with
 the exception of 4 systems (without \ion{Fe}{ii} lines)
 % with \ion{Zn}{ii}, but , 
  published by Pettini et al. (1994, 1997) and
 Boiss\'e et al. (1998). 
% 
% From the analysis of Fig. \ref{sample}   we can derive the
% following considerations.
 %
%First, 

The updated sample includes more than twice  the measurements
of the one discussed by Boiss\'e et al. (1998) but,
%, which included 17 measurements and several limits. 
in spite of this increase,  
column densities  above the original
threshold % $N_\mathrm{Zn} > 10^{13.1}$ atoms cm$^{-2}$
(line AB in the figure) are not   found. 
%
%This limit is   remarkably similar to the
%one found by Boiss\`e et al. with a half number of measurements. 
%
% Second,
The lack of DLAs with $N_\mathrm{Zn} > 10^{13.1}$ atoms cm$^{-2}$
is surprising, since  Milky Way lines of sight with
\ion{Zn}{ii} measurements do show values above this limit
in a significant fraction of cases 
% REVISION 3
(Fig. \ref{sample}). 
%
%
Studies of gamma-ray bursts (GRBs) observed
inmediately after the explosion demonstrate that zinc column densities
above the threshold exist and can be detected in extragalactic absorbers\footnote{
%
Absorbers detected in front of GRBs, however, may have 
different properties from classical DLAs; 
%% FINAL ADDITION
a claim for gray extinction in GRB/DLAs has been made by
Savaglio et al. (2003). 
%
}
% when the background source is sufficiently bright 
(Savaglio et al. 2003;
the only case with measured $N_\mathrm{H}$ is indicated
with a star in Fig. \ref{sample}). 

%
These results confirm that  some selection effect is preventing the  
detection of
values of $N_\mathrm{Zn}$ above the threshold 
when the background source is faint. 
%The bias  is  not of  instrumental origin, since we are able to measure
%column densities above the threshold. 
%
%
%An intrinsic cutoff at high metallicity cannot explain  the lack of systems
%% in Fig. \ref{sample}, but not 
%in the triangle ABC, since the lines of constant metallicity to the left of the line AC
%intersect observational data points. 
%% where such systems would have
%%metallicities inside the observed range.   
%
The remarkable similarity between the DLA cutoff at 
$N_\mathrm{Zn} = 10^{13.1}$ atoms cm$^{-2}$  
and  the ISM turning point  
% of  the derivative $\partial A_V / \partial (\log N_\mathrm{Zn}) $ 
at  $N_\mathrm{Zn} \simeq 10^{13.2}$
 suggests that the rapid rise of the extinction  
above the turning point  can be  responsible for the cutoff. 
%
This is consistent with the conclusion of the previous section
that  the extinction per unit zinc column density of many DLAs
is similar to that of the local ISM. 
%%
%{\em The sharpness of the cutoff suggests that the
%extinction per unit zinc column density 
%is approximately constant among DLAs. 
%%
%All together, these arguments support
%the notion that the parameter $G$, which
%describes the properties of the dust grains, does not vary
%dramatically among different galaxies. } 

%Given the faintness of quasars, compared to the brightness
%of the Galactic stars observed in   interstellar studies, a rapid rise of the
%extinction would be critical.  
%
%The lack of a clear cutoff with
%increasing   $N_\mathrm{H}$ or   metallicity
%in Fig. \ref{sample} % the ($\log N_\mathrm{Zn},\log N_\mathrm{H}$) plane
% is consistent with our previous conclusion, based on Eq. (\ref{ext_zNH}),
% that it is hard to disentangle the dependence of the extinction
% on  $N_\mathrm{H}$ and [Zn/H].  


\begin{figure}
   \centering
 \includegraphics[width=7.5cm,angle=0]{1570f4.eps}
% \hspace{1.5cm}
%\includegraphics[width=6.cm,angle=0]{Vmaghist.eps}
\caption{Normalized frequency distribution  of quasar apparent magnitudes 
for the 
%DLA sample of Fig. \ref{sample}.  
% Bottom panel: full sample. Top panel: 
 sub-sample of
 DLAs with $\log N_\mathrm{Zn} > 12.53$ (shaded histogram)
 and $\log N_\mathrm{Zn} \leq 12.53$ (light histogram).}
\label{Vmaghist}
\end{figure}


If the     extinction increases with $N_\mathrm{Zn}$, we   
expect to find evidence of this effect also below the threshold,
in the range    
$\log N_\mathrm{Zn} \la 13.1$
%i.e.  in the interval % of $ N_\mathrm{Zn}$ 
where we do detect DLAs.
To search for this evidence we analysed the frequency distribution
of quasar apparent magnitudes of the zinc sample. 
We used the $V$ magnitude, which is measured in most cases.
% REVISION 3
In 4 quasars without $V$ measurements we adopted the magnitude
of nearby optical bands. 
% END REVISION 3
In practice, we compared the behaviour of the two sub-samples
with  $\log N_\mathrm{Zn}$ below and above the median
value, $(\log N_\mathrm{Zn})_\mathrm{median}= 12.53$. 
%The resulting distributions are represented
%as a light and a shaded histogram, respectively, in %the top panel of  
%Fig. \ref{Vmaghist}.
%
The differences between the two distributions, shown in Fig. \ref{Vmaghist}, are consistent
with a rise of the quasar extinction with increasing $N_\mathrm{Zn}$.
%
The analysis  of the three bins with more statistics 
% between $V$\,=\,16 and  19 mag, 
 indicates that the maximum of the sub-sample with high $N_\mathrm{Zn}$ (shaded histogram) is 
shifted  by $\approx 1$ bin
relative to the maximum of the other sub-sample (light histogram). 
 A Kolmogorov-Smirnov test shows that there is only a 9.7\% level of
probability that the two distributions are drawn from the same parent
population.
%
The shift to fainter magnitudes suggests that even at $\log
N_\mathrm{Zn} \lsim 13. 1$ dust extinction is already present,
affecting more the high-$N_\mathrm{Zn}$ sub-sample. 
%
The magnitude of this effect can only be explained by invoking
a redshift amplification of the extinction, % of a factor of $\gsim 3$,
as predicted by  relation (\ref{AlamHzeta}), since the extinction
expected at rest-frame is  only $\simeq 0.35$ mag 
at $\log N_\mathrm{Zn} = 13. 1$ (Fig. \ref{figAV_NZn}). 
%
  
  
\section{ Mathematical formulation of the bias }

In   Eq. (\ref{AlamHzeta})
the variation of the depletion in DLAs is accounted for
by the term $f_\mathrm{Fe}(Z)$, which is determined by the metallicity.
The term $G$ is approximately constant, at least in galaxies
with metallicity similar to those of the SMC and Milky Way.
Possible variations of $G$
in galaxies of very low metallicity do not affect the prediction of the extinction
since, in any case, $f_\mathrm{Fe}(Z)$ vanishes when $Z \ll Z_{\sun}$.   
For these reasons, relation (\ref{AlamHzeta}) allows us to estimate 
the extinction of a DLA system using, in practice, only its \ion{H}{i} column density
and metallicity. 
%In fact, 
%
%For the term $G$, we assume that it is similar to 
%
%convert the frequency distribution
%of hydrogen column densities and metallicities into a frequency distribution
%of extinctions. 
%$f_{N_\mathrm{Zn}} $  into the frequency distribution of the quasar extinctions
%due to intervening DLAs. 
We use this property to derive a mathematical formulation  
of the obscuration bias  based on the study of the 
 distribution functions of
$N_\mathrm{H}$ and $Z$ in DLA systems. 
%
% [start revision1] 
%In practice, we consider the extinction due to the bulk of the population
%of observed DLAs, which is peaked at   redshift $z \sim 2.3$.
%We neglect the extinction due to DLAs with Ly\,$\alpha$ absorption below
%the atmospheric cutoff ($z \lsim 1.8$), for which it is practically
%impossible to determine   $N_\mathrm{H}$ and $Z$. 
%Luckily, there are two factors that make less important the extinction
%contribution of low-redshift DLAs:
%(1) the redshift dependence of the extinction, shown in relation (\ref{AlamHzeta});
%(2) the intrinsic  reduction of the number density of DLAs at
%low redshift (Rao \& Turnshek 2000; Ellison et al. 2004).
%The fraction in dust should not significantly increase with cosmic time below $z \lsim 1.8$
%because $f_\mathrm{Fe}$ already approaches  a value close to unity well before
%  $z \sim 2$ (Vladilo 2004). Instead, 
%DLAs at very high $z$ probably do not yield a strong contribution
%to the extinction  since there is evidence for a drop  
%of $f_\mathrm{Fe}$ at the earliest stages of chemical evolution (see Eq. 10 above). 
%In summary, by  considering only the extinction of DLAs in the   interval
%$1.8 \lsim z \lsim 3.0$ 
%we can obtain a reasonable estimate of the overall effect of quasar obscuration
%and, at the same time, make the best use of available observations
%without     modelling the  extinction properties in redshift intervals
%that cannot be tested observationally. 
%% START REVISION 2
%{\bf
In practice, we determine the fraction of DLAs  
that are missed as a consequence of their own extinction.
%, where
%the extinction is computed from (\ref{AlamHzeta})
%given $N_\mathrm{H}$,  $Z$ and $z$. 
We then assume that multiple DLA absorbers in a given line of sight
play a negligible role in the estimate of the obscuration bias. 
%With these empirical considerations in mind, we have derived a mathematical
%formulation of the extinction bias  which considers  
%DLAs in a given redshift interval as the only
%sources of quasar extinction. 
%
In this way,  as we show in the Appendix, we   derive
a   mathematical formulation of the   bias
which only makes use of observable statistical distributions
of DLAs and quasars.
% together with the empirical-analytical relation (\ref{AlamHzeta}).
%
To quantify the obscuration effect we need   a good statistics
of the observed distributions and this requirement limits
the redshift range where  our method can be presently applied,
since the bulk of DLAs data are currently concentrated
in $1.8 \lsim z \lsim 3.0$. 

An advantage of our formulation 
is that it does not require a knowledge of 
% (but does not make predictions on) 
the intrinsic luminosity function of quasars or
 the geometrical distribution of the neutral gas in DLA galaxies. 
%
We refer to   FP93 for a   mathematical
treatment of the effect of quasar obscuration
%  REVISION 3 fottnote eliminated
%\footnote
%{
%%This is equivalent to ignore multiple DLA absorbers in a given line of sight. 
%%
%The way multiple absorptions are treated in the formulation by FP93 can be understood
%from the derivation of their Eqs. (5) and (6); this derivation can be found in Eqs. (18)-(21) of
%Fall \& Pei (1989), where the probability of finding more than one absorber in the
%interval $(z, z+\delta z)$ is an
%infinitesimal of order $(\delta z)^2$.
%}
%
 which describes the relation
with the quasar luminosity function and the geometrical distribution of the gas
in the intervening galaxies.   
%}
%%%%%%%%% END REVISION 2
% As an advantage over previous work,
%In our formulation the extinction is computed  making
%use of   statistical distributions   that can be tested
%observationally.  In particular, the metallicity distribution can be
%probed more easily than the distribution of dust-to gas ratios
%employed in the formulation by Fall \& Pei (1993). 
 %%%%
% [end revision1]

 
\subsection{ True and biased distributions }

%%% REVISION 3
 %
We   consider the DLAs in  the redshift interval 
$(z_\mathrm{min},z_\mathrm{max})$
that are detectable in a survey with limiting magnitude $m_\ell$.
%
We call
$f_{N_\mathrm{H}} d N_\mathrm{H}$   the  
  number of  such  DLAs with  column densities between
$N_\mathrm{H}$ and  $N_\mathrm{H}+dN_\mathrm{H}$  
and $f_{Z} \, dZ$ the    number of   those with
metallicity between $Z$ and $Z+d \,Z$.
%
The distributions $f_{N_\mathrm{H}}$ and $f_{Z}$ are defined in absence
of obscuration bias and we call them  the "true" distributions.
%
When we consider the effect of the  
 extinction generated by     these DLAs, 
the number of detectable systems
  will be  
$f^\mathrm{b}_{N_\mathrm{H}} d N_\mathrm{H}$ 
and
$f^\mathrm{b}_{Z} \, dZ$.
%
We call $f^\mathrm{b}_{N_\mathrm{H}}$ and $f^\mathrm{b}_{Z}$ the "biased" distributions. 
%
In the Appendix we derive the following relations between true and biased
distributions  
[see Eqs.  (\ref{fb_H}) and (\ref{fb_zeta})],
%
 \begin{equation}
 f^\mathrm{b}_{N_\mathrm{H}}  \simeq 
{
\int_{0}^{\infty}%_{Z_\mathrm{min} }^{Z_\mathrm{max} }
  \mathcal{B}_{m_\ell}  ( N_\mathrm{H}, Z) \,
f_{Z}  \, d\, Z
 \over 
\int_{0}^{\infty}%_{Z_\mathrm{min} }^{Z_\mathrm{max} }
f_{Z}  \, d\, Z
 } ~  f_{N_\mathrm{H}} ~~
 \label{fb_H_paper}
\end{equation}
% 
and 
%
%In a similar way, by integrating  Eq. (\ref{HZdistr}) in column density, 
%it is easy to obtain a   relation for  the biased distribution
%of metallicities, i.e.
%
\begin{equation}
 f^\mathrm{b}_{Z}  \simeq   
{
\int_{N_\mathrm{DLA}}^{\infty} 
  \mathcal{B}_{m_\ell}  ( N_\mathrm{H}, Z) \,
f_{N_\mathrm{H}}  \, d\, N_\mathrm{H} 
 \over 
\int_{N_\mathrm{DLA}}^{\infty} 
f_{N_\mathrm{H} }  \, d\, N_\mathrm{H}
 } ~  f_{Z} ~~.
 \label{fb_zeta_paper}
\end{equation}
%where $N_\mathrm{DLA}=10^{20.3}$ atoms cm$^{-2}$
%is the definition threshold of DLA systems.  

For simplicity  we omit  the dependence of the equations
% of the functions   on   the integration limits
on $z_\mathrm{min}$ and $z_\mathrm{max}$.
We assume that
   the statistical distributions  of interest       smoothly vary
inside the redshift interval, so that we can approximate them   
with their value at mean redshift  $\overline{z}=(z_\mathrm{max}-z_\mathrm{min})/2$,
%
  where     
% $(\overline{z}- {\Delta z\over 2}, \overline{z}+ {\Delta z\over 2})$
$\overline{z}$   
  is  close to  the peak of the observed distribution.


The "bias function"
$
\mathcal{B}_{m_\ell} (N_\mathrm{H}, Z) 
\equiv 
B_{m_\ell} (N_\mathrm{H}, Z, \overline{z}) 
%\label{BigCalB_paper}
$
that transforms the   true distributions      
into the biased ones is given by the relation
[see Eq. (\ref{BigBHZz})]
\begin{equation}%\left[
B_{m_\ell}  ( N_\mathrm{H}, Z,z) \equiv
{
  \int_\circ^{m_{\ell}-A_\lambda(N_\mathrm{H},Z, z)} \,  
%
 n(m; z)  \, d m \,
%
  \over
  \int_\circ^{m_\ell}  \, n(m; z)  \,  dm \, ,
 \label{BigBHZz_paper}
 } 
\end{equation}
where
$A_\lambda(N_\mathrm{H},Z, z)$ is the DLA extinction
and $n(m; z)$ the quasar magnitude distribution. 

 In the next section we show how to estimate $n(m; z)$, while
 in Section 4.3  we present a procedure
 for deriving the true distributions from Eqs.  (\ref{fb_H_paper}) and (\ref{fb_zeta_paper}),
 starting from the observed distributions. 
%

In deriving these equations
%Eqs. (\ref{fb_H_paper}) and (\ref{fb_zeta_paper})   
we made the assumption that the
distributions of $N_\mathrm{H}$     and $Z$
are statistically independent. We discuss this assumption in Section 4.4.
%   revision 3 next sentence eliminated
%   In the Appendix   we test the consistency of the above mathematical
% formulation  with the equations derived by FP93 for the ideal case
% of a constant dust-to-gas ratio and power-law quasar luminosity function
% (Section A.1.2). 
%
 
 
\subsubsection{ The quasar magnitude distribution }
 
We call  $n(m; z) \, dm$   the
  number  of quasars  with apparent magnitude  $m \in (m,m+dm)$
observable beyond   redshift   $z$  all over the sky.
This  number is defined 
in absence of quasar obscuration and we call
 $n(m; z) \, dm$  the "true" distribution.
 %
%To use Eqs.  (\ref{fb_H}) and (\ref{fb_zeta}) we need to estimate
%the unbiased frequency distribution of quasar apparent magnitudes 
%$n(m; \overline{z})$
%which enters in the definition  of  the term 
%$\mathcal{B}_{m_\ell}  ( N_\mathrm{H}, Z,)$. 
%%[see Eq. (\ref{BigBHZz})].
%
Our goal is to infer the true   distribution from the observed one, 
% REVISION 3
which must equal
the distribution  biased by the obscuration effect, $n^\mathrm{b}(m; z)$.
%
The  relation between true and biased distribution is discussed 
 in the Appendix (Section A.2),
where we derive the equations Eq. (\ref{nb_m}), i.e.
 \begin{equation}
n^\mathrm{b}(m; z)
\simeq 
\left[ 1-\overline{\mathcal{F}}_1 \right] \, n(m; z) 
+ \, \overline{\mathcal{F}}_1 \, n^\mathrm{b}_1(m;z) ~,
 \label{nb_m_paper}
\end{equation}
and Eq. (\ref{p_m}), i.e.
\begin{equation}
n^\mathrm{b}_1(m; z) = K 
\int_{-\infty}^{\infty} 
n(m^\prime; z) ~ 
f_{A_\lambda}\!(m-m^\prime;  \overline{z}_e) ~ dm^\prime ~.
\label{p_m_paper}
\end{equation}
%
In these relations 
$\overline{\mathcal{F}}_1$ is 
the fraction of quasars with one foreground DLA,
$n^\mathrm{b}_1(m; z) $   the magnitude distribution
of quasars with one foreground DLA, 
  $K$ is a normalization factor and
$f_{A_\lambda}(A_\lambda; \overline{z}_e)$ %\, dA_\lambda$ 
  the frequency
distribution of the extinctions % $A_\lambda = m-m^\prime$
of   the DLAs in the redshift range $z \in (0,\overline{z}_e)$
($\overline{z}_e$
is the mean redshift of the quasars in the survey). 
%
In Section 4.3, we show how to   use Eqs. (\ref{nb_m_paper}) and (\ref{p_m_paper}) 
to infer  $n(m; z)$ (see also Appendix A.2).
%%
%To solve these equations we  estimate  $\overline{\mathcal{F}}_1$,
%$f_{A_\lambda}(A_\lambda; \overline{z}_e)$
%and $K$ as we explain below. 

%%%% END REVISION 3


  
 \subsection{Obscuration fraction}
 
% We call {\em obscuration fraction} the fraction of DLA  systems 
% of a   survey with limiting magnitude $m_\ell$ 
% lost as a consequence of the   extinction bias. 
 % carried on . 
 We define the obscuration fraction  using the mathematical relations
 between the true and biased distributions. 
 We call {\em total obscuration fraction} 
 the quantity
 \begin{equation}
\Phi_{m_\ell} 
= 1 - 
{
\int^\infty_{N_\mathrm{DLA}}  f^\mathrm{b}_{N_\mathrm{H}} \, d N_\mathrm{H} 
\over 
\int^\infty_{N_\mathrm{DLA}}  f_{N_\mathrm{H}} \, d N_\mathrm{H} 
}
%= 1 - 
%{
%\int  f^\mathrm{b}_{Z} \, d\, Z
%\over 
%\int  f_{Z} \, d\, Z 
%}
%= 1 -
%{
%\int   \mathcal{B}_\ell (N_\mathrm{Zn}) \, f_{N_\mathrm{Zn}}  \, d N_\mathrm{Zn} 
%\over 
%\int  f_{N_\mathrm{Zn}} \, d N_\mathrm{Zn} 
%}
~.
\label{BigPhi}
\end{equation}
The total obscuration fraction
  represents the   number   ratio
of   DLAs missed due to quasar obscuration
in a survey with limiting magnitude $m_\ell$. 
The dependence on the limiting magnitude enters in
$f^\mathrm{b}_{N_\mathrm{H}} $ through Eqs. (\ref{BigBHZz}) and (\ref{fb_H}).
The total obscuration fraction can also be derived 
using the true and biased distributions of metallicity. 
% 
% The {\em cumulative obscuration fraction}, i.e. 
% the total fraction of DLA systems that {\em cannot} be detected
% owing to the obscuration effect can instead be derived 
% by comparing the areas subtended by an intrinsic distribution
% and by the corresponding biased distribution. 
% For instance, 
%the total number of {\em detectable}   systems  
%can be   estimated from the integral   $\int f^\mathrm{b}_{N_\mathrm{Zn}} \, d N_\mathrm{Zn}$, 
%while the total number of  systems
% from $\int f_{N_\mathrm{Zn}} \, d N_\mathrm{Zn}$.  
%In this case the total obscuration fraction is given by  
%\
%Similar equations to derive $\Phi_\ell$
%can be obtained by integrating the $N_\mathrm{H}$ and
%$Z$ distributions.  Obviously, the same value of 
%$\Phi_\ell$ must be obtained independently of the particular distribution function
%chosen to perform the calculation. 

  
We call  {\em  obscuration fraction}, $\phi_{m_\ell}$, 
the fraction by number of  systems obscured at a particular value     
of column density or metallicity.
% of the   distribution functions $f_x$ discussed above
% ($f_{N_\mathrm{H}}$, $f_{Z}$ and $f_{N_\mathrm{Zn}}$).
% To measure $\phi(x)$ we use
% the "biased-over-unbiased" ratio of the distribution function estimated
% at a particular value of $x$. 
 For instance, the fraction of systems obscured at a specific
  value of $N_\mathrm{H}$ is
 \begin{equation}
 \phi_{m_\ell} (N_\mathrm{H}) = 1 - 
 {f^\mathrm{b}_{N_\mathrm{H}}  \over f_{N_\mathrm{H}}} 
% =
% 1 - \mathcal{B}_\ell (N_\mathrm{Zn})     
 ~~.
 \label{phi}
 \end{equation}
%The meaning of this expression is clear from Eq. (\ref{BigBias_Zn}).
% The function $\mathcal{B}_\ell (N_\mathrm{Zn})$ represents
%the fraction of DLA/quasar pairs that {\em can} be detected at a given value
%of $N_\mathrm{Zn}$.    Therefore, its complement to unit   represents  
%the fraction of pairs that {\em cannot} be detected. 
%Because  $\mathcal{B}_\ell (N_\mathrm{Zn})$ is a function of the limiting magnitude, 
%also $\phi_\ell (N_\mathrm{Zn})$ depends on $m_\ell$, as indicated by the index $\ell$. 
Similar expressions can be derived % from Eqs. (\ref{fb_H}) and (\ref{fb_met}) 
to define the  obscuration fractions 
$ \phi_{m_\ell} (Z)$ and $ \phi_\ell (N^\mathrm{t}_\mathrm{Zn})$, 
i.e. the fraction of systems missed at a
particular value of $Z$ and $N^\mathrm{t}_\mathrm{Zn}$. 


\subsection{ Solution of the equations  }
 
 
%The biased distribution functions
%$f^\mathrm{b}_{N_\mathrm{H}}$ and $f^\mathrm{b}_{Z}$
%can be determined  from  observational surveys
%{\bf 
%within the same limits of validity of the approximation 
%(\ref{approximation}). 
%}
%  
The true   functions 
$f_{N_\mathrm{H}}$ and $f_{Z}$ are the unknown
quantities in Eqs. (\ref{fb_H_paper}) and (\ref{fb_zeta_paper}). 
%
The procedure that we follow to solve these equations
consists in 
(i) adopting trial functions $f_{N_\mathrm{H}}$ and $f_{Z}$,
(ii) estimating  $n_m \equiv n(m; \overline{z})$,
(iii) computing $f^\mathrm{b}_{N_\mathrm{H}}$ and $f^\mathrm{b}_{Z}$
from  the equations
%Eqs. (\ref{fb_H}), (\ref{fb_zeta}) and (\ref{nb_m})
and (iv) comparing these
biased distributions with the observed ones.
%
These steps are repeated by changing the trial functions
until the predicted biased functions match the observed ones.  
%
%% START REVISION 3 next sentence eliminated
%{\bf
%This procedure is correct within the limits of validity of  the approximation 
%(\ref{approximation}). 
%}
% END REVISION 3
%
We briefly describe each step of this procedure.

In the first step, 
we adopt analytical functions in parametrical form
as the trial functions $f_{N_\mathrm{H}}$ and $f_{Z}$.
%of the true
%distributions % $f_{N_\mathrm{H}}$ and $f_{Z}$ 
%  revision 3 next sentence eliminated
%
%The choice of the shape of the functions is driven
%by basic physical arguments and by the shape of
%the empirical  distributions 
%in the range of $N_\mathrm{H}$ and $Z$ 
%where the extinction is   negligible.
%
%  (low column densities and low metallicities). 
%
%We choose functions with the lowest number of parameters
%which satisfy the above criteria. 
%
Once we choose   the   shape, the problem
of determining the true functions $f_{N_\mathrm{H}}$ and $f_{Z}$ 
is equivalent to that of determining the values of their parameters.
%In order to generate trial functions $f_{N_\mathrm{H}}$ and $f_{Z}$ 
%we therefore generate a set of trial values of their parameters. 
%
 
%%% START REVISION 2
%{\bf 
In the second step we use the trial  functions
 $f_{N_\mathrm{H}}$ and $f_{Z}$
to derive the  distribution of extinctions
%$f_{A_\lambda}(A_\lambda; \overline{z}_e)$ 
from Eq. (\ref{AlamHzeta}).
%All  these distributions are defined 
%to include the DLAs with redshift 
%$z \in (z_\mathrm{min}, z_\mathrm{max})$. 
We use this extinction distribution
% as an approximation of  $f_{A_\lambda}(A_\lambda; \overline{z}_e)$ 
in Eq. (\ref{p_m_paper}). 
We then iterate Eqs. (\ref{nb_m_paper}) and  (\ref{p_m_paper}),
starting with a trial function\footnote{
The observed distribution  $n^\mathrm{b}(m; z)$ can be used
as the starting trial function of $n(m; z)$.}  
%
  $n(m; z)$, until we find  a $n(m;z)$
  that yields a perfect agreement between
$n^\mathrm{b}(m; z)$ and  the  
  distribution $n^\mathrm{obs}(m; z)$
measured from the surveys. 
% revision 3 next part eliminated
%and  make a preliminary estimate of
%$n^\mathrm{b}_1(m; z)$ from   Eq. (\ref{p_m_paper}).
%We   insert this   
%$n^\mathrm{b}_1(m; z)$ 
% in Eq. (\ref{nb_m_paper}), together with       
%the estimates of $\overline{\mathcal{F}}_1$ and 
%$n^\mathrm{b}(m; z)$.
%This last distribution is estimated from
%In this way we obtain from  Eq. (\ref{nb_m_paper}) an improved guess of  $n(m; z)$. 
%%
%Proceeding by iterations we obtain a function $n(m; z)$
%such that  the calculated distribution $n^\mathrm{b}(m; z)$
%can match $n^\mathrm{obs}(m; z)$ with any degree of accuracy desired. 
% end revision 3
%
%} 
%%%%% END REVISION 2
%With this choice of $n_m$ 

In the third step we  use $n(m; \overline{z})$ to derive    
$\mathcal{B}_{m_\ell}  ( N_\mathrm{H}, Z)$
from  Eq. (\ref{BigBHZz_paper}) % and  (\ref{BigCalB_paper})
and, finally, the biased 
distributions  %$f^\mathrm{b}_{N_\mathrm{Zn}}$, 
$f^\mathrm{b}_{N_\mathrm{H}}$ and $f^\mathrm{b}_{Z}$ 
from Eqs.  (\ref{fb_H_paper}) and (\ref{fb_zeta_paper}). 

%{\bf
In the last step  we compute the $\chi^2$ deviation between
the predicted distributions
$f^\mathrm{b}_{N_\mathrm{H}}$ and $f^\mathrm{b}_{Z}$ 
and the corresponding observed distributions.
%}

By iterating the steps we determine
the  parameters of $f_{N_\mathrm{H}}$ and $f_{Z}$
which allow     
$f^\mathrm{b}_{N_\mathrm{H}}$ and $f^\mathrm{b}_{Z}$ 
to best fit the empirical distributions.  
The search  is done via $\chi^2$ minimization. 
Since we are not guaranteed a priori of the unicity of the solution,
we carefully investigate the variation of $\chi^2$ in the parameter space
to make sure the minimum is unique.  
%
 


\subsection{Are $N_\mathrm{H}$ and
 $Z$ statistically independent   ?}

  
The mathematical formulation presented above
%derivation of Eqs. \ref{convolution}, \ref{fb_H}, and \ref{fb_met}
relies on the assumption that $N_\mathrm{H}$ and
 $Z$  are statistically independent of each other in the true population of DLAs.  
%
The local ISM offers an example in support of this independence, 
since a broad range of \ion{H}{i} column densities,
including the high values typical of DLAs,   are found at   constant  
metallicity, $Z \simeq Z_{\sun}$.
Similarly,  a DLA system of given metallicity $Z \simeq Z_\circ$
will display a distribution of  \ion{H}{i} column densities, if
sampled along random lines of sight. 
This  distribution will reflect the mass spectrum of interstellar clouds,
which is likely to be determined by some physical mechanism,  
such as interstellar turbulence (see discussion in  Khersonsky \&
Turnshek 1996 and refs. therein), rather than by the  metallicity.  
In this sense, it is reasonable to assume that   
  $f_{N_\mathrm{H}}$ and $f_{Z}$  are independent distributions.
  
  Independent support of these   arguments comes
from the    
hydrodynamic simulations of DLAs evolving in $\Lambda$CDM
cosmology (Cen et al. 2003), 
which indicate that metallicity and hydrogen 
column density are almost completely uncorrelated.  
%{\bf
Similar results are found from the  cosmological SPH simulations
performed by Nagamine et al. (2004).
%}

Notwithstanding, a trend is expected because the metallicity  
correlates with the star formation rate, 
which in turn increases with the gas density in the host galaxy. 
Therefore, lines of sight with high $N_\mathrm{H}$ may have a
tendency to be associated with higher metallicities.
A   weak evidence for this trend was found
by Cen et al. (2003), but
the effect is particularly small at redshift $z \simeq 2-3$.  
%
%
 If the metallicity is correlated with $N_\mathrm{H}$, 
the effect of the obscuration is   more important than 
in the ideal case considered above. In this case
  the application of our mathematical formulation 
  gives a conservative estimate of the bias. 
%

 
\subsection{Gravitational magnification}

Competing with the obscuration effect, is the possible gravitational magnification 
of the background quasar by the absorber itself. 
Many studies over the years have attempted to estimate
the magnitude of this phenomenon for both DLAs (Smette et al. 1997, Le
Brun et al. 2000) and metal absorption lines (Vanden
Berk et al. 1996, M\'enard \& P\'eroux 2003). The results show that for the DLAs
selected at optical wavelengths, gravitational effects are small.
%
%In principle, the gravitational lensing could be  formally
%included in our equations by adding a magnification term
%in the bias function   (\ref{BiasFun}). 
%In practice, this would require introducing a number
%of extra parameters not constrained by the empirical
%distributions   considered here.  
%
In the present application of our method, 
based on DLAs selected at optical wavelength, we therefore neglect  
the magnification effect.
% revision 3 next sentence elminiated
%We note that the  magnification {\em due to other absorbers}, if present,
%is automatically accounted for by the method.
%In fact such an effect would change the  observed distribution
%of apparent magnitudes, which is the input required by our method
%to derive the magnitude distribution not affected by extinction, $n_m$.
% 
   

\section{Implementation of the procedure}

We   describe the statistical distribution functions   adopted here
as a first example of   implementation of the  procedure. 
% 
For the intrinsic distributions we only need to adopt
a functional dependence (i.e. a shape),  
leaving to the procedure the task of
determining the parameter values. 
 

\subsection{ The empirical distributions}

 We consider the sub-sample of
DLA systems with available measurements of \ion{Zn}{ii} column densities,
for which it is possible to derive metallicities corrected
for depletion effects   (Vladilo 2004). 
%The observed frequency distribution of
%metallicities was estimated from the zinc sample,
%taking into account dust depletion effects (Vladilo 2004).
%where the effects of dust depletions are taken into account.
The differences between the corrected metallicities and
the [Zn/H] values taken at face value are small, with a mean value
 of $+0.12$ dex.  
%The total redshift interval considered corresponds to an interval
%of cosmic time of xxx Gyr, equivalent to about ...\% of the age
%of the universe. ...
%
Given the size  of the zinc sub-sample ($\sim 40$ systems;
see list of references in Table 1 of Vladilo 2004) we can estimate 
the frequency distribution of metallicities 
 with a Poisson statistical error $\la 10\%$
 in only a few bins.
%of the empirical frequency distributions.
To make the best use of the available data we  selected  
the systems in the redshift range $1.8 \leq z \leq 3$,
where most of the $N_\mathrm{Zn}$ measurements have
been obtained so far.  
The mean redshift of the sample (28 systems)
is $z \simeq 2.3$.
% corresponding
%to an age of the Universe of $\simeq 2.9$ Gyrs
%for current cosmological parameters. 
% The redshift interval corresponds
%to a cosmic time interval of $\simeq$ 1.5 Gyr, starting at about
%$\simeq 2.2$  Gyr after the Big Bang.
% (look-back time between $\simeq 10.0$ and 
%$\simeq 11.5$ Gyrs before present)
%By excluding systems at $ z > 3$ we should be safe from
% dramatic changes in the metallicity distribution expected
%at very high redshift, where we get closer  to the onset
%of metal production in the first protogalaxies. 
%
 % 
To derive the empirical distribution 
of $N_\mathrm{H}$ we used the surveys by
Storrie-Lombardi \& Wolfe (2000) and
P\'eroux et al. (2003),
which have more statistics  than the zinc sub-sample. 
Selecting the DLAs in the range $1.8 \leq z \leq 3$ 
we obtained a  sample of $59$ systems.   
 

Then
$\log N(\ion{H}{i})$ and [Zn/H]=$\log (Z/Z_{\sun})$, 
we transformed the derived
distributions in linear space, $N(\ion{H}{i})$ and $Z$,
for later comparison with the distributions $f_\mathrm{H}$ and $f_Z$.
%
In computing  the  biased distributions
with our procedure, we   took into account the fact that
the typical  limiting magnitude of the  $N_\mathrm{H}$ surveys
($m_\ell \sim 19.5$)
is somewhat larger than that of the  
metallicity surveys ($m_\ell \sim 19$), owing to the different requirements
in spectral resolution.  
In fact, Eqs. (\ref{fb_H}) and (\ref{fb_zeta}) 
can be solved independently of each other, inserting in each
case the appropriate value of $m_\ell$. 
 

\subsubsection{The frequency distribution of quasar magnitudes}

To derive   the empirical distribution of quasar magnitudes
we used the data of the Sloan Digital Sky Survey (SDSS)  (Schneider et al. 2003).
The distribution was determined for two photometric bands  well representative
of the visual part of the spectrum, namely the
$g^\prime$ and $r^\prime$ bands, with effective wavelengths 
$\lambda=0.48 \mu$m and $0.62 \mu$m, respectively
(Fukugita et al. 1996). 
In each band we binned  the data   
at   steps of  0.5 mag,  counting all the quasars located beyond    
the typical absorption redshift of our sample
(in practice, we adopted $ 2 \leq z_e \leq 5 $). 
%
% revision 3
An example of
empirical distribution computed for the $g^\prime$ band 
is shown in Fig. \ref{plotQSOmag} (diamonds with error bars). 
%
After binning the data, we performed
a polynomial fit to the observed $\log n$ - $\log m$ distribution, up to the 
magnitude $21.5$, for which the statistics   and the
completeness of the SDSS sample are still good. 
%
% revision 3
This fit was adopted as a smooth model of  the observed distribution
in the  magnitude range $ 17 \leq m \leq 21.5$.   
An example of   fit is   shown in Fig.
\ref{plotQSOmag} (solid line).
%
The limit $m \sim 21.5 $ is sufficient
for studying   the effects of quasar obscuration on the statistics
on DLAs  since  the typical limiting magnitude
of DLA surveys is currently $m_\ell \lsim 21$. 
 
  
\begin{figure}
  \centering
 \includegraphics[width=7.cm,angle=0]{1570f5.eps}
\caption{ 
%%% REVISION 3
Diamonds: 
distribution of the SDSS $g^\prime$
  apparent magnitudes  for
quasars in the   interval $2 \leq z_e \leq 5$. 
Smooth lines: example of a
pair of model distributions       in which
the "true" model (dashed line) is   determined
from the biased model 
that fits the observations (solid line),
% by means Eqs. (\ref{nb_m_paper}) and (\ref{p_m_paper}), 
once  the  distribution of DLA extinctions  
and $\overline{\mathcal{F}}_1$ are specified
(see Section 4.1.1). 
}
 \label{plotQSOmag}
\end{figure}




\begin{center} 
\begin{table*}  
\caption{  Results. }
\begin{tabular}{cccccccccc}
\hline \hline
 Ext.curve &  Band & $\beta$ & $\alpha$ & 
$\log ( {Z_\ast \over Z_{\sun} }) $ & 
$\Phi_{22} (m_\ell) ^a $ & $\Phi_{23} (m_\ell) ^b $ &
$\log ( {<Z> \over Z_{\sun}}) $& 
$\Omega_{22}^a$ & $ \Omega_{23}^b$ \\
& & & & & $19.0 / 19.5$ & $19.0 / 19.5$ & & $(10^{-3})$ & $(10^{-3})$ \\
\hline
MW  & $r^\prime$ & 1.56 & $-0.44$ & $-0.13$ & $0.41 / 0.34$ &  $0.44 / 0.38$ & $-0.37$ & $2.3$ & $~7.1$ \\

MW  & $g^\prime$ & 1.51 & $-0.40$ & $-0.10$ & $0.48 / 0.42$ & $0.51 / 0.46$ & $-0.31$ & $2.8$ & $~9.5$\\

SMC & $r^\prime$ & 1.61 & $-0.46$ & $-0.19$ & $0.33 / 0.27$ & $0.37 / 0.30$ &  $-0.44$ & $1.9$ & $~5.5$ \\

SMC & $g^\prime$ & 1.48 & $-0.38$ & $-0.07$ & $0.52 / 0.46$ & $0.56 / 0.51$ & $-0.27$ & $3.1$ & $11.3$\\
\hline
\end{tabular} 
\\
$^a$   
  Upper limit of integration in Eq. (\ref{BigPhi})  or (\ref{Omega})
  truncated at $\log N_\ion{H}{i}=22$. 
\\  
$^b$   
  Upper limit of integration in Eq. (\ref{BigPhi})   or (\ref{Omega})
  truncated at $\log N_\ion{H}{i}=23$. 
\\  
\end{table*}
\end{center}

 

 
\subsection{ The shape of the true  $N_\mathrm{H}$ distribution }
 
One way to approximate
 the true shape of   $f_{N_\mathrm{H}}$
is to assume a 
geometrical distribution of the gas   in DLA galaxies. 
% 
For   planar disks with radial  exponential profiles 
$N_{\bot }(r) = N_{\bot 0} \exp (-\alpha \, r)$, 
the expected frequency distribution is
$f_{N_\mathrm{H}} \propto N_\mathrm{H}^{-1} ~ \ln N_\mathrm{H}$
for $N_\mathrm{H}\ll N_{\bot 0}$
and 
$f_{N_\mathrm{H}} \propto N_\mathrm{H}^{-3} $
for $N_\mathrm{H}\gg N_{\bot 0}$
(FP93).
For  central column densities $N_{\bot 0} \ga 3 \times 10^{21}$ cm$^{-2}$
or much larger, as inferred by FP93, this gives  $f_{N_\mathrm{H}} \propto N_\mathrm{H}^{-1} ~ \ln N_\mathrm{H}$
for most of the column-density range typical of DLAs. 

A realistic approximation of $f_{N_\mathrm{H}}$  
should also take into account the fact that, in a given galaxy,
the interstellar clouds show a spectrum of masses, $m_c$, and sizes, $r_c$, 
which is not accounted for by a smooth radial profile. 
%
A large number of studies indicate that the mass spectrum of 
Milky-Way interstellar clouds follows a power law $f(m_c) \sim m_c^\gamma$,
with $\gamma = -1.5 \pm 0.2$ over a 6 orders of magnitude range
of masses (Scalo \& Lazarian 1996 and refs. therein). 
%
From this mass spectrum, together with a relation between 
the internal density of the cloud, $n$, and the cloud size 
we can infer a column-density distribution.
Adopting $n \sim r_c^p$ with $-1.2 < p \leq 0$
(Scalo \& Lazarian 1996 and refs. therein) we find\footnote{
With the adopted relation we find   
$m_c \sim N_\mathrm{H}^{p+3 \over p+1}$
and then derive the lower limit of $\beta$ assuming 
$p=0$.
}
that the column-density distribution should follow a power law
$f_{N_\mathrm{H}} \sim N_\mathrm{H}^{-\beta}$ with
$\beta \geq 2.5$. If this column-density spectrum applies to
DLAs, we would expect a decline of column densities faster
than that predicted by the exponential profile model. 

%%   revision 3 next sentence eliminated
%Modelling   the real distribution $f_{N_\mathrm{H}} $ in DLAs
%while taking into account  both a  smooth radial profile and 
%a column-density spectrum   for different types
%of galaxies,  
%would yield
%a complex analytical expression with a large number
%of unconstrained parameters. 
%On the other hand, 
%% end revision 3
%

Both the exponential profile model and
the cloud mass-spectrum hypothesis suggest that the
true distribution may be approximated with a power law.
% at least over the range of column densities
%typical of DLAs, which span less than two orders of magnitude.
 The hydrodynamic computations of DLAs by Cen et al. (2003) 
yield a simulated true distribution of $N_\mathrm{H}$
consistent with  this conclusion. 
 
 
%
%%
%Independent support to this conclusion comes from the fact that 
% the frequency distribution of \ion{H}{i} column densities
%%$N_\mathrm{H}$ 
%in quasar absorbers  follows  a power-law %%\begin{equation}
%%$f_{N_\ion{H}{i}} \propto N_\ion{H}{i}^{-\beta} $
%%%\label{f_H}
%%%\end{equation}
%%%with $\beta \simeq -1.46$
% over  
%more than 9 orders of magnitude in column density %$N_\mathrm{H}$, 
%%from the Ly\,$\alpha$ forest % , with  $\beta \simeq 1.1 [?]$, 
%%to % Lyman Limit systems and 
%%DLA systems
%(Tytler 1987; Storrie-Lombardi \& Wolfe 2000).
%The typical value of the spectral index
%is $\beta \simeq 1.5$,
%even if optically thick absorbers are better represented by
%$\beta \simeq 1.3$ (Petitjean et al. 1993). 
%%For  the 
%%A simultaneous fit with a single power law over the whole range  
%%yields $\beta \simeq 1.5$, but the data are better represented by
%%two power laws with a break 
%%at $N_\ion{H}{i} \simeq 10^{16}$ atoms cm$^{-2}$
%%and  %$\beta \simeq 1.83$  and 
%%in the high column-density regime
%Since these values of $\beta$ apply also to the column density regime
%inmediately below the DLA threshold, where the extinction
%is   negligible,
%this empirical evidence suggests that a power law can be
%a good approximation of the true distribution.
%%$f_{N_\mathrm{H}}$ in DLAs.  
%%
% 

%However, detailed studies of the very high 
%column-density interval indicate that the 
%frequency of DLAs decreases faster than a power law % with $\beta \simeq 1.32$
%and is matched by a function
%%
%The parameter $N_{\star}$ represents the turning point of 
%the $N_\mathrm{H}$ distribution.     
%
%with %$f_\star=0.03$, 
%$\log N_{\mathrm{H},\star} = 21.39$, and $\beta=1.52$
%Here we adopt the working hypothesis   that
%the true distribution of DLAs follows a simple power law,
%as observed in quasar absorbers with negligible extinction,
%and that the  fast decline   
%is due to  the onset of the quasar extinction when 
%$N_\mathrm{H}$ is sufficiently high. 
%%The hypothesis that the true
%%distribution is a power law is 
%%consistent with the observations of quasar absorbers of lower 
%%$N_\ion{H}{i}$, for which the extinction is  negligibile.
%In the application of our procedure we do not make
%any assumption on the value of $\beta$, which is instead
%determined by the procedure itself.  
%%By adopting the power law  (\ref{f_H}) 
%%we only need to determine the parameter $\beta$ because
%%the normalization factor is not required in the solution of our
%%problem.   

In light of these considerations, we adopt   
the simple   law 
$f_{N_\mathrm{H}} \sim N_\mathrm{H}^{-\beta}$
for the true distribution, with 
the parameter $\beta$ to be determined by the method itself. 
%
If this is a good approximation, % of  $f_{N_\mathrm{H}}$, 
%we should find  
%$\beta \approx 1.3/1.5$ for consistency for the value typical
%of quasar absorbers with negligible extinction.
%In addition,  
 the resulting biased distribution $f^\mathrm{b}_{N_\mathrm{H}}$
% obtained starting from a simple power law %(\ref{f_H}) % in Eq. (\ref{fb_H})
must reproduce the observed fast decline of the number of DLAs
with column density, which %, at variance with low-$N_\mathrm{H}$ absorbers,
 is commonly fitted by a   Schechter-type function
\begin{equation}
f_{N_\mathrm{H}} \propto % = {\left( f_\star \over N_{\mathrm{H},\star} \right) } 
{\left( N_\mathrm{H} \over N_\ast  \right)}^{-\beta} \, 
e^{-\left( N_\mathrm{H} \over N_{\ast} \right)} ,~
\label{schechter}
\end{equation}
%
    characterized by a fast decline at $N_\mathrm{H} > N_{\ast}$
(Pei \& Fall 1995; Storrie-Lombardi \& Wolfe 2000; 
P\'eroux et al. 2003).  

  
 
\subsection{ The shape of the  true metallicity distribution  }
 
%\subsubsection{The metallicity distribution function}
The arguments   leading to the choice of   $f_{Z}$
    can be summarized as follows. 
The observed distribution  of [Zn/H] measurements
peaks at [Zn/H] $\simeq -1.1$ dex and does not show
systems below $\approx -2$ dex
or above solar metallicity  (Pettini et al. 1997).   
Although this distribution is  likely biased at
both ends, we do expect a genuine
%The absence of systems at very low metallicity 
%is probably due to the $N_\mathrm{Zn}$
%observational cutoff below the detection limit of the \ion{Zn}{ii} resonant
%lines. %  most often used in this class of investigation. 
%On the other hand, the lack of systems  at high metallicity
%may result from the obscuration bias.  
%In spite of these observational biases, we expect to exist
%an intrinsic 
decline in the number of DLAs below and
above some critical values of metallicity. 
At   [Zn/H] $\lsim -1.5$ dex  the zinc sample
shows a decrease of the number of DLAs
 in a region of the 
($N_\mathrm{Zn}$, $N_\mathrm{H}$)
plane not affected by   any   bias
(the triangle DEF in Fig. \ref{sample}). 
At high metallicity it is reasonable to predict a decrease since we
expect a natural decline  of the
number of systems with higher and higher star formation rates.  
%
These arguments indicate that the true  
distribution of [Zn/H] starts from a negligible value
at low metallicity,  shows a rise around $\approx -1.5$ dex
and declines after reaching a maximum.
%
% REVISION 3
We have been particularly careful in modelling
this trend  with an analytical function
because the results, and in particular the mean metallicity,
depend on the adopted functional form. 
%
To choose the function we started by making an educated
guess of the shape of the metallicity distribution based
on our knowledge of the statistics of galaxies and DLAs. 
%
We then made sure that the adopted function satisfies two requirements:
%
(1)  that it is a "conservative" one, i.e.  a function that does not overpredict
the effects of the obscuration;
%%
%We also made sure 
%that 
(2) that the predicted extension of the function at high metallicities,
where the obscuration is important,
is not affected by the poor knowledge of the low end of the distribution. 
%does not affect our estimate of the high end,  
%where the obscuration is important. 
%
%On the basis of these requirements,
After considering several possibilities, % among which a lognormal distribution,
 we adopted the function
 %
%For the same reasons, we prefer functional form with  
%a mild rise in the range of low metallicities. In this way we minimize the number
%of systems lost due to observational bias other than
%quasar obscuration. 
%
% 
\begin{equation}
f_Z
 ~ \propto ~ (Z/Z_\ast)^\alpha ~
 e^{-Z/Z_\ast} ~,
 \label{MetalDistFunction}
\end{equation} 
which satisfies very well the above requirements, as we explain below. This function is
% 
similar to the one proposed by Schechter (1976) for
describing the luminosity function of galaxies. 

To make an educated guess for the shape of the   distribution %(first criterion)
we started from  the  metallicity-luminosity relation in galaxies.
% 
Evidence is building up that this  relation is valid not only in the local
Universe  (e.g. Lamareille et al. 2004), but also in DLA systems
(Ledoux et al. 2005). 
The empirical relation   between the logarithmic metallicity 
and  the absolute magnitude
implies that  the linear metallicity $Z$
increases linearly  with the luminosity. The luminosity
in turn follows a Schechter distribution both
in the local Universe (e.g. Cuesta-Bolao \& Serna 2003),
at redshift $z \simeq 0.1$ (e.g. Blanton et al. 2003) and, apparently, up
to   $z \simeq 3.5$ (Poli et al. 2003). 
Combining these different pieces of evidence, it is reasonable
to assume that $Z$ may follow  a   Schechter distribution in DLA systems. 

%
The     requirement (1) of a "conservative" function   
is equivalent to make sure that         decline at high metallicity 
is fast.  
In fact, the faster the decline of the "true" distribution, the lower
 the estimated number of missed DLAs.
% and  the mean metallicity.    
%
Luckily, the Schechter  satisfies this requirement
since it provides a very   fast decline
at $Z \!\!>\!\! Z_\ast$. In fact,
the decline is faster than in       any polynomial in $Z$
and in a lognormal distribution. 

The   requirement (2) implies that we should   avoid functions
for which a single parameter specifies
the behaviour at both ends of the distribution.  %
%the shape at high metallicities,
%where we expect the obscuration,
%and the shape at low metallicities
%are determined by the same parameter(s). 
%%
%With the third criterion one can model the high-metallicity tail
%of the distribution independently of the observed behaviour at
%low metallicities. 
%%This criterion is important because
%%at very low metallicities we have a poor statistics. 
%Thanks to this criterion, the statistical error due to the fit
%at very low metallicities, where the statistics is poor, will not propagate at high metallicities,  
%where the obscuration can be important. 
The Schechter function    satisfies   this requirement
since % for the Schechter function (\ref{MetalDistFunction}), 
the parameter   $\alpha$, 
which in practice controls the extension of the distribution at low metallicities,
has a negligible effect on the exponential decrease at $Z >  Z_\star$.  
%
%Therefore the poor statistics of the measurements at
%low metallicity does not affect the
%determination of $Z_\star$ and hence of  $<\!\! Z \!\!>$.  
%
For a lognormal distribution, instead, 
the width parameter
% (and the extension
%of the distribution at high metallicities)
 is affected 
by the fit   of the distribution at the low-metallicity end,
which is poorly constrained by the observations. 
The resulting error on the width parameter 
will affect the extension of the distribution at high metallicities and therefore
  the estimate of the obscuration effect. 
%
The same considerations apply to any distributions that,
like the lognormal one, are  specified by a centroid and  a width.
%
For these types of functions,
the uncertain behaviour of the low end of the metallicity distribution
affects the determination of the obscuration effect. 
  


%
%When $\alpha \!\!>\!\! -1 $,
%the Schechter function provides
%the rise at low metallicities of the logarithmic distribution 
%$f_{(\log Z)}= (\log_e 10) ~ 10^{(\log Z)} ~
%f_Z (10^{(\log Z)}) $, 
%  expected from the   measurements of
%[Zn/H] = $\log (Z/Z_{\sun})$. 
%%

% end revision 3

%
%%With the choice of the function (\ref{MetalDistFunction})
%%we need only two 
%The parameters $\alpha$ and $Z_\ast$ 
%%to specify the metallicity distribution.  
%%We do not make any assumption on the values
%%of these parameters, which 
%are determined by
%the application of the procedure.  
 
 
 
 
\section{Results and discussion}

In Table 1 we summarize the results obtained   from the application
of our procedure.   
The quasar extinctions were   estimated in the
 photometric bands $r^\prime$ and $g^\prime$ 
for an absorption redshift $z=2.3$. 
%
The differences between the results obtained
 for the two   %$g^\prime$ and $r^\prime$ 
 bands give  an estimate of the uncertainty 
% in the determination of the bias effect 
due to the lack of a homogeneous set of visual magnitudes % in a single visual band
for all the quasars of the adopted surveys. 
%
The obscuration fraction   in the $r^\prime$ band is more conservative
than that in the $g^\prime$ band, given the increase
of the extinction with decreasing effective wavelength.
The results for the $g^\prime$ band ($\lambda=0.44 \mu$m) are more
comparable to those derived 
in previous studies of the   obscuration bias,  
which   considered the $B$ band  ($\lambda=0.44 \mu$m;
Ellison et al. 2001).
 
Two types of dust extinction curves were considered, namely the MW
curve by Cardelli et al. (1988) and the SMC curve by Gordon et al. (2003). 
The presence or absence of the 2175\AA\ extinction bump in DLAs
is irrelevant for most of our results, because at $z=2.3$
the extinction curves in the $r^\prime$ and $g^\prime$ bands 
do not sample the bump
 (Fig. \ref{ext_z}).  The only marginal exception is the 
combination of the $r^\prime$ band with the MW-type extinction.
 If the bump is completely absent in DLAs, the MW results
 become closer
to     the SMC results for the $r^\prime$ band. 
 
For each combination of extinction type  and photometric band   
we derived the best-fit values of the parameters 
% $\beta$, $\alpha$ and $Z_\ast$ 
via   $\chi^2$ minimization
in the intervals 
$1.0 \leq \beta \leq 2.0$, 
$-1.0 \leq \alpha \leq 0.0$, and
$-1.0 \leq \log (Z_\ast/Z_{\sun}) \leq +0.5$.  
%
The best-fit parameters are listed in Table 1. 
%
We made sure the minimum is unique
by careful inspection of the variation of $\chi^2$ 
in the specified parameter space. 
The typical fit errors 
of $\beta$, $\alpha$ and $\log (Z_\ast/Z_{\sun})$
are $\simeq \pm 0.12$, $\sim \pm 0.25$  and $\simeq \pm 0.15$ dex,
respectively. 
 % revision 3
% {\bf
% The   total error
% on $\log (Z_\ast/Z_{\sun})$, including the uncertainties of the
% empirical distributions,  
% is 
% %$\simeq \pm 0.2$, $\sim \pm 0.4$ and 
% $\simeq \pm 0.3$ dex.
% % respectively. 
% }
 %


In Fig. \ref{fits_mw} % and \ref{fits_smc} 
we show the  true and biased distributions (dashed and solid lines) 
of the  best-fit solution  for the   case of
MW-type extinction and $g^\prime$ band. 
The empirical distributions (circles with error bars)
are well reproduced by the computed biased distributions. 
Similar results  are found for all the   cases
considered in Table 1.
%In panel (c) we display the $f_{N_\mathrm{Zn}}$ and
%$f^\mathrm{b}_{N_\mathrm{Zn}}$  distributions resulting from the application
%of the procedure, without no attempt to match 
%the corresponding empirical $N_\mathrm{Zn}$ distribution.  
%The comparison between adopted and observed distributions
%for the best-fit solution is shown in Figs. \ref{fits_mw}
%and \ref{fits_smc} ,
%respectively. 




\begin{figure*}
   \centering
 \includegraphics[width=7.5cm,angle=0]{1570f6a.eps} 
 \hspace{0.5cm}
  \includegraphics[width=7.5cm,angle=0]{1570f6b.eps} 
% \hspace{0.2cm}
%  \includegraphics[width=5.7cm,angle=0]{plotfitZn_mw.eps} 
  \caption{  Frequency distributions of 
  of \ion{H}{i} column densities (left panel) and metallicities (right panel)
  in DLAs. 
%Input parameters %for the intrinsic distributions
% are indicated in the labels.
%   $\beta = 1.5$; [Zn/H]$_\circ = -0.7$ dex; $\sigma(\mathrm{[Zn/H]}) = 0.5$ dex.
Dashed lines:   true distributions.
Solid lines:  distributions biased by  quasar obscuration.
% computed  for 
%the limiting magnitude $m_\ell$ of the corresponding surveys. 
Circles: empirical distributions obtained from DLA surveys.  
%  {\bf a)} $N_\mathrm{H}$ distribution;  $m_\ell=19.5$.
%   {\bf b)}   metallicity distribution;  $m_\ell=19.0$.
%     {\bf c)} $N_\mathrm{Zn}$ distribution;  $m_\ell=19.0$.
By imposing that the biased distributions best fit the observed ones,
we infer the true distributions (see Section 4.4). The solution shown in the figure was
derived for DLAs with a MW-type extinction and quasars observed
in the $g^\prime$ band.
  }
         \label{fits_mw}%
 \end{figure*}

The effect of DLA extinction on the shape of
distribution function of quasar apparent magnitudes 
%is generally modest, at least up  $m \sim 21$ in the visual.
%
% after the \ion{H}{i} column density
%and metallicity distributions are specified. 
%%   
can be seen in Fig. \ref{plotQSOmag},
where we show  
%{\bf
the adopted distribution $n_m \equiv n(m;z)$ (dashed line)
and the predicted biased distribution 
$n^\mathrm{b}_m \equiv n^\mathrm{b}(m;z)$ (solid line)
%}
for the $g^\prime$ band and MW-type extinction.     
%
The agreement between  $n^\mathrm{b}_m$ and
the empirical distribution (diamonds) demonstrates the    
capability of the procedure to account for the bias
in a self-consistent way. 
%%%%%% START REVISION 2
%{\bf
The results are very stable for variations of $\pm 0.1$
of the parameter
$\overline{\mathcal{F}}_1$ around the adopted value
$\overline{\mathcal{F}}_1 = 0.5$. 
%
Fig. \ref{plotQSOmag} shows that 
the obscuration effect does not change substantially
the shape of the distribution of apparent magnitude of the quasars, at least
within the current values of limiting magnitude of the surveys.
%
This is due to the fact that a large fraction of DLAs has   relatively
low \ion{H}{i} column density, as shown in Fig. \ref{fits_mw} and, as a consequence,  
 low extinction. 
%
The modest bias of the quasar statistics
does not imply that the bias is unimportant
for the DLAs statistics, as we discuss below. 
%}
%%%%%% END REVISION 2

%
%
Fig.  \ref{fits_mw}{\bf a} is reminiscent of Fig. 8 by FP93
where these authors compare the model and empirical distributions of $N_\mathrm{H}$ and
$k$. In fact, the dust-to-gas ratios $k$ is defined by FP93 as an extinction per unit
\ion{H}{i} column density and is therefore almost equivalent to a metallicity, as one
can see in  our Eq. (\ref{AlamHzeta}). 
The fact that
the metallicity distribution is much better constrained
by the observations than the  $k$ distribution   is an advantage of our approach.  
In FP93 the model distribution of $N_\mathrm{H}$  is derived 
using an exponential \ion{H}{i} radial profile;
in our work the model
% the conversion from adopted to biased distributions of $N_\mathrm{H}$ and
%metallicity $Z$ is done
%taking into account self-consistently the adopted distributions.
is   meant to be the simplest analytical approximation
of the true distribution in the range probed by the
observations. % with the lowest number of free parameters. 
%%%%%%%%%%%%%%%% }
%[end revision]


\subsection{The \ion{H}{i} column density distribution}
 
From the analysis of  the \ion{H}{i} column density distribution
we derive two main results.  
First, the typical value of  $\beta$ that we find, $\beta \simeq 1.5$, %in all cases 
is remarkably
similar to that measured in quasar absorbers of lower column 
densities, not affected by obscuration bias
(Tytler 1987; Petitjean et al. 1993; Storrie-Lombardi \& Wolfe 2000).
%
%[start revision]
%%%%%%%%%% {\bf 
Second, the   biased distribution
$f^\mathrm{b}_{N_\mathrm{H}}$ 
derived from a simple power law
successfully reproduces  
the shape of the empirical distribution. 
%%%%%%%%%%%%%%}
%[end revision]
% usually fitted
%with the more complex Schechter-type function (\ref{schechter}). 
%
Both results indicate  that the fast decline of the number of absorbers
observed in the DLA regime is due to the obscuration effect,
the true distribution of column density being instead consistent
with a simple power law, as for the majority of quasar absorbers. 
%
Quite interestingly, the typical
values of $\beta$ that we derive are intermediate between the predictions of
the exponential profile model of FP93, $\beta \sim 1$, and the values
expected from the mass-spectrum of interstellar clouds, $\beta \ga 2.5$
(see Section 5.2.2).  This seems to indicate that both effects should
be taken into account in the full theoretical treatment of the true distribution
$f_{N_\mathrm{H}}$.  
  The hydrodynamic simulations of DLAs by Cen et al. (2003)
yield a slope slightly below 2, broadly consistent with our results.  

 %[start revision]
%%%%%%%%%%%%%% {\bf 
The approximation of the true distribution with a simple power law   
 should not be extrapolated
 in the range of % extremely high
 column densities not sampled by the observations
 ($\log N_\ion{H}{i} \gsim 22$). 
An intrinsic faster decline in that range must occur in order to avoid
an infinite value of $\Omega_\mathrm{DLA}$ (see Section 6.5). 
A faster decline is expected  in the exponential profile model  
of disk gas distribution when  $N_\ion{H}{i}$
approaches the central value $N_{\bot 0}$ (FP93).
Also the onset of physical mechanisms 
specific of very high density environments 
may induce a sharp drop of the distribution  at
 very values of  $N_\ion{H}{i}$ (Schaye 2001). 
The present results suggest that the genuine fast decline    
of the $N_\ion{H}{i}$ distribution may lie beyond the range
probed by the observations.  
In principle, one could model this intrinsic decline with a Schechter function,
but the  position of the "knee" would be unconstrained.  
In the following we introduce        
an upper cutoff in the column density distribution  
when we need to estimate quantities affected by the
high end of the $N_\ion{H}{i}$  distribution. 
%%%%%%%%%%%%%%%%%}
%[end revision]
 
%

\subsection{The metallicity distribution}
  
 % revision 3
%
The results on the metallicity distribution depend on the adopted model
for the shape of the distribution. As discussed in Section 5.3, we
believe that the results obtained from a Schechter function
are more conservative and reliable than those derived from other functions
(polynomials, lognormal or other functions parametrized in terms
of "centroid" and "width"). 
In addition to the considerations given in Section 5.3 we note here that
the position of the   "knee"  $Z_\star$ is sufficiently well  constrained
by the observational data (Fig. \ref{fits_mw}{\bf b}).
%Since the decline of the Schechter function is extremenly fast after the "knee",
%the obscuration of missed DLAs predicted by our method in the metallicity
%range not sampled by the observations is certainly small. 
%
%For instance, for a lognormal function   we find  
%a best-fit solution centered at [Zn/H]$= -0.1$ dex
%for a MW extinction curve and obscuration estimated in the $g^\prime$ band.
%However, as we explain in Section 5.3, the results based on a lognormal function
%(and other functions)
%are less conservative and reliable than those based  
%  on a Schechter distribution, adopted here (Table 1).
%%
%%because half of the lognormal curve covers
%%an interval of metallicity currently unconstrained by the
%%observations. 
%The Schechter distribution, in addition of being an  
%educated guess of the metallicity distribution of DLAs, offers
%the following advantages: (1)

%(2) the very   fast decline  at  $ Z > Z_\star$, guarantees that
%the number of DLAs predicted to exist in the range not constrained
%by the observations is small;
%(3)  the predicted behaviour at  high metallicities, where the obscuration
%is important, is not affected
%by our poor knowledge of the low end of the metallicity distribution. 

The main result of the analysis of the metallicity distribution
is that a non-negligible fraction of DLAs   exist
with near solar metallicity.  
%
%%%%%%%%%%%%%}
%[end revision]
 % 
In previous work
the metallicity distribution of DLAs has been
compared with that  of different stellar populations of the Milky Way  
to cast light on the nature of DLA galaxies. 
Pettini et al. (1997) found that 
the  [Zn/H] distribution observed in DLAs does not
resemble that of any Milky-Way population.
% a part from
%a marginal overlap with the distributions of halo stars
%and thick disk stars.  
%
The true metallicity distribution of DLAs inferred in the present work
changes significantly  these conclusions.
%
In Fig. \ref{metdis} we compare the DLA distributions obtained
for the best-fit solutions listed in Table 1 with the  stellar
distributions published by Wyse \& Gilmore (1995)
for the thick and thin disk of the Milky Way
(the same template used by Pettini et al. 1997). 
%
One can see that the true DLA distribution  
envelops   those   of both disk populations. 
%
This result is consistent with the paradigm that DLA galaxies
are progenitors of present-day disk galaxies, even though
there is plenty of room in the distribution for contributions
from metal-poor galaxies. 
%
We cannot derive more information from these results 
because the exact shape of the true distribution is  
uncertain, particularly in the tails, owing to the
poor statistics of the surveys. 
%
In any case, the present results bring fresh support to
the most commonly accepted paradigm on the
nature of DLA galaxies, suggesting that most disks
%[start revision]
%%%%%%%%%%%%%{\bf 
may already be
%%%%%%%%%%%%}
%[end revision]
in place at $z \simeq 2.3$, 
consistent with what happened in the Milky Way.   

  
 


\subsection{The total obscuration fraction}

%The fraction of lost DLAs is a key indicator of the importance of the bias. 
In Table 1 we list the total obscuration $\Phi_{22}$ 
computed adopting $N_\ion{H}{i}=10^{22}$ atoms cm$^{-2}$
as upper limit of integration\footnote{
The highest measured column density in a DLA system
is currently $N_\ion{H}{i}=10^{21.85}$ atoms cm$^{-2}$  (Prochaska et al. 2003).
Column densities up to $10^{22}$ atoms cm$^{-2}$ are found 
in the Milky Way ISM (Dickey \& Lockman 1990). 
}
in  Eq. (\ref{BigPhi}). This is equivalent to truncate
the \ion{H}{i} distribution  in the range of column densities not probed
by the observations. 
The true obscuration factor might be higher than
$\Phi_{22}$ if DLA systems exist also beyond such limit. 
%$N_\ion{H}{i}=10^{22}$ atoms cm$^{-2}$.
An estimate of this uncertainty can be appreciated in Table 1, where
we also give $\Phi_{23}$, obtained integrating the  
  distributions up to  $N_\ion{H}{i}=10^{23}$ atoms cm$^{-2}$.
%Typical differences are in the order of ...

The total obscuration fraction   lies in the range
$\Phi_{22} \sim 0.33/0.52$
for the magnitude limit typical of high-resolution surveys
($m_\ell \sim 19.0$).
 For spectroscopic surveys of moderate resolution,
with $m_\ell \sim 19.5$, we obtain $\Phi_{22} \sim 0.27/0.46$. 
These figures are  underestimated by only $\simeq 3/5\%$ if DLAs
exist with the same power law,
up to $N_\ion{H}{i}=10^{23}$ atoms cm$^{-2}$.

The fraction of missing systems that we find is consistent
with the range 0.23 to 0.38 estimated 
by  PF95 at $z \sim 3$. 

Our estimate of total obscuration fraction  can be
compared with  that obtained by Ellison et al. (2001)
from the analysis of an unbiased sample of radio-selected quasars.
By comparing their number density of DLAs, $n(z)$, with that obtained by
Storrie-Lombardi \& Wolfe (2000) from    
  an optically-selected (biased) sample, Ellison et al.
conclude that the unbiased $n(z)$  is about $\sim 50\%$ larger
than the biased one. 
Since the Storrie-Lombardi \& Wolfe survey has a magnitude
limit typical of moderate resolution surveys, this result should be
compared with our obscuration fractions derived for $m_\ell = 19.5$,
which indicate that the true number is between $\sim 40\%$ and $\sim 90\%$
larger than the biased one. The general agreement with the result
found by Ellison et al. represents an important test of validity
of our procedure.  



\begin{figure}
  \centering
 \includegraphics[width=7.cm,angle=0]{1570f7.eps}
\caption{ 
Distribution functions of the logarithmic metallicity
normalized to unity. Curves: 
true distribution of DLAs at $z \simeq 2.3$ inferred
with our method. 
Dashed (dotted) lines:  MW-type (SMC-type) extinction;
red (green): results for the $r^\prime$ ($g^\prime$) band.
% (in black and white the red curves are the two darker ones). 
Histograms: Milky-Way
stars belonging to the thick disk (shaded area)
and thin disk (thick line)
populations. 
}
 \label{metdis}
\end{figure}



\begin{figure*}
  \centering
 \includegraphics[width=7.cm,angle=0]{1570f8a.eps}
  \hspace{0.2cm}
 \includegraphics[width=7.cm,angle=0]{1570f8b.eps}
\caption{ {\bf (a)} Fraction of obscured DLAs  and
{\bf (b)} number density of DLAs predicted as a function of the
limiting magnitude for the best-fit solutions of Table 1. 
Dashed and dotted lines:  MW-type and SMC-type extinction, respectively.
 Red and green: results for the $r^\prime$ and $g^\prime$ bands.
%(in black and white the red curves are the two darker ones). 
%The number density predictions are obtained adopting an unbiased total number
% $n^\mathrm{t}(z)=0.4$.
 Stars with error bars: CORALS results from Ellison et al. (2001). 
%Solid line: $\Phi_{23}$ in the case of highest obscuration
%($g^\prime$ band and SMC-type dust).  
% 
 }
 \label{maglimPhi}
\end{figure*}


\subsection{Number density versus limiting magnitude}

One prediction of the obscuration bias is that the
fraction of missed DLAs must decrease with increasing limiting magnitude
of the survey. 
The variation predicted by our computations  
is shown in Fig. \ref{maglimPhi}a for the different
cases considered in Table 1.  
The   obscured fraction decreases, but does not vanish
at least up to $m_\ell = 21.5$, where $\Phi_{22} \sim 0.1/0.2$. 
This variation of   $\Phi$ versus $m_\ell$ offers an important
observational test % on the existence of the bias,
based on the measurement of the number density
%\footnote{
%The number density is the total number of DLAs divided
%by the total intervening redshift interval covered by the survey. 
%}
 of DLAs, $n(z)$, 
at increasing values of  $m_\ell$   in an unbiased sample. 
Ellison et al. did find an increase of $n(z)$ with $m_\ell$
which supports the existence of the bias. 
%While this general result  supports our predictions,
%the interpretation of the detailed behaviour of $n(z)$ is less clear since
The number density increases in the range   $19 \la m_\ell \la 20$,
but flattens at $m_\ell \ga 20$, while we predict
that it should keep increasing at least up to   $m_\ell \la 21.5$. 
%
However, taking into account the experimental error bars
  there is no real  discrepancy.
To demonstrate this, we plot in Fig. \ref{maglimPhi}b 
   $n(z) = n^\mathrm{t}(z) \, (1-\Phi)$
for an assumed total density $n^\mathrm{t}(z)=0.4$. 
One can see that these predictions agree with the measurements
of $n(z)$ performed by Ellison et al. 
If  $n^\mathrm{t}(z)=0.4$, the true number of
systems   would be $\sim 90\%$ larger  than the   biased $n(z)$ 
measured by Storrie-Lombardi \& Wolfe.  
This figure of obscuration is still consistent with our estimate.
The only "disagreement" with Ellison et al. would be on the interpretation
of the $n(z)$ versus $m_\ell$ plot, for which we claim
that there is a steady increase rather than a plateau.  
More stringent measurements are crucial for clarifyng
this issue.  
%
It is clear, in any case, that the combination of our treatment
of the bias with studies of unbiased samples offers 
a powerful tool for a quantitative estimate of the obscuration effect.
 
%
 
 
 
\subsection{ The gas content of DLAs }

By using our  best-fit $N( \ion{H}{i})$   distribution functions  
 % for the MW-type and SMC-type extinctions
we can   compute the total contribution of the gas in DLAs
to the critical density of the Universe, 
\begin{equation}
\Omega_\mathrm{DLA} =
{ H_0 \mu \, m_\mathrm{H} \over c \, \rho_\mathrm{crit}} \,
\int^\infty_{N_\mathrm{min}} 
N_\mathrm{H} \, f_{N_\mathrm{H}} \, d N_\mathrm{H} ~.
\label{Omega}
\end{equation}
If $f_{N_\mathrm{H}} $ is a power law with $\beta < 2$, as in our case, 
the integral diverges and we must introduce an upper cutoff.
In Table 1 we give the values $\Omega_{22}$ and $\Omega_{23}$
obtained integrating up to $N_\mathrm{H} = 10^{22}$ and $10^{23}$
atoms cm$^{-2}$, respectively. The value $\Omega_{22}$ 
is conservative since the true distribution $f_{N_\mathrm{H}}$ may
extend above  $N_\mathrm{H} \sim 10^{22}$ cm$^{-2}$.
The values that we find for   different    input parameters
lie in a relatively narrow range, namely 
$1.9 \times 10^{-3} \leq  \Omega_{22} \leq 3.1 \times 10^{-3}$.

To estimate the missed fraction of \ion{H}{i} mass, we compare these values
of $\Omega_{22}$ with  
% measured without taking into account the bias,
$ \Omega^\mathrm{obs}_\mathrm{DLA} \simeq 0.9 \times 10^{-3}$,
the  biased  value   obtained  integrating our   \ion{H}{i} sample 
% at $1.8 \leq z \leq 3$
with the commonly adopted Schechter function   (\ref{schechter}).   
Since   $\Omega_{22}$ is a conservative estimate,
this comparison indicates that at $z \sim 2.3$
the true comoving mass of DLAs is at least
a factor of 2 higher than that obtained by magnitude-limited
surveys.  

The values of $\Omega_{22}$ that we derive agree well  
with the value $ \Omega_\mathrm{E01} \simeq 2.6 \times 10^{-3}$
measured by Ellison et al. (2001) from the analysis of their "dust-free" sample. 
 In addition, they agree with the value 
$\Omega_\mathrm{DLA} \simeq 3 \times 10^{-3}$
derived at $z \simeq 2.3$ by Cen et al. (2003)
 from their hydrodynamic simulations. 


% revision 3 next two pars eliminated
%
%Should DLAs exist beyond the observational limit
%$N_\mathrm{H} = 10^{22}$ atoms cm$^{-2}$,  
%the total gas content implied by our computations would be larger 
%than $\Omega_{22}$. 
%However, our extrapolated values  $\Omega_{23}$,
%listed in Table 1, are
%   larger than allowed by the CORALS data
% and by the hydrodynamic simulations.  %$ \Omega_\mathrm{E01}$
%We conclude that
%the distribution of   \ion{H}{i} column densities must drop
%with a steeper slope
%between $10^{22}$ and $10^{23}$ atoms cm$^{-2}$. 

%The true $\Omega_\mathrm{DLA}$ is probably slightly    
%higher than   $\Omega_{22}$. This would give a value
%slightly higher than $ \Omega_\mathrm{E01}$,  
%in line with the slight excess of $n(z)$ that we have discussed above  
%(Fig. \ref{maglimPhi}b). 

%
% end revision 3

%On the other hand,  the CORALS data, 
%do not allow for much higher values of $\Omega_\mathrm{DLA}$
%and we  use this result to put an upper limit to the
%  cutoff of $f_{N_\mathrm{H}}$. 


% 
 

   
\subsection{ The metal  content of DLAs }

The column-density weighted metallicity of DLA systems 
is used to measure the degree of metal 
enrichment of the population of DLAs as a whole (Pettini et al. 1999) and  
to estimate the mean cosmic metallicity of the high-redshift
Universe (Pei \& Fall 1995; Cen et al. 2003).    
The expression commonly used in literature to measure the weighted metallicity
from a finite set of column densities, 
 $ <\! (\mathrm{Zn/H})_\mathrm{DLA} \!> =
{  ( \sum_i N_{\mathrm{Zn},i}  ) /  (\sum_i N_{\mathrm{H},i} )}$,
can be put in the form
%
\begin{equation}
<\!  ( \mathrm{Zn} / \mathrm{H}  )_\mathrm{DLA} \!> =
 %
{  
\int N_\mathrm{Zn} ~ f_{ N_\mathrm{Zn} } ~ d N_\mathrm{Zn}  
 \over
\int N_\mathrm{H} ~ f_{ N_\mathrm{H} } ~ d N_\mathrm{H}  
}~,
\label{MeanZnH}
\end{equation}
to measure the same quantity from the frequency distribution functions
$f_{N_\mathrm{Zn} }$ and $f_{N_\mathrm{H} }$ normalized to unit area. 
If the metallicity $Z$ and the \ion{H}{i} column density are independent
variables, as we assume in our mathematical formulation, 
then  it is easy to show that
$<\! (\mathrm{Zn/H})_\mathrm{DLA} \!>$ equals the expectation value 
$<\! Z \!> =  \int Z ~ f_Z ~ d\, Z $ estimated from the   
distribution $f_Z$ normalized to unit area. 


%%    revision 3 
%{\bf 
%The mean value $<\!\! Z \!\!>$  
%is very sensitive to the behaviour of the metallicity distribution at   high $Z$,
%but not at low $Z$ (the low-metallicity tail of the distribution
%gives  a little contribution to the mean metallicity). 
%%
%If we   model
%the metallicity distribution with an ideal analytical function 
%in which a parameter $p_l$ controls the  
%  low-metallicity end   and a parameter 
%$p_h$ constrols the     high-metallicity end,
%we can estimate $<\!\! Z \!\!>$  from $p_h$, without introducing
%the uncertainties inherent to the  determination of $p_l$.
%%
%This is important because
%the paucity  of low metallicity measurements
%makes uncertain the determination of the low-metallicity end. 
%%  
%These considerations provide a further reason for choosing
%a Schechter function, rather than a lognormal function,
%as a model of the metallicity distribution 
%(see Section 5.3). 
%%
%In fact, for the Schechter function (\ref{MetalDistFunction}), 
%the parameter $Z_\star$ controls the
%the behaviour at high $Z$ while    $\alpha$
%controls the extension of the distribution at low metallicities.
%%
%Therefore the poor statistics of the measurements at
%low metallicity does not affect the
%determination of $Z_\star$ and hence of  $<\!\! Z \!\!>$.  
%%
%For a lognormal distribution, instead,  the mean value $<\! Z \!>$  
%depends both on the centroid and  the width of the distribution.
%%
%Since     these parameters are   affected by the behaviour of the
%empirical distribution at all metallicities, including the low-metallicity end,
%the determination of $<\! Z \!>$   using a lognormal function 
%is   affected
%by the poor statistics at low metallicity
%and therefore less reliable than in the case of the Schechter function. 
%%
% }
 

In Table 1 we list the     mean metallicity, normalized in
the usual form $\log (<\! Z \!>/ Z_{\sun})$,  
%%%%%%%%%%%%{\bf 
obtained for a Schechter metallicity distribution.
%%%%%%%%%%%%%% }
%[end revision]
The mean metallicity
lies between $-0.44$ and $-0.27$ dex below
the solar level for the different dust models and photometric bands
considered in Table 1. 
% revision 3
Considering the total error of $\sim -0.3$ dex in our estimate
(fit errors plus statistical errors of the empirical distributions)
we conclude that the mean metallicity is    
  $ \gsim -0.7$ dex at 1 $\sigma$ level.
%
These values are   higher
 than the direct measurements of weighted metallicity of DLAs,
 which give  $\sim -1.1$ dex 
at the typical redshift of our sample (Pettini et al. 1999; Prochaska et al. 2003). 
%
%
This large difference is probably be due to the fact 
that the weighted metallicity is extremely
sensitive to  the number of  high column-density
DLAs  included in the average, which are  
the DLAs  more affected by   obscuration.   
%
% % revision 3
Our results are marginally consistent   with the mean
weighted metallicity  
  $\left[ <\!  \mathrm{(Zn/H)_{DLA}} \!> \right] = -0.88 \pm 0.21 $ dex obtained from the  
  CORALS metallicity survey (Akerman et al. 2005).


 

    \begin{figure*}
   \centering 
     \includegraphics[width=5.7cm,angle=0]{1570f9a.eps} 
 \hspace{0.2cm}
  \includegraphics[width=5.7cm,angle=0]{1570f9b.eps} 
 \hspace{0.2cm}
  \includegraphics[width=5.7cm,angle=0]{1570f9c.eps} 
  \caption{  Fraction of obscured DLAs as a function
  of  {\bf (a)} \ion{H}{i} column density,
  {\bf (b)} metallicity $Z$, and {\bf (c)} \ion{Zn}{ii} column density
  for the best-fit solutions of Table 1.
 %
 Dashed and dotted lines:  caption as in Fig. \ref{maglimPhi}.
% MW-type and SMC-type extinction, respectively.
% Red and green curves: results for the $r^\prime$ and $g^\prime$ bands (the red curves are
% the two lower ones in each panel).     
  }
 \label{plotPhi}%
    \end{figure*}




Quite interestingly, the high mean metallicity that we derive
may help to solve the "missing metal" problem 
pointed out by  Wolfe et al. (2003) 
in their work on the Star Formation Rates (SFRs) in DLAs: 
%
the mass of metals produced by $z \simeq 2.5$, inferred from their SFRs,
is 30 times larger than detected  in absorption in DLAs. 
%
%Our mean metallicity is 5 to 6 times larger than the canonical
%value obtained from direct measurements, but could be even higher if
%%
%the true metallicity distribution drop less sharply than
%the Schechter  function (\ref{MetalDistFunction}).
%%[start revision]
%%%%%%%%%%{\bf
%A similar estimate of the obscuration effect on
%the mean metallicity of DLAs  at $z \sim 2.2$  
% was  predicted by Pei \& Fall (1995).  
 %%%%%%%%%%%%%% }
%[end revision]
    
  


\subsection{Observational thresholds due to obscuration bias }

Our method allows us to estimate, for the first time,  
the  fraction of obscured  DLAs as a function
of \ion{H}{i} column density and metallicity $Z$, and also as
a function of the total zinc
column density. 
These  obscuration fractions
$\phi_\ell(N_\mathrm{H})$, $\phi_\ell(Z)$
and $\phi_\ell(N^\mathrm{t}_\mathrm{Zn})$, shown in Fig. \ref{plotPhi},
should only be interpreted in a statistical sense. However, they
provides us a powerful indication of how the statistical
distributions of DLAs are distorted by the obscuration bias.
In particular, they provide a means of quantifying the existence of
observational cutoffs induced by the bias itself. 
 
The analysis of $\phi_\ell(N^\mathrm{t}_\mathrm{Zn})$, shown in panel {\bf (c)},
gives a quantitative explanation for the lack of DLAs systems above the
obscuration threshold in \ion{Zn}{ii} column density
originally proposed by Boiss\'e et al. (1998). We estimate 
that    over $60\%$ of DLAs are missed
at $N^\mathrm{t}_\mathrm{Zn} = 10^{13.2}$ atoms cm$^{-2}$,
and this fraction rises rapidly to   $\sim 95\%$ if the 
$N^\mathrm{t}_\mathrm{Zn}$ increases by $\simeq 0.5$ dex.  
The threshold is rather independent of the adopted input parameters.
%% FINAL ADDITION
Only for  peculiar
extinction curves without UV rise (gray extinction) we may expect that
the threshold can be crossed. 
%%%
% revision 3 next sentences eliminated
%The cutoff effect is reinforced by the fact that the observed column density
%slightly underestimates $N^\mathrm{t}_\mathrm{Zn}$
%due to dust depletion (Vladilo 2004). As a consequence,  
%  a true column density $N^\mathrm{t}_\mathrm{Zn}$
%slightly  above the threshold may 
%yield a measurement $N_\mathrm{Zn}$ slightly below the threshold.  
%This depletion effect is small on average ($\simeq 0.1$ dex), but it can
%be higher when the quantity of dust is relatively high and
%the extinction bias is more important. 
% end revision 3

%[ start revision]
%%%%%%%%%%%%%%%%{\bf 
We can use our results to estimate how many DLAs should be observed
in an unbiased survey in order to  detect cases with high metal column density. 
From the typical distribution $f_{N_\mathrm{Zn}}$ that we infer,
we estimate that the number of DLAs      
  in the range between the detection limit
and the Boiss\'e's threshold ($11.5 \lsim \log N(\ion{Zn}{ii}) \lsim 13.1$) 
is about 24 times larger than the number of systems
in a range of similar extension above the threshold ($13.1 < \log N(\ion{Zn}{ii}) \lsim 14.7$). 
This means that one is not guaranteed to detect one DLA above the threshold 
even with $\sim 30$ unbiased spectroscopic observations. 
This is currently an observational challenge because the high extinction of the DLAs above
the threshold makes very hard to perform high-resolution spectroscopy, even
for the quasars which lie at the bright end of the true distribution
of apparent magnitudes. 
%%%%%%%%%%%%%%%}
%[end revision]


 
A natural consequence of the cutoff at high metal column densities,
in conjunction with the artificial DLA threshold 
$N_\ion{H}{i} \geq 10^{20.3}$ atoms cm$^{-2}$,
is the existence of an observational cutoff at high metallicities. 
From panel {\bf (b)} one can see that about $90\%$ of systems are
missed at solar metallicity and it would be practically impossible
to detect systems with oversolar metallicity,  
if they exist. Also worth of mention is
the fast rise of the obscuration with increasing metallicity, which
%$\log (Z/Z_{\sun})$ 
distorts dramatically our perception of the metallicity 
distribution of DLA systems. The fact the different curves plotted
in panel {\bf (b)} lie close to each other indicates that
these results are rather
independent of the parameters adopted. 

%The  obscuration fraction of metallicity
% $\phi_\ell(Z)$, shown in panel b, increases
%quite fastly in the interval of metallicities typical of DLA systems, 
%and approaching $\sim 100\%$ obscuration at about solar
%metallicity, where a non-negligible fraction of systems still exists,
%as indicated by the intrinsic distribution (dotted lines). 
%The obscuration is higher for an SMC-type extinction, but the rise
%is faster for a MW-type extinction, indicating that  the
%obscuration bias 

 
%
Also the distribution of \ion{H}{i} column densities is
severely distorted by the extinction bias, as shown in
panel {\bf (a)}. The curves of $\phi(N_\mathrm{H})$ obtained
for different parameters are not tightly close to each other, but
the general behaviour is quite similar in all cases. 
The obscuration  starts from a low value 
in the sub-DLA   regime and rises steadily  
over the DLA regime,  
approaching  $\sim 80\%$  at $N_\ion{H}{i} \sim 10^{22}$ atoms cm$^{-2}$.
This is the approximate value of the observational threshold
for detection of high column density DLAs. 
The rise of    $\phi(N_\mathrm{H})$,
in conjunction with the genuine decrease of the true   distribution
$f_{N_\mathrm{H}}$,     provides a natural explanation
for the  non-detection of DLAs 
at $N_\ion{H}{i} \ga 10^{22}$ atoms cm$^{-2}$.  
%
This  makes it unnecessary to invoke the sudden transition
from atomic to molecular hydrogen proposed by Schaye (2001)
in order to explain  this empirical cutoff.   
%
In any case, if  fully molecular clouds exist as 
predicted by Schaye, they  would   be
characterized by a high column density of dust grains.
The fact that  quasar absorbers with $N(\mathrm{H_2}) > N(\ion{H}{i})$
have not yet been detected may indicate that also this type of absorber
is missed due to    obscuration.  
 
    
\section{Summary and conclusions}
 
 
 Starting from
  the theoretical relation between the  interstellar extinction, $A_\lambda$,
and  the  column density of a volatile metal, $N_{\mathrm{X}_v}$,
we investigated the detailed 
relation between $A_\lambda$ and  $N_\mathrm{Zn}$ 
  in DLAs. 
%
%isolating the contributions of interstellar abundances, 
%depletions and dust grain properties. 
%% (Eqs. 4, 5 and 6).  
%We calibrated this relation using local ISM measurements of $A_\lambda$
%and zinc column densities. 
%We find that $A_\lambda$ increases exponentially with $\log N_\mathrm{Zn}$,
%with a characteristic turning point $\log N_\mathrm{Zn} \simeq 13.2 $ dex   above which the extinction undergoes a fast rise, acting as a barrier against
%detection of faint sources (Fig. 1). 
%%This turning point  is remarkably similar to the  $\log N_\mathrm{Zn}$ 
%%threshold for detection of DLAs  tentatively attributed to quasar obscuration
%%  by Boiss\'e et al. (1998).
%
%We investigated
%in detail the 
We derived in this way  the Eq. (\ref{AlamHzeta}), which gives
the quasar extinction in the observer's frame  as a function of 
the DLA hydrogen column density, $N_\mathrm{H}$, metallicity,
$Z = N_\mathrm{Zn}/N_\mathrm{H}$, fraction in dust of iron, $f_\mathrm{Fe}$, and a factor $G$, 
which describes   the dust grain properties. 
%
 For the first time, the metallicity evolution of the 
dust-to-metal ratio in DLAs is explicitly accounted for in this type of relation,
using an expression  $f_\mathrm{Fe} = f_\mathrm{Fe}(Z)$ 
obtained from a previous study of depletions (Vladilo 2004). 
%
We argue that the factor $G$ may not   vary
substantially in   interstellar environments 
with moderately low metallicity ($Z \ga 0.1 \, Z_{\sun}$)
and derive
a value of 60\% the Milky-Way value   for SMC-type dust. 
%
Possible variations of $G$ at the early stages of chemical evolution
do not affect the extinction   
because $f_\mathrm{Fe}(Z)$ vanishes
when $Z \ll Z_{\sun}$.
% and the predicted extinction is, in any case,
%negligible. 
% using recent extinction
%measurements of the SMC (Gordon 2003).   
% revision 3 next sentence eliminated
%For  DLAs  with $Z \ga 0.1 \, Z_{\sun}$
%and $z \ga 1$ 
%the extinction per unit column density of zinc atoms
%predicted  by Eq. (\ref{AlamHzeta})  
% is similar, or even larger,
%than that typical of the Milky-Way ISM. 
%The   extinction    
%% increases exponentially with $\log N_\mathrm{Zn}$  and 
%attains $\ga 1$ magnitude
%at redshift $z \ga 2$   when   
%$\log N_\mathrm{Zn} \ga 13.5 $ dex (Fig. 2). 

We searched for empirical evidence of the rise
of the   extinction with  metal column density in DLAs. 
We did find an increase of the quasar magnitude with $N_\mathrm{Zn}$, 
consistent with this expectation
%for the sub-sample of DLAs with $N_\mathrm{Zn}$  measurements 
(Fig. \ref{Vmaghist}). 
 
%[start revision2]
%%%%%%%%%%%%%
%{\bf
Starting from Eq. (\ref{AlamHzeta}), 
we derived a mathematical formulation aimed at  estimating 
how the frequency distributions
of \ion{H}{i} column densities and metallicities of a sample of DLAs
are biased as a result of the obscuration 
generated by the DLAs of the same sample.
We ignore % in this way 
%the extinction due to DLAs 
%with Ly\,$\alpha$ absorption below the atmospheric cutoff.
%%Our mathematical formulation computes the  extinction
%%due the bulk of known DLAs, in the redshift interval $1.8 \lsim z \lsim 3.0$.
%%Contributions from DLAs at $z \lsim 1.8$ are almost unconstrained
%%from the observational point of view and are note considered.
the   obscuration due to low-redshift DLAs 
along the same line of sight, showing that this contribution
 is relatively low. 
%This obscuration   is expected to be relatively
%low because of the redshift dependence of the extinction (12) and because
%of the intrinsic decline in the number density of DLAs at low $z$.  
%Other sources of extinction are not considered. 
% revision 3 next sentence eliminated
%The core of the  formulation is represented by   Eqs. 
%  % (\ref{HZzdistr_paper}),
% (\ref{fb_H_paper}) and (\ref{fb_zeta_paper}),
%% and (\ref{BigBHZz_paper}), 
% which  give the ratio of biased over
%unbiased frequency distributions of column densities and
%metallicities at a given redshift. 
% revision FINAL
At variance with the formulation of PF95, where
a constant dust-to-metal ratio was adopted for all DLAs,   we adopted
a
metal-dependent dust-to-metal ratio $f_\mathrm{Fe}(Z)$. 
% end revision FINAL
We cross-checked the validity of our formulation making use
of the equations derived by FP93 for the case of a power law
quasar luminosity function and constant dust-to-gas ratio (Section A.1.2).

%Eq.s (\ref{fb_H}) and (\ref{fb_zeta}), which relate
%the unbiased distributions
%$f_{N_\mathrm{H}}$ and $f_Z$
%%of \ion{H}{i} column densities, , and metallicities,
%to the
%biased distributions, $f^\mathrm{b}_{N_\mathrm{H}}$
%and $f^\mathrm{b}_Z$. 
%
We presented a practical procedure for 
%solving Eqs.
%(\ref{fb_H_paper}) and (\ref{fb_zeta_paper}) 
%and 
recovering the unbiased distributions of column densities and
metallicities, $f_{N_\mathrm{H}}$ and $f_Z$, 
using the empirical distributions obtained from magnitude-limited surveys.
%}
%[end revision2]
%%%
%%%%
%$f^\mathrm{obs}_{N_\mathrm{H}}$
%and $f^\mathrm{obs}_Z$.  
%
The unbiased distributions are modelled with simple analytical
expressions in parametric form.    
% revision 3 next sentence eliminated
%The
% parameters are determined  via $\chi^2$ minimization
% of the residuals   between the empirical   
%and the computed biased distributions.  
% end revision 3
%
% revision 3 next sentence eliminated
%The method does not require a knowledge of the intrinsic
%luminosity function of quasars, but only of the frequency distribution
%of apparent magnitudes. 
% end revision 3
The bias induced by   the extinction
on the quasar 
magnitude distribution is accounted for self-consistently by the procedure. 
 %At variance with previous work, % proposed to estimate the obscuration bias,
%our 
% revision 3 next sentence eliminated
%In addition, the method does not require    making assumptions
%on the geometrical distribution of the neutral gas
%in DLA galaxies.  
% end revision 3
% from theoretical
%considerations. Therefore our results do not depend on the choice of a  
%particular model of DLA galaxies. 
A unique characteristic of the method is
the possibility of recovering the true metallicity distribution
starting from an educated guess of the functional form of the distribution.
% revision FINAL
We have shown that using a Schechter function 
for the metallicity distribution appears to provide  more conservative
and reliable results
than other functions (e.g. lognormal or polynomials) for the estimate
of the bias (Section 5.3). 
%


% We applied our procedure to the current sample of $N(\ion{H}{i})$ and $N(\ion{Zn}{ii})$ measurements in DLAs 
%at redshift $1.8 \leq z \leq 3.0$, with mean redshift
%$z \simeq 2.3$, where most of  the
%spectroscopic measurements are presently concentrated. 

%[start revision2]
%%%%%%%%%%%%%
%{\bf
We applied  our method to the   sample of DLAs 
in the redshift interval $1.8 \leq z \leq 3$, 
with mean redshift  $<\!\! z \!\!> \simeq 2.3$,
where the bulk of   spectra are available. 
%
%}
%[end revision2]
% inferred from magnitude-limited surveys.
%
%In this first application we did not attempt
%to disentangle evolutionary effects  because  binning the data in  
%redshift     would have given rather  inaccurate empirical
%distributions.  
%
%
The  extinctions were computed in the $g^\prime$ and $r^\prime$
photometric bands of the SDSS, considering both a 
MW-type and an SMC-type 
%% FINAL ADDITION
average extinction curve.
%%
% revision 3 next sentence eliminated
%The presence or absence of the 2175\,\AA\ extinction bump 
%in DLAs at $z=2.3$ does not
%affect the estimates of the quasar extinction  in these bands.
% end revision 3
%
%%%%%% revision 3
We found that the effect of the obscuration
on the quasar magnitude distribution is modest (e.g. Fig. 5).
On the other hand,
% even ignoring  the extinction at $z <1.8$,      
  the bias plays an important role
in shaping the statistical distributions of DLAs. 
%%%%%%%%% END REVISION
%
% The results of this exercise can be summarized as follows.

The unbiased distribution function of \ion{H}{i} column densities is  
successfully approximated by a power law 
$f_{N_\mathrm{H}} \propto N_\mathrm{H}^{-\beta}$,
with $\beta \simeq 1.5$. 
This simple law, in conjunction with the bias effect, 
is able to fit the observed decline of the   number DLAs
with $N_\mathrm{H}$
%[start revision]
%%%%%%%%%%%%{\bf
   up to $N( \ion{H}{i}) \sim 10^{21.8} $ atoms cm$^{-2}$. 
The faster drop  of the distribution  expected at very high column
densities (FP93; Schaye 2001) probably lies outside the
range currently probed by the observations. 
%%%%%%%%%%%%%%%%%}
%[end revision] 
%the commonly adopted   
%$f_{N_\mathrm{H}} \propto 
%% = {\left( f_\star \over N_{\mathrm{H},\star} \right) } 
%{\left( N_\mathrm{H} / N_\ast  \right)}^{-\beta} \, 
%e^{-\left( N_\mathrm{H} / N_{\ast} \right)} $. 
%
%{\bf
%The  distribution function $f_{N_\mathrm{H}}$ inferred
%with our method is   steeper than that modelled by Pei \& Fall (1993).  
%}
The   slope   $\beta \simeq 1.5$ is remarkably similar to the value 
which characterizes most of quasars absorbers with lower column densities,
for which the extinction bias is negligible. 
%Instead, by fitting our sample with the Schechter function,
%we derive $\beta = 1.05$ [Celine, pls check], which is substantially different from
%the   spectral index  of the other quasar absorbers. 
% revision 3 next sentence eliminated
%
%These results suggest the existence of a continuity in the 
%$N(\ion{H}{i})$ frequency distribution
%of quasar absorbers, which becomes evident only after
%accounting for the selection effects. 
% end revision 3
% 
The value of the slope  suggests that the mass spectrum
of interstellar clouds may play an important role in shaping
the distribution. 
 
 
The observed distribution of metallicities is  
reproduced modelling the unbiased distribution with a 
Schechter function  
$f_Z ~ \propto ~ (Z/Z_\ast)^\alpha ~
 e^{-Z/Z_\ast} $, with $\alpha \simeq -0.4$ 
 and $\log ( Z_\ast/Z_{\sun} ) \simeq -0.15$ dex. 
 %
Once converted into logarithmic scale, this function  
peaks at [Zn/H] $\sim -0.3/-0.4$ dex, indicating that
a non-negligible fraction of DLAs exist with near solar metallicity.
%It is impossible to adopt a simpler analytical expression
% because we have evidence for a genuine rise and subsequent decrease of
% the $f_{\log Z}$ distribution function.
%
% REVISION 3 NEXT SENTENCE ELIMINATED
%The fast decline of the Schechter function at $Z > Z_\ast$  
%minimizes the risk of  overestimating the number of  DLAs  missed
%at [Zn/H] $\ga 0$. 
%
% END REVISION 3
%
%not sampled by the observations. 
%In fact, the "knee" that we find,
%$\log ( Z_\ast/Z_{\sun} ) = -0.15 $  dex, lies 
%in an interval which can be tested by the observations. 
The true metallicity distribution of DLAs envelops the distributions
of the thick-disk and thin-disk Milky-Way  stellar populations.  
%
This result is in line with the paradigm that DLA galaxies
are progenitors of present-day disk galaxies. However, in
the inferred  distribution there is also room for
a relevant contribution from metal-poor galaxies.  

The  mean weighted metallicity that we derive ,
%  $\log(<\!\!Z\!\!> \! /Z_{\sun}) 
 $ \simeq -0.4/ \! -0.3$ dex,    
%which can be directly compared with, but is 
is significantly higher than 
the   column-density weighted metallicity
$<\!\! \mathrm{[Zn/H]} \!\!> = -1.1$ dex
measured at $z \simeq 2.3$. 
% revision 3 next sentence eliminated
%This result shows  
%  the importance of taking into account the
%bias even if the obscuration fraction by number is not exceedingly high. 
% end revision 3
 This high value of mean metallicity may  help
to solve the  discrepancy between the metallicity observed in DLAs
and that predicted on the basis of their SFRs (Wolfe et al. 2003). 

The fraction by number of obscured DLAs, $\Phi$, decreases
with increasing limiting magnitude of the survey, $m_\ell$.
%at redshift $z \sim 2.3$.
%
We obtain    $\Phi \ga 0.4/0.6$ at $m_\ell \la 18.5$, a limit
representative of  high resolution spectroscopic surveys, 
carried out with 4-m class telescopes. 
The corresponding figures
for high-resolution surveys in
10-m class telescopes  are  $\Phi \simeq 0.35/0.55$
($m_\ell \simeq 19.0$).
% revision 3 next sentences eliminated
%The osbcuration is more moderate for $\ion{H}{i}$ surveys of lower 
%spectral resolution,
%for which we estimate   $\Phi \simeq 0.3/0.5$ at $m_\ell \simeq 19.5$. 
%%
%The predicted obscured fraction keeps decreasing with $m_\ell$, 
%but does not vanish
%at least up to $m_\ell = 21.5$, where $\Phi \simeq 0.1/0.2$. 
% end revision 3

% revision 3 next sentences eliminated
%
%The application of our procedure   allowed us to estimate 
%the comoving \ion{H}{i} mass  in DLAs 
%and the mean metallicity at $z \simeq 2.3$. 
%We obtain    $\Omega_\mathrm{DLA} \simeq 2 / 3 \times 10^{-3}$, 
%a comoving mass density at least a factor 2 higher  than 
%the biased $\Omega_\mathrm{DLA}$ estimated by magnitude-limited
%\ion{H}{i} surveys with $m_\ell \sim 19.5$. 
%
% end revision 3



% revision 3
An important result of our work is the explanation (Fig. \ref{plotPhi}c)  of the
observational limit $N_\mathrm{Zn} \sim 10^{13.2}$ atoms cm$^{-2}$ 
(Boiss\' e et al. 1998) in terms of very simple physics (Section 2), with no tuning
of the local dust parameters. 
%  FINAL CHANGE 
Alternative explanations  would require 
  an additional mechanism   able
to produce the exact same limit predicted by the extinction. 
The simplicity of the extinction model  
favours the existence of the obscuration. 
%
The existence of the threshold $N_\mathrm{Zn} \sim 10^{13.2}$ atoms cm$^{-2}$ 
is fundamental for reconciling 
the predictions of galactic models 
(Prantzos \& Boissier 2000, Hou et al. 2001; Churches et al. 2004)
and cosmological simulations (Cen et al. 2003, Nagamine et al. 2004)
with the observations of DLAs. 
% FINAL ADDITION
Only for gray extinction curves, without UV rise,
we may expect that the threshold 
can be crossed. 
%
%   END REVISION 3


%  revision 3  NEXT paragraph eliminated
%%
%To deny the presence of extinction in DLAs one should assume that  
% the factor $G$, 
%which describes the intrinsic properties of dust grains,
%%in Eq. (12), 
%vanishes in DLAs with metallicity similar to that of the SMC
%or the Milky Way. 
%%is $\ll 1$, at variance with the value
%%estimated   in the Milky Way ($G=1$) and in the SMC ($G \simeq 0.6$).
%%While we are not in the position to determine the true value of $G$ in DLAs,
%The price to pay for this arbitrary assumption % $G \ll 1$
%is relatively high. % in terms of economy of hypothesis. 
%In fact, in this case we would need to 
% find alternative explanations for
%(1)  the observed
%cutoff at  $N_\mathrm{Zn} \sim 10^{13.2}$ atoms cm$^{-2}$ 
%and $\log N_\mathrm{H} \sim 10^{22}$ atoms cm$^{-2}$;
%%which are
%%predicted so well by the present analysis (Fig. \ref{plotPhi}c);
%%
%(2)  the increase of apparent magnitude versus
%$\log N_\mathrm{Zn}$ (Fig. \ref{Vmaghist}); 
%%
%(3)  the gradual steepening  
% of the $N(\ion{H}{i})$  distribution in the DLA regime;
%%predicted in a natural way by the extinction effect;
%(4)   the  peculiar   slope of the $N(\ion{H}{i})$  distribution  
%of DLAs, different from that of other optically thick quasar absorbers;
%%naturally explained by the our results yield a spectral index remarkably similar to
%%that of  other quasar absorbers;
%(5)   the peculiar metallicity distribution of DLAs,
% different from  that of any     stellar population;
%(6) the very large amount of missing metals suggested by
%studies of SFRs in DLAs; % (Wolfe et al. 2003);
%%(5) explain why the grain parameters  
%%should conspire to yield strong variations of $G$ 
%%in spite of the general arguments in opposite sense discussed in Section 2.1.  
%(7) the discrepancies between the results of cosmological simulations
%and the actual measurements of DLAs. % (Cen et al. 2003) 
%%
%

% revision 3
%
Our results on the magnitude of the obscuration effect 
are broadly consistent with those 
presented by  FP93 and PF95, 
with some important differences. 
%
The modest bias of quasar statistiscs that we find is at variance with
the predictions of FP93, which allowed up to 70\% of quasars
to be obscured. 
%
Contrary to FP93,  we do not expect a significant contribution
of DLA extinction at redshift $z \gsim 3$
in spite of the   rise of $A_\lambda$ with $z$. 
%
In fact, our model of dust fraction  $f_\mathrm{Fe}(Z)$ vanishes
at very low metallicity so that the extinction  (\ref{AlamHzeta})  
vanishes at $z \gsim 3$, when the metallicity $Z \lsim 0.01 Z_{\sun}$.

% end revision 3


A crucial test of  our results is the comparison with 
studies of radio-selected samples of quasars/DLAs.  
Our determinations are consistent with the estimates
of   obscuration fraction, number density   and $\Omega_\mathrm{DLA}$ 
obtained from the    CORALS
survey (Ellison et al. 2001).
% REVISION 3
Our prediction of mean metallicity is marginally consistent
with the recent estimate of mean weighted metallicity
of the CORALS sample
(Akerman et al. 2005). 

% 
%However, we predict  that the number density of DLAs, $n(z)$,
%must keep increasing with the limiting magnitude,  at least
%up to $m_\ell = 21$. As we show in Fig. \ref{maglimPhi}b, this
%prediction is perfectly consistent with the actual measurements
%of $n(z)$ versus $m_\ell$ obtained so far with the CORALS survey, 
%{\bf provided $n(z) \simeq 0.4$ at $z=2.3$.}
%
%   REVISION 3 NEXT SENTENCE ELIMINATED
%This  broad agreement indicates that
%the amount of extinction, i.e. the factor $G$, 
%has not been grossly overestimated in our analysis. 
% END REVISION 3

% revision 3
Accurate estimates of the obscuration effect will be possible
as soon as the  samples with metallicity
measurements   become sufficiently large for   
precise statistical analysis. 
 % 
%The combination of  our type of analysis  with  
%studies of unbiased surveys
%%, 
%%or with accurate hydrodynamic simulations,  
%will   yield accurate results on the  unbiased  
%properties  of DLAs.

% revision 3 next sentence eliminated
%For instance, in spite of the limited statistics,  
%we have already been able  
%  to constrain the number density of DLAs at
%$z \sim 2.3$,  $n(z) \approx 0.4$ (Fig. 8b), 
%and to predict a drop of
%the column density distribution function in the interval
%$22 < \log N_\mathrm{H }<  23 $ % not directly sampled by the observations
%(Section 6.5). 
% end revision 3

% revision 3 next paragraph eliminated
% The results of our analysis are generally consistent with the
%predictions of  hydrodynamic simulations of evolving
%DLAs in   $\Lambda$CDM  cosmology (Cen et al. 2003). 
%These simulations indicate 
%that the metallicity and the hydrogen column density are not
%correlated, lending support to the hypothesis used to develop our  
%mathematical formulation. 
% end revision 3




%In summary, the quantification of the extinction bias
%provides a natural explanation for 
% (1) the observed
%cutoff at  $\log N_\mathrm{Zn} \sim 13.2$ dex, which is
%predicted so well by the present analysis (Fig. \ref{plotPhi}c);
%(2)  the steepening  of the $N(\ion{H}{i})$  
%distribution in the DLA regime.
% 
% 
 
 
% revision 3 next sentence eliminated
 
%Larger spectroscopic surveys of  DLAs,  with accurately
%defined magnitude limits, will allow us to estimate with    
%accuracy  the true distribution functions of hydrogen column
%density and metallicity at different redshifts.
%The evolution of the true distributions will in turn offer
%fresh clues  for unveiling the nature        
%of   DLAs and their associated galaxies. 
 
% end revision 3

% obscuration fractions,
%comoving mass density and mean metallicities of DLAs.
%Unveiling the true distribution of metallicity  will   
%cast new light on the nature of the high-redshift galaxies
%responsible for DLA absorption. 

 %The parallel improvement of accuracy of the statistical
%measurements obtained  from unbiased, radio-selected surveys
%will provide very stringent tests on the validity of the results
%obtained with the present type of analysis. 
  
%The combination of the present type of analysis with
%independent measurements of $\Phi$ and $\Omega_\mathrm{DLA}$
%will allow us to have a more detailed insight on the true
%distribution of column densities and metallicities of DLAs and hence
%on the nature of the associated high-redshift galaxies. 

 \begin{acknowledgement}
%%%%%%%%{\bf 
This work has benefitted from interactions with  
 Irina Agafonova, Patrick Boiss\'e, Miriam Centuri\'on, Sergei Levshakov, Paolo Molaro, Pierluigi Monaco,
 Sara Ellison and Paolo Tozzi. 
 We warmly thank  
 Arthur Wolfe for his remarks on the first part of the manuscript.
% and     
% Michael Fall for his comments on the mathematical formulation.   
%%%%%%%% }
\end{acknowledgement}


% START REVISION 2
%{\bf
 
\appendix

\section{Mathematical formulation}

\subsection{ True and biased distributions }

%[start revision 2]
%{\bf
%
To derive our mathematical formulation
we define the frequency distributions of DLAs and quasars 
most appropriate for taking into account the causes and effects
of the extinction bias. 
%
To account for the DLA extinction via Eq. (\ref{AlamHzeta})
we are interested in the number   of DLAs 
per unit hydrogen column density and metallicity.
%
To account for  the dimming of the quasar  we are interested
in the number of quasars per unit of apparent magnitude.
The apparent magnitude, $m$, can be directly compared with the limiting magnitude of the observational survey, $ m_{\ell}$. 

%In the formulation of the problem we distinguish between
%"potential DLAs" 
%(any \ion{H}{i} layer with $N(\ion{H}{i}) \geq 10^{20.3}$ atoms cm$^{-2}$
%that lies along a random line of sight with or without a background quasar),
%% DLAs (the potential DLAs that do have a background quasar) 
%and "detectable DLAs" 
%(the potential DLAs 
%with a background quasar with apparent magnitude $m \leq m_{\ell}$).  


We call $h(N_\mathrm{H},Z,z) \, d N_\mathrm{H} \, dZ\, dz \, d\omega$
the   number  of neutral hydrogen layers
%
  with column density
$N_\mathrm{H} \in (N_\mathrm{H}, N_\mathrm{H}+dN_\mathrm{H})$ and
metallicity $Z \in (Z, Z+d \,Z)$ located in the  
spherical shell of redshift $z \in (z,z+dz)$
in the infinitesimal solid angle $d\omega$ 
% REVISION 3
in a random direction
%% $\vec{r}$.
%We omit the dependence on the direction
%% $\vec{r}$
%since.
(we assume that the gas layers are distributed
isotropically). 
 
We call $q(m,  z_e) \, dm  \, dz_e $
the true number  of quasars  
with apparent magnitude  $m \in (m,m+dm)$  
in the   shell of
emission redshift $z_e \in (z_e, z_e+d z_e)$ all over the sky 
(we assume an isotropic distribution of quasars). 
% 
We  define        
$n(m; z, z_{e}^\mathrm{max})
\equiv \int_{z}^{z_{e}^\mathrm{max}} q (m,  z_e) \, d z_e$,
where $z$ is the absorption redshift of a foreground \ion{H}{i} layer    
and $z_{e}^\mathrm{max}$ the maximum redshift of the quasars in the survey. 
For simplicity of notation we omit hereafter
the dependence on $z_{e}^\mathrm{max}$. 
%
Therefore  $n(m; z) \, dm$ is the
total number  of quasars  with magnitude  $m \in (m,m+dm)$
observable beyond   redshift   $z$  all over the sky.
This is the true number
in absence of quasar obscuration.
%In the next section 4.1.1 we show how to obtain this number    
%from the number observed in quasar surveys. 

We call $\delta\omega_\mathrm{q} (z)$
%$\delta^{\prime}\omega_\mathrm{q}(z_e)$ 
%the solid angle subtended 
%in the observer's optical band quasars at redshift $z_e$. 
the average
solid angle subtended in the observer's optical band
by quasars in the redshift interval $z < z_{e} \leq z_{e}^\mathrm{max}$.
%will be $\delta\omega_\mathrm{q} (z)
%\equiv {1 \over \left( z_\mathrm{max}-z\right)} \int_{z}^{z_\mathrm{max}} 
%\delta^{\prime}\omega_\mathrm{q}(z_e)\, d z_e$.

With the above definitions, the number of DLAs 
with    
$N_\mathrm{H} \in (N_\mathrm{H}, N_\mathrm{H}+dN_\mathrm{H})$,
 $Z \in (Z, Z+d \,Z)$ and redshift $z \in (z,z+dz)$
in front of one single background quasar  with $m \leq m_{\ell}$ is 
$$h(N_\mathrm{H},Z,z) 
\, d N_\mathrm{H} \, dZ\, dz \, \delta\omega_\mathrm{q}(z) $$ 
provided $N_\mathrm{H}=N(\ion{H}{i}) \geq 10^{20.3}$ atoms cm$^{-2}$.
%
The number of DLAs in the same differential element    
$d N_\mathrm{H} \, dZ\, dz$
%$N_\mathrm{H} \in (N_\mathrm{H}, N_\mathrm{H}+dN_\mathrm{H})$,
% $Z \in (Z, Z+d \,Z)$ and redshift $z \in (z,z+dz)$
that are detectable 
in front of the $n(m; z)  \, dm$
background quasars with $m \in (m,m+dm)$ is
 $$h(N_\mathrm{H},Z,z) 
\, d N_\mathrm{H} \, dZ\, dz \, \delta\omega_\mathrm{q}(z) \times \, n(m; z)  \, dm \, ,$$ 
provided $m \leq m_{\ell}$. 
%
By integrating in $dm$ up to the limiting magnitude $m_\ell$
we calculate the total number of  DLAs   that are   detectable
in the survey.
We first consider the ideal case in which  no quasar extinction
  is present. 
%If we set $A_{\lambda} (N_\mathrm{H},Z; z)=0$, we obtain
In this case the unbiased number of detectable DLAs  
in $d N_\mathrm{H} \, dZ\, dz$ is
 \begin{eqnarray}
\lefteqn{
g_{m_\ell}(N_\mathrm{H},Z,z) \, d N_\mathrm{H} \, dZ\, dz    \equiv {} }
                  \nonumber\\
& &
{}            h(N_\mathrm{H},Z,z)   \,
\delta\omega_\mathrm{q}(z)
   \left[ \int_{\circ}^{m_{\ell}} n(m; z) \, dm \right]  d N_\mathrm{H} \, dZ\, dz    
                  \, .
\label{truedetDLAs}
\end{eqnarray}
% 
We call $g_{m_\ell}(N_\mathrm{H},Z,z)$ the true distribution of 
detectable DLAs in $d N_\mathrm{H} \, dZ\, dz$. 

We then consider  the quasar extinction $A_{\lambda} (N_\mathrm{H},Z; z)$   
due to the DLAs  in % the   differential element   
$d N_\mathrm{H} \, dZ\, dz$  (Eq. \ref{AlamHzeta})
 ignoring other sources of extinction. 
From this we define the bias function 
%%$N_\mathrm{H} \in (N_\mathrm{H}, N_\mathrm{H}+dN_\mathrm{H})$,
%% $Z \in (Z, Z+d \,Z)$ and redshift $z \in (z,z+dz)$
%that are  detectable, taking into account their own extinction,
%in the quasars with $m \in (m,m+dm)$ is 
%$$b_{m_\ell}(N_\mathrm{H},Z, m,z) \, h(N_\mathrm{H},Z,z) 
%\, d N_\mathrm{H} \, dZ\, dz \, \delta\omega_\mathrm{q}(z)  \, n(m; z)  \, dm ,$$
%where 
%%To account for the extinction  we introduce the bias function
\begin{equation}
b_{m_\ell} (N_\mathrm{H},Z, m,z) \equiv
\left\{
\begin{array}{ll}
1 & ~~ \mbox{if $m \leq{m}_{\ell} -A_{\lambda} (N_\mathrm{H},Z; z)   $ } \\
0 & ~~ \mbox{if $m > {m}_{\ell} -A_{\lambda} (N_\mathrm{H},Z; z)  $ } \\ 
\end{array} ~.
\right.
\label{BiasFun}
\end{equation} 
%where    $A_{\lambda} (N_\mathrm{H},Z; z)$    is the   extinction
%(Eq. \ref{AlamHzeta})  due to the DLAs with
%$N_\mathrm{H} \in (N_\mathrm{H}, N_\mathrm{H}+dN_\mathrm{H})$,
% $Z \in (Z, Z+d \,Z)$,
%and   $z \in (z,z+dz)$.
%
% $N_\mathrm{H} \in (N_\mathrm{H}, N_\mathrm{H}+dN_\mathrm{H})$,
%% $Z \in (Z, Z+d \,Z)$ and redshift $z \in (z,z+dz)$
The number of DLAs  in  
$d N_\mathrm{H} \, dZ\, dz$ that,
in spite of their own extinction, are detectable 
in front of the $n(m; z)  \, dm$
background quasars    is 
$$b_{m_\ell}(N_\mathrm{H},Z, m,z) \, h(N_\mathrm{H},Z,z) 
\, d N_\mathrm{H} \, dZ\, dz \, \delta\omega_\mathrm{q}(z)  \, n(m; z)  \, dm .$$
%
% 
By integrating in $dm$ we obtain the total number of   DLAs  
in   $d N_\mathrm{H} \, dZ\, dz$
that, in spite of their own extinction,  are detectable in the survey  
\begin{eqnarray}
\lefteqn{
g_{m_\ell}^\mathrm{b}(N_\mathrm{H},Z,z) \, d N_\mathrm{H} \, dZ\, dz    \equiv 
   h(N_\mathrm{H},Z,z) \,
\delta\omega_\mathrm{q}(z) \times
{} }
                  \nonumber\\
{}    &  \times
        \left[ \int_{\circ}^{m_{\ell}-A_\lambda(N_\mathrm{H},Z, z)}  \, 
        n(m; z) \, dm \right]  d N_\mathrm{H} \, dZ\, dz   \, .&   
\label{biasdetDLAs}
\end{eqnarray}

The number of DLAs in the survey that are
missed as a consequence of their own extinction is
$\left( g_{m_\ell}-g_{m_\ell}^\mathrm{b} \right) 
\, d N_\mathrm{H} \, dZ\, dz $.
%
The fraction of DLAs % in  $d N_\mathrm{H} \, dZ\, dz$
missed as a consequence of their own extinction is  
\begin{equation}
\varphi_{m_\ell} (N_\mathrm{H},Z, z) \,
\equiv
{ g_{m_\ell}-g_{m_\ell}^\mathrm{b} \over g_{m_\ell}}
= 1 - B_{m_\ell}  ( N_\mathrm{H}, Z,z) \, ,
\label{philost}
\end{equation}
%
where
\begin{equation}%\left[
B_{m_\ell}  ( N_\mathrm{H}, Z,z) \equiv
{
  \int_\circ^{m_{\ell}-A_\lambda(N_\mathrm{H},Z, z)} \,  
%
 n(m; z)  \, d m \,
%
   \over
  \int_\circ^{m_\ell}  \, n(m; z)  \,  dm 
 \label{BigBHZz}
 } \, .
\end{equation}

The ratio of the integrals of the   magnitude distribution
in Eq. (\ref{BigBHZz}) % in this expression of our formulation 
is reminiscent
of the ratio of the integrals of the quasar luminosity function
in Eq. (3) of   FP93.
%
This ratio represents, in a sense, the core relation
from which the bias effect is estimated.

The relation between the true distribution
% $g_{m_\ell}(N_\mathrm{H},Z,z)$ 
and the distribution
biased by its own  DLAs is
%
\begin{equation}  
g_{m_\ell}^\mathrm{b}(N_\mathrm{H},Z,z)  
= B_{m_\ell}  ( N_\mathrm{H}, Z,z)  ~
g_{m_\ell}(N_\mathrm{H},Z,z) \, .
\label{HZzdistr}
\end{equation}
%
This   relation %  show that the fraction $\varphi_{m_\ell}$
% of DLAs lost as a consequence of their own extinction 
is independent of   the average solid angle
$\delta\omega_\mathrm{q} (z)$.   
The normalization factor of $n(m,z)$ is also
irrelevant because it is eliminated in Eq. (\ref{BigBHZz}).

To make practical use of Eq. (\ref{HZzdistr}) 
we need to understand the relation between the biased distribution
$g_{m_\ell}^\mathrm{b}(N_\mathrm{H},Z,z) $
and the observed distribution of DLAs in 
$d N_\mathrm{H} \, dZ\, dz$, that we call
$g_{m_\ell}^\mathrm{obs}(N_\mathrm{H},Z,z)$.
%
To derive an expression for the observed distribution
we must consider all the sources of obscuration,
including the DLAs at $z^\prime < z$ that lie in the
same lines of sight of the DLAs at redshift $z$.
%
If these additional sources of obscuration are negligible,
then the expression (\ref{HZzdistr}) is a good
approximation  of the observed distribution (see A.1.1), i.e.
%
\begin{equation}
g_{m_\ell}^\mathrm{obs}(N_\mathrm{H},Z,z) 
\simeq
g_{m_\ell}^\mathrm{b}(N_\mathrm{H},Z,z) \, .
\label{approximation}
\end{equation}
%
%In    we show that this approximation
%is sufficiently good for  
%a simplified treatment of the obscuration bias. 
%
{\em 
Using the above approximation, the treatment of the obscuration bias
is self-contained, in the sense that  given the true distributions 
of column densities and metallicities at a redshift $z$, 
the observed distributions of the same quantities are uniquely determined. 
In this way we can study the  
obscuration effect in a redshift interval sampled by the observations,
even if we do not have   good statistics of the DLAs at lower redshifts. 
}
%
    
%
% Eq. (\ref{HZzdistr})  
%The above relations consider   only DLAs
%in an infinitesimal redshift interval. 
To compare the distributions with the observations 
we   integrate Eq. (\ref{HZzdistr}) % them %Eq. (\ref{HZzdistr}) 
in $dz$  over  the interval 
$(z_\mathrm{min},z_\mathrm{max})$ with  
% $(\overline{z}- {\Delta z\over 2}, \overline{z}+ {\Delta z\over 2})$
$\overline{z}=(z_\mathrm{max}-z_\mathrm{min})/2$ 
close to the peak of the observed distribution.
If  the statistical functions  of interest  show a smooth variation  
inside the interval we can approximate them   
with their value at  $z=\overline{z}$.
% e.g. $n(m,z) \simeq n(m; \overline{z})$.
%$A_{\lambda} (N_\mathrm{H},Z, z) \simeq 
%A_{\lambda} (N_\mathrm{H},Z; \overline{z})$ and
%$B_{m_\ell}  ( N_\mathrm{H}, Z, z)
%\simeq B_{m_\ell}  ( N_\mathrm{H}, Z; \overline{z})$.
%
% With this approximation, by integrating Eq. (\ref{HZzdistr}) in $dz$
%over the redshift interval 
%$(z_\mathrm{min},z_\mathrm{max})$
We obtain         
%
\begin{equation}  
f^\mathrm{b}(N_\mathrm{H},Z)  
\simeq \mathcal{B}_{m_\ell} (N_\mathrm{H}, Z)  ~
f(N_\mathrm{H},Z) ~,
\label{HZdistr}
\end{equation} 
where 
 \begin{eqnarray} 
f^\mathrm{b}(N_\mathrm{H},Z) &
\equiv &  
\int_{z_\mathrm{min}}^{z_\mathrm{max}}     
g^\mathrm{b}(N_\mathrm{H},Z, z) \ dz  
  \nonumber \\ 
f(N_\mathrm{H},Z)~ &
\equiv &  
\int_{z_\mathrm{min}}^{z_\mathrm{max}}    
g(N_\mathrm{H},Z, z) \ dz 
\label{intgz}
\end{eqnarray}
and
\begin{equation} 
\mathcal{B}_{m_\ell} (N_\mathrm{H}, Z) 
\equiv 
B_{m_\ell} (N_\mathrm{H}, Z, \overline{z}) ~.
\label{BigCalB}
\end{equation}
%
For simplicity   we omit   the dependence of these
functions   on $\overline{z}$,
$z_\mathrm{min}$ and $z_\mathrm{max}$.

 
 
We now  transform the distribution $f(N_\mathrm{H},Z)$
into a distribution of column densities, $f_{N_\mathrm{H}}$,
and a distribution of metallicities, $f_{Z}$, defined in such a way that
$f_{N_\mathrm{H}} d N_\mathrm{H}$ is the  
true number of detectable DLAs with  column densities between
$N_\mathrm{H}$ and  $N_\mathrm{H}+dN_\mathrm{H}$ 
in the   interval $(z_\mathrm{min},z_\mathrm{max})$
and $f_{Z} \, dZ$ the true  number of detectable DLAs with
metallicity between $Z$ and $Z+d \,Z$ in the same redshift interval.  
%
We call $f^\mathrm{b}_{N_\mathrm{H}}$
and $f^\mathrm{b}_{Z}$ the corresponding biased distribution functions.

To proceed further we  assume that $N_\mathrm{H}$ and $Z$  are independent
variables\footnote{
A similar assumption was done by Fall \& Pei (1993; Appendix) to   
separate the distribution of column densities from that of dust-to-gas ratios.
}
 (this assumption is discussed in Section 4.4).
%
This implies that  
$f(N_\mathrm{H},Z) \, d N_\mathrm{H} \, d\, Z=
f_{N_\mathrm{H}} \, d N_\mathrm{H} \, f_Z \, d\, Z$. 
 We substitute this expression in the right hand of Eq. (\ref{HZdistr}). 
%
We then integrate that equation  in metallicity and obtain  
%
 \begin{equation}
\int_{0}^{\infty} f^\mathrm{b}(N_\mathrm{H},Z) \, dZ \simeq 
\left[ \int_{0}^{\infty} %_{Z_\mathrm{min} }^{Z_\mathrm{max} }
  \mathcal{B}_{m_\ell}  ( N_\mathrm{H}, Z) \,
f_{Z}  \, d\, Z \right]   ~  f_{N_\mathrm{H}} ~~.
 \label{fb_H_a}
\end{equation}
%
Apart from a constant factor, the left term of this equation represents
the biased distribution $ f^\mathrm{b}_{N_\mathrm{H}}$. 
The constant   
can be determined by imposing that in absence of bias
% [i.e., when $\mathcal{B}_{m_\ell}  ( N_\mathrm{H}, Z)=1$] 
the true and biased distributions must be equal.
From this we derive the relation 
%
 \begin{equation}
 f^\mathrm{b}_{N_\mathrm{H}}  \simeq 
{
\int_{0}^{\infty}%_{Z_\mathrm{min} }^{Z_\mathrm{max} }
  \mathcal{B}_{m_\ell}  ( N_\mathrm{H}, Z) \,
f_{Z}  \, d\, Z
 \over 
\int_{0}^{\infty}%_{Z_\mathrm{min} }^{Z_\mathrm{max} }
f_{Z}  \, d\, Z
 } ~  f_{N_\mathrm{H}} ~~.
 \label{fb_H}
\end{equation}
% 

In a similar way, by integrating  Eq. (\ref{HZdistr}) in $d N_\mathrm{H}$, 
we obtain the   relation 
%
\begin{equation}
 f^\mathrm{b}_{Z}  \simeq   
{
\int_{N_\mathrm{DLA}}^{\infty} 
  \mathcal{B}_{m_\ell}  ( N_\mathrm{H}, Z) \,
f_{N_\mathrm{H}}  \, d\, N_\mathrm{H} 
 \over 
\int_{N_\mathrm{DLA}}^{\infty} 
f_{N_\mathrm{H} }  \, d\, N_\mathrm{H}
 } ~  f_{Z} ~~,
 \label{fb_zeta}
\end{equation}
where $N_\mathrm{DLA}=10^{20.3}$ atoms cm$^{-2}$.
%is the definition threshold of DLA systems.  

  


\subsubsection{Condition of validity of Eq. (\ref{approximation}) }

We start by
  deriving the relation between the true number
of detectable DLAs in absence of obscuration,
$g_{m_\ell}(N_\mathrm{H},Z,z) \, d N_\mathrm{H} \, dZ\, dz$,
%the "biased" number detectable when the only source of extinction
%is that of the DLAs,
%$g_{m_\ell}^\mathrm{b}(N_\mathrm{H},Z,z) \, d N_\mathrm{H} \, dZ\, dz$,
and the number observed when all sources of obscuration are considered,
$g_{m_\ell}^\mathrm{obs}(N_\mathrm{H},Z,z) \, d N_\mathrm{H} \, dZ\, dz$.

At redshift $z$ the observed number equals the true number  
minus the fraction of DLAs % at redshift $z$ 
missed  as a consequence of their own extinction, 
$\varphi = \varphi_{m_\ell} (N_\mathrm{H},Z, z) $,
minus the fraction of DLAs at redshift $z$ 
missed due to the extinction of other DLAs at $z^\prime < z$
that lie in the same line of sight, $\varphi^\prime
=\varphi_{m_\ell}^\prime (N_\mathrm{H},Z, z)$.
This implies
\begin{equation}
%\begin{eqnarray}
%\lefteqn{
g_{m_\ell}^\mathrm{obs}(N_\mathrm{H},Z,z)  =
% {} }
%  \nonumber\\
%& &
%{}     
%~~~~~~
g_{m_\ell}(N_\mathrm{H},Z,z) \times
\left( 1 - \varphi  -
\varphi^\prime \right) \, .
\label{A1}
\end{equation}
%\end{eqnarray}
%
 In the term $\varphi^\prime$ 
 we should not count the DLAs
at redshift $z$ that are missed due to their own extinction, since they
are already counted in the term $\varphi$.
We should instead count only the fraction 
$\left( 1 - \varphi \right)$
of DLAs at redshift $z$ that are not missed
due to their own extinction.
Of this residual fraction, we must in turn take the fraction
of DLAs at redshift $z$ that have an additional DLA at $z^\prime < z$  
in the same line of sight, $F(z^\prime,z)$.
Of the resulting fraction, we must finally consider 
the mean fraction of DLAs at redshift $z^\prime < z$ that are
missed due to their own extinction, $\overline{\varphi}
(z^\prime \!\! < \!z)$. 
%
From these considerations we obtain the relation 
%
\begin{equation}
\varphi^\prime =
\left( 1 - \varphi \right) \times F(z^\prime,z)
\times \overline{\varphi}(z^\prime \!\!  < \!z) . 
\label{A2}
\end{equation}

From Eqs. (\ref{philost}) and (\ref{HZzdistr}) we have %that allows us to neglect the extinction of foreground DLAs
%in the treatment of our problem and, as a consequence, 
%makes valid the relation 
$ 
g_{m_\ell}^\mathrm{b}(N_\mathrm{H},Z,z) =
g_{m_\ell}(N_\mathrm{H},Z,z) \times
\left( 1 - \varphi % (N_\mathrm{H},Z, z)  
\right)$. Comparing with  Eq. (\ref{A1})
one can see that the approximation of
 Eq. (\ref{approximation}),
$g_{m_\ell}^\mathrm{obs}(N_\mathrm{H},Z,z) 
\simeq
g_{m_\ell}^\mathrm{b}(N_\mathrm{H},Z,z) $,
 is valid when
\begin{equation}
\varphi^\prime \ll 
1- \varphi % (N_\mathrm{H},Z, z) 
\label{A3}
\end{equation}


By combining Eq. (\ref{A2}) with  (\ref{A3}) we obtain that
\begin{equation}
F(z^\prime,z) \times \overline{\varphi}_{m_\ell}(z^\prime \!\! < \!z) 
 \ll 1
\label{A4}
\end{equation}
describes the limits of validity of the approximation
(\ref{approximation}). 
%
We estimate the validity of this condition at $z \simeq 2.3$, the mean
absorption redshift of the surveys. 

From a recent summary (Ellison et al. 2004)  of the unbiased number density of DLAs,
$n_\mathrm{DLA}(z)$, we calculate
that the probability of intersecting 2 DLAs in a random line of sight
up to $z \simeq 2.3$ is $\approx 0.15$. 
We take this value as  an estimate of $F(z^\prime,z)$.

By comparing the number densities of biased and unbiased surveys
there is room for no more than $\approx 50\%$ of missed DLAs at 
$z \simeq 2.3$ (Ellison et al. 2001, 2004). This represents an upper limit
to $\overline{\varphi}_{m_\ell}(z^\prime< 2.3) $ because the  
extinction in the observer's frame tends to decrease
with decreasing redshift and therefore
the missed fraction ${\varphi}_{m_\ell}(z^\prime<z)$ tends to diminish
at lower redshifts. 

All together, we expect therefore
$F(z^\prime,z) \times \overline{\varphi}_{m_\ell}(z^\prime<z) \la 0.08$
at $z \simeq 2.3$ and lower values  
at lower redshifts. 
We conclude that the approximation (\ref{approximation}) is good, even if it may slightly  underestimate   the
obscuration fraction. 

%%
%Even assuming that $\approx 50\%$ DLAs are lost at
%$z^\prime \lsim 2.3 $ and that the probability of finding two DLAs
%in the same quasar is   $\approx 15\%$ for $z^\prime \lsim 2.3 $,
%we find that the inequality is satisfied to a good approximation.
%%
%The condition is satisfied even better at lower redshift for two reasons.
%One is the increase of the
%extinction in the observer's frame
%with increasing redshift, discussed in the derivation of Eq. (12),
%which tends to diminish the lost fraction ${\varphi}_{m_\ell}(z^\prime<z)$
%at lower redshifts. 
%The other is the ithe intrinsic  reduction of the number density of DLAs at
%low redshift (Rao \& Turnshek 2000; Ellison et al. 2004), which tends to
%diminish the fraction $F(z^\prime,z)$. 
%%
%

%To do so we assume that the mean fraction of DLAs that are lost
%due their own obscuration has a modest evolution, with
%a mild increase with redshift, i.e.
%\begin{equation}
%\overline{\varphi}_{m_\ell}(z^\prime<z) \la  \varphi_{m_\ell}(z).
%\label{A5}
%\end{equation}
%This assumption is supported by 
%Combining the relations (\ref{A2}) and (\ref{A4}) it is easy to see
%that the condition (\ref{A3}) is equivalent to

%%
%This inequality is  satisfied when
%  at least one of the two  following conditions are satisfied: 
%(1) $F(z^\prime,z) \ll 1$, (2)
%$\left[ 1 - \varphi_{m_\ell}(z) \right] \ll 1$. 
%The first  condition is true
%at low $z$, where the probability that an additional DLA
%at $z^\prime < z$ is intercepted becomes very low.
%The second condition is satisfied when the fraction of DLAs lost at redshift $z$
%as a consequence of their own extinction is very high
%($\varphi_{m_\ell}(z) \approx 1$). 
%This second case is very important because it guarantees that
%when we find a strong bias effect the predicted biased
%distribution is a good approximation of the observed distribution.
%Only when we find a weak bias and the redshift is relatively high,
%the approximation is less good. Even in this case, however, the approximation
%yields a conservative estimate of the bias. 
% 




\subsubsection{Test of Eqs.  (\ref{BigBHZz}),  (\ref{HZzdistr}) and  (\ref{fb_H}) }

To test  our formulation we  cross-checked its predictions in a particular case considered by Fall \& Pei (1993).  Namely, we compared the predictions of our Eq. (\ref{HZzdistr}), calculated at constant metallicity $Z$, 
with those of Eq. (21) by FP93, which is calculated at a constant 
dust-to-gas ratio $k$. This test is equivalent to insert in Eq.   (\ref{fb_H})
a Dirac delta distribution of metallicities. 
%
To make this comparison we transformed the power-law distribution of quasar luminosities 
used by FP93 [their Eq. (8)] into a magnitude distribution  $n(m;z)$
for our Eq. (\ref{BigBHZz}). 
We adopted $\beta=2$  and $\xi({\lambda \over 1+z})=2.78$ as common values in both equations. 
In Eq. (21; FP93) we adopted a value of  $k=0.04$ consistent with our choice of 
$G=1$, 
$Z/Z_{\sun}=0.1$ and $f_\mathrm{Fe} (Z)=0.52$ 
for this particular test. 
In Fig. \ref{figAppB} we compare the resulting ratio
$f_o(N,z)/f_t(N,z)$ predicted from Eq. (21) by FP93  
with the corresponding ratio
$g_{m_\ell}^\mathrm{b}(N_\mathrm{H},Z,z)/
g_{m_\ell}(N_\mathrm{H},Z,z)$ predicted from our Eq.  (\ref{HZzdistr}),
which equals the term $B_{m_\ell}  ( N_\mathrm{H}, Z,z)$ of
Eq.  (\ref{BigBHZz}). Our predictions
(empty circles) perfectly match those obtained with the equations
derived by FP93 (filled diamonds). 
%
%Even if the limiting magnitude appears in Eq.  (\ref{BigBHZz}), 
Our results
are independent of $m_\ell$ provided
$m_\ell \gg 1$ and $m_\ell \gg A_\lambda$.
For  $m_\ell \approx 19$ - $20$ 
these conditions are well  satisfied. 

%
%A remarkable aspect of this result is its independence 
%of the limiting magnitude  of the survey, $m_\ell$. 
%%
%In the formulation developed by FP93
%this is the consequence of the particular situation considered for
%the derivation of their Eq. (21). 
%%
%In our formulation this results from the fact that
%$m_\ell 
% is not obvious a priori because
%$m_\ell$   appears in the limits of integration. 
%%
%The fact that our Eq. (\ref{BigBHZz}) gives a result  identical
%to the predictions of FP93, independently of the adopted $m_\ell$,  
%is a consequence of the particular shape of the $n_m$ distribution
%considered. 
 

    \begin{figure}
   \centering 
     \includegraphics[width=7cm,angle=0]{1570f10.eps}  
  \caption{ A test of validity of the equations  
  presented in Section 4.1. 
Circles: 
predictions of our Eq. (\ref{BigBHZz}).
Diamonds: 
$f_o(N,z)/f_t(N,z)$ predicted from Eq. (21) by Fall \& Pei (1993).
The adopted parameters are given in Appendix A.1.2.
  }
 \label{figAppB}%
    \end{figure}



%}
% END REVISION 2


\subsection{The quasar magnitude distribution}

% REVISION 3
We derive the relation
between     the "true" magnitude distribution of quasars, $n(m; z)$, 
and the observed one, which is biased by the obscuration effect.
%
%
We consider only the obscuration due to DLA systems. 
%
%The mathematical treatment of this problem can be simplified 
%considering  that the large majority of quasars either does not have
%a foreground DLA or it does have only  one. Only a minority of lines of sight
%will intersect two or more DLAs (see A.1.1). 
% revision 3 next sentence eliminated
%Even if this fraction is not completely
%negligible, it will not change substantially the overall shape of the
%observed frequency distribution of apparent magnitudes. 
%The shape is all that we need because,  as we show in Eq.  (\ref{BigBHZz}), 
%the normalization factor of $n(m; z)$ is irrelevant for estimating
%the bias.   
%
We start by defining
%with the following
%definitions. By analogy with       the true
%frequency distribution $q(m,z_e)$ defined in the previous section,
%we call 
$q^\mathrm{b} (m,  z_e) \, dm  \, dz_e $
the observed (biased)
number  of quasars  with   magnitude  $m \in (m,m+dm)$
at redshift $z_e \in (z_e, z_e+d z_e)$.
%
We then divide the quasars according to the number $i$
of DLAs that     lie in their direction.
We call $\mathcal{F}_i(z_e)$ the fraction of quasars at redshift $z_e$
that have $i$ intervening DLAs. 
%
Depending on the value of $i$, the distribution
of apparent magnitudes of the quasars will be different. 
%
We call $q^\mathrm{b}_i (m,  z_e) \, dm  \, dz_e $
the biased number of quasars  with $i$ foreground DLAs,
  magnitude  $m \in (m,m+dm)$
and redshift $z_e \in (z_e, z_e+d z_e)$.
%
%%%%%%%%% REVISION 3  Eliminated next equation
%From these definitions we have 
%\begin{equation}
%q^\mathrm{b} (m,  z_e) = 
%\sum_{i=0}^{\infty} \mathcal{F}_i(z_e) \, q^\mathrm{b}_i (m,  z_e) ~,
%\label{nprimeb_sum}
%\end{equation}
%with      
%$\sum_{i=0}^{\infty} \mathcal{F}_i(z_e)=1$.
%
We now assume that 
the fraction of quasars with two or more  
  DLAs   is negligible (see A.1.1), i.e. that $\mathcal{F}_{i\geq 2}(z_e) \ll \mathcal{F}_0(z_e)$
and $\mathcal{F}_{i\geq 2}(z_e) \ll \mathcal{F}_1(z_e)$. 
%
%
%Therefore we keep only the first two terms of the sum
%in Eq. (\ref{nprimeb_sum}) and, 
From the condition 
$\sum_{i=0}^{\infty} \mathcal{F}_i(z_e)=1$, we obtain
$\mathcal{F}_0(z_e) \simeq 1-\mathcal{F}_1(z_e)$.
%
This gives
\begin{equation}
q^\mathrm{b}(m, z_e)
\simeq 
\left[ 1-\mathcal{F}_1(z_e) \right] \, q(m,z_e)  
+ \mathcal{F}_1(z_e) \, q^\mathrm{b}_1(m, z_e)  \, 
 \label{nbprime_m}
\end{equation}
where $q(m,z_e) = q^\mathrm{b}_0(m, z_e) $  
because the true distribution, $q(m,z_e)$, must equal the
distribution of quasars without foreground DLAs,
$q^\mathrm{b}_0(m, z_e)$.
%%%%%%%%% END REVISION 3 
 
We now integrate Eq. (\ref{nbprime_m}) in $dz_e$ over the   interval
$(z ,z_e^\mathrm{max})$, where $z$
is the absorption redshift and  $z_e^\mathrm{max}$
the maximum redshift of the quasars in the survey. 
%
We assume   $\mathcal{F}_1(z_e)$ to
vary smoothly in the interval, so that it can be
approximated with its   mean value 
$\overline{\mathcal{F}}_1$. 
% = 
%\int_{z}^{z_e^\mathrm{max}} \mathcal{F}_1(z_e) \, d z_e /
%\int_{z}^{z_e^\mathrm{max}}   \, d z_e$. 
%
This gives
\begin{equation}
n^\mathrm{b}(m; z)
\simeq 
\left[ 1-\overline{\mathcal{F}}_1 \right] \, n(m; z) 
+ \, \overline{\mathcal{F}}_1 \, n^\mathrm{b}_1(m;z) ~,
 \label{nb_m}
\end{equation}
%
where  
$n^\mathrm{b}(m; z)
\equiv \int_{z}^{z_{e}^\mathrm{max}} q^\mathrm{b} (m,  z_e) \, d z_e$
and
$n^\mathrm{b}_1(m; z)
\equiv \int_{z}^{z_{e}^\mathrm{max}} q^\mathrm{b}_1 (m,  z_e) \, d z_e$.
% in analogy with our   definition of $n(m; z)$.
%

 %%%% REVISION 3
We now derive an independent relation
for $n^\mathrm{b}_1(m;\overline{z})$. 
%the apparent-magnitude
%distribution of quasars with one foreground DLA.
%
The apparent magnitude of the  quasars with one DLA,
 $m$, equals   the
sum of the  "true"   magnitude in absence of extinction, $m^\prime$,
plus the DLA extinction in the observer's frame,  $A_\lambda$.
%
The frequency distribution of the sum $m =m^\prime+A_\lambda$
is given by the convolution of the frequency distributions
of $m^\prime$ and $A_\lambda$, since these are independent variables. 
%
This implies
\begin{equation}
n^\mathrm{b}_1(m; z) = K 
\int_{-\infty}^{\infty} 
n(m^\prime; z) ~ 
f_{A_\lambda}\!(m-m^\prime;  \overline{z}_e) ~ dm^\prime ~,
\label{p_m}
\end{equation}
where    $K$ is a normalization factor,
$f_{A_\lambda}(A_\lambda; \overline{z}_e)$ %\, dA_\lambda$ 
is the frequency
distribution of the extinctions % $A_\lambda = m-m^\prime$
of all the DLAs in the redshift range $z \in (0,\overline{z}_e)$,
and $\overline{z}_e$
is the mean redshift of the quasars in the survey,
$\overline{z}_e \in (z ,z_e^\mathrm{max})$.


The fraction  
$\overline{\mathcal{F}}_1$ 
 is estimated
from  statistical surveys.  From the statistics 
of quasars with $\overline{z}_e  \simeq 3.0$ one finds,
in the redshift path  observable from ground ($z \gsim 1.8$,
$\overline{z} \simeq 2.3$),
   that $\approx 40\%$   of the quasars have one single DLA  
 (see P\'eroux et al. 2003).  
The  true fraction 
$\overline{\mathcal{F}}_1$ will be slightly higher than $0.4$
owing to the limitations of the wavelength coverage of the spectra. 
We adopt here $\overline{\mathcal{F}}_1 \simeq 0.5 \pm 0.1$. 



%
We   assume that $f_{A_\lambda}(A_\lambda; \overline{z}_e)$
can be approximated with the distribution of extinctions
in the redshift range where most DLAs are observed,
$z \in (1.8,3.0)$. 
In this way the problem of estimating 
$f_{A_\lambda}(A_\lambda; \overline{z}_e)$ is brought back
to that of estimating the  distributions of column densities and metallicities
% of DLAs 
in the same redshift range, $f_{N_\mathrm{H} }$ and $f_{Z}$.
%From these distributions we   calculate the distribution
%of extinctions from Eq. (\ref{AlamHzeta}). 
Assuming that 
$N_\mathrm{H}$ and $Z$  are statistically independent variables,
on the basis of Eq. (\ref{AlamHzeta})
we obtain the extinction distribution by convolving 
$f_{N_\mathrm{H} }$ and $f_{Z}$.

The normalization factor $K$  % $n^\mathrm{b}_1(m; z)$  
is determined 
from the condition $\int   n^\mathrm{b}_1(m; z) \, dm = 
\int   n(m; z) \, dm$,
which, from the integration in $dm$ of Eq. (\ref{nb_m}), is equivalent
to 
$\int   n^\mathrm{b}(m; z) \, dm = 
\int   n(m; z) \, dm$.
These conditions guarantee that the total number of quasars
% in the adopted redshift interval  
is  independent of the bias, since the only
effect of the extinction is a redistribution of the  
apparent magnitudes.


% revision 3   NEXT TWO PARagraphs ELIMINATED
%

%The fact that we approximate the distribution of
%extinctions  at $z \in (0,\overline{z}_e \simeq 3.0)$,
%$f_{A_\lambda}(A_\lambda; \overline{z}_e)$,
%with the distribution   
%at $z \in (1.8,3.0)$ yields a deficit of  
%the lower values of extinction. In fact, the DLAs  
% lost in this approximation
%have statistically lower extinctions
%because $A_{\lambda} (N_\mathrm{H},Z; z)$
%decreases at lower redshifts.  
%The evolution of the dust content   should
%not increase systematically the extinction at $z < 1.8$ because    
%the fraction in dust $f_\mathrm{Fe}$ of Eq. (\ref{AlamHzeta}) 
%is already "saturated",
%i.e. $\approx 1$,  in DLAs at $z \approx 2$ (Vladilo 2004).
%The deficit of low values of extinction is probably modest because
%the number density of DLAs also decreases at lower redshifts
%(Rao \& Turnshek 2000; Ellison et al. 2004). 

%
%The other approximation made in this section,
%i.e. ignoring quasars with multiple DLA systems,
%yields instead a deficit of the higher values of extinction because,
%in this case, the DLAs  lost in the approximation
%have statistically higher extinctions.
%Also this effect is modest given the low number of quasars
%with multiple DLAs. 

% END REVISION 3
%

%In Section 4.3, we show how one can use Eqs. (\ref{nb_m}) and (\ref{p_m}) 
%to estimate the true distribution $n(m; z)$ by iterations.


 
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Shimasaku, K., \& Schneider, D.P. 1996, \aj, 111, 1748

\bibitem[]{} Gordon, K.D., Clayton, G.C., Misselt, K.A., Landolt, A.U., \& Wolff, M.J.
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2001, \aap, 370, 23


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\bibitem[]{} Khersonsky, V.K., \& Turnshek, D.A. 1996, \apj, 471, 657


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\bibitem[]{} Lamareille, F., Mouhcine, M., Contini, T., Lewis, I., \&
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\bibitem[]{} Le Brun, V., Smette, A., Surdej, J., \& Claeskens, J.-F.
2000, \aap, 363, 837



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2005, in Probing Galaxies through Quasar Absorption Lines, Proc. IAU Coll. No. 199,
eds. P.R. Williams et al., in press (astro-ph/0504402)


\bibitem[]{} Ledoux, C., Petitjean, P.,  \& Srianand, R. 2003, \mnras, 346, 209


%
%\bibitem[2002]{L&02} Lehner, N., Gry, C., Dembach, K.R. et al.
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% Garc�a-Lorenzo,�B., Oscoz,�A., \& Serra-Ricart,�M. 2002, �\apj, 574, 719
 
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%
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\bibitem[]{} Vanden Berk, D.E., Quashnock, J.M., York, D.G., \& Yanny, B.
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\bibitem[2002]{V02b} Vladilo, G. 2004, \aap, 421, 479

%
%\bibitem[]{} Vladilo, G.
%2005, in Probing Galaxies through Quasar Absorption Lines, Proc. IAU Coll. No. 199,
%eds. P.R. Williams et al., in press (astro-ph/ ???????  )


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\bibitem[]{} Wolfe, A.M., Turnshek, D.A., Smith, H.E., \& Cohen, R.D.
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\end{thebibliography}

 
\end{document}

 
astro-ph0502137.extracted_bib
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\bibitem[]{} Ledoux, C., Petitjean, P.,  \& Srianand, R. 2003, \mnras, 346, 209


%
%\bibitem[2002]{L&02} Lehner, N., Gry, C., Dembach, K.R. et al.
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%\bibitem[2001]{MLDBC01} Molaro, P., Levshakov, S. A., D'Odorico, S., Bonifacio, P.,
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1999, \apj, 522, 604


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\bibitem[]{} Pettini, M., Smith, L.J.,   King, D.L., \& Hunstead, R.W.
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%2002a, \apj, in press (astro-ph/0206296)

%
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%et al. %J.M., Tytler, D., Wolfe, A.M., Kirkman, D., Lubin, D., \& Suzuki, N.
%2002b, \apj, in press (astro-ph/0202140)

\bibitem[]{} Prochaska, J.X., \& Wolfe, A.M. 2002, \apj, 566, 68

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%1999, 346, 785

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\bibitem[2002]{V02b} Vladilo, G. 2004, \aap, 421, 479

%
%\bibitem[]{} Vladilo, G.
%2005, in Probing Galaxies through Quasar Absorption Lines, Proc. IAU Coll. No. 199,
%eds. P.R. Williams et al., in press (astro-ph/ ???????  )


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