1 Introduction

Perhaps it is something of a paradox, but large ground-based optical telescopes like Keck and VLT routinely do absorption-line spectroscopy of the gas in high-redshift objects which surpasses what can be done for the interstellar medium (ISM) of our own Galaxy near the Sun. Viewing neutral gas at high redshift makes the dominant ion stages (C II, Fe II, etc.) accessible, to say nothing of Lyman series in H I and D I and the Lyman and Werner bands of ${\rm H}_2$. The high spectral resolution and large collecting areas of modern ground-based instrumentation cannot easily be matched in space-based instruments, and certainly not for the same cost.

Thus we may have exquisitely detailed and sensitive spectra of systems which cannot be imaged, whose nature is therefore left to be inferred from their patterns of gas kinematics (Prochaska & Wolfe 1997, 1998; Haehnelt et al. 1998; Ledoux et al. 1998; McDonald & Miralda-Escudé 1999) and ionization (Wolfe & Prochaska 2000a,b). So it is with most damped Lyman-$\alpha$ systems, defined as those absorption-line systems having N(H I) $\ge 2\times 10^{20}~{\rm cm}^{-2}$ when seen against the emission of background QSO's. Although well-formed galaxian systems can and do harbor some of them at low redshift, damped Lyman-$\alpha$ systems are for the most part believed to be protogalactic objects, perhaps in disk systems (Prochaska & Wolfe) or perhaps in the ongoing merger of protogalactic clumps (Haehnelt et al. 1998). The numbers are such that damped Lyman-$\alpha$ systems seem to contain at least as many baryons as can be found in the local Universe now, and it is of great interest to understand how these baryons become recognizable nearby objects within a relatively short redshift interval.

The absorption-line gas in damped Lyman-$\alpha$ systems is easily discussed in the same terms, using the same physical processes, that are employed locally. Evidence that the rest-frame optical/uv radiation field is comparable to that near the Sun (Levshakov et al. 2002; Molaro et al. 2002; Petitjean et al. 2000) and the presence of metals at a low but non-negligible level 0.1-0.01 Solar, makes it possible (or, perhaps, merely hopeful?) to talk about the "interstellar medium'' in these systems. The low metallicity and general underabundance of dust (which is not directly observed in any one object but can be inferred statistically (Fall & Pei 1993; Pei et al. 1991) or from patterns of gaseous abundances (Boisse et al. 1998)) may render the gaseous medium in damped Lyman-$\alpha$ systems only more extreme versions of those in local dwarf systems like the LMC and SMC. But it has also been argued that the apparent recognizability of patterns in the absorption spectra has been over-interpreted (Izotov et al. 2001), and that the similarity of intermediate and low ion kinematics (Al III and C II or Fe II; C IV and C II behave differently) could mean that substantial ionization corrections to the metallicity are needed (however, see Vladilo et al. 2001, for a contrary opinion).

At the present time there seem to be several lines of evidence suggesting that, unlike the local ISM (where low-altitude neutral gas is perhaps 2/3 cool and 1/3 warm and the overall ratio including high-altitude material is 1/2 and 1/2) the gas in damped Lyman-$\alpha$ systems is more predominantly warm. Examples include the high H I spin temperatures inferred from comparison of $\lambda21$ cm and Lyman-$\alpha$ absorption (Wolfe & Davis 1979; Carilli et al. 1996a,c; Chengalur & Kanekar 2000; Kanekar & Chengalur 2001) (but see Lane et al. 2000), the similarity of low and intermediate ion kinematics mentioned in the preceding paragraph, and the very low column densities of ${\rm H}_2$ discussed here. Along lines of sight with reddening E_B-V > 0.05 mag (N(H) $> 3 \times 10^{20}~{\rm cm}^{-2}$) in local Copernicus spectra, it is always the case that a few percent or more of the neutral hydrogen is molecular. By contrast, the molecular hydrogen fraction in damped Lyman-$\alpha$ systems is 2-4 orders of magnitude lower (i.e. 10^-4 - 10^-6), which can be interpreted as meaning that the gas temperature must be above 3000 K (Petitjean et al. 2000) Lanzetta et al. (1989) used N(${\rm H}_2$)/N(H) to constrain the dust/gas ratio toward Q1337+113, similar to the approach taken in this work.

Here we consider the formation of molecular hydrogen in a gas which is in two-phase thermal equilibrium at low metallicity. Perhaps because ${\rm H}_2$ has been observable so rarely in the local ISM there is not a big literature on this subject, but the extant 1970's-era Copernicus observations are well-explained in this way (Liszt & Lucas 2000) using modern shielding factors for radiative dissociation (Lee et al. 1996). The only surprise (if there indeed is one) is the low densities that are required to start ${\rm H}_2$ formation locally, and the fact that even a "standard'' H I cloud (Spitzer 1978) should have a molecular fraction of 10-30% deep inside. Local diffuse clouds also have surprisingly high abundances of complex polyatomic species which follow immediately upon the presence of ${\rm H}_2$ (Liszt & Lucas 1996; Lucas & Liszt 1996), a phenomenon which is not well understood but which can be used to account for the observed abundancess of simpler species such as CO (Liszt & Lucas 2000).

Figure 1: Ionization and thermal equilibrium calculations in atomic gas. The pressure P/k is shown as a function of the density of H-nuclei n(H), for four sets of parametric variations. a) Upper left; the metallicity, (dust/gas, C/H, O/H etc.) varies in steps of 2 from 4 times to 1/32 times its reference value; b) Upper right, carbon and oxygen are removed (depleted) from the gas phase in steps of 2; c) Lower left, the "interstellar'' (ambient) radiation field (ISRF) varies from 4 times to 1/32 times the reference value; d) Lower right, the impinging flux of soft X-rays is scaled.

In the course of this work, we consider the ionization and fine-structure excitation of carbon, and the inferences which may be drawn from observations of carbon in the damped Lyman-$\alpha$ systems. While the importance of carbon to the physical state of the gas and the conditions for forming ${\rm H}_2$ cannot be stressed too highly, the extended discussion here in fact arose because of something of a coincidence. Searching the literature, it quickly became evident that there is a high degree of overlap between the two relatively scant datasets for carbon (chiefly, C I, C II, and C II*) and ${\rm H}_2$.

The plan of the current discussion is as follows. In Sect. 2 we lay out the basics of a calculation of two-phase equilibrium essentially following Wolfire et al. (1995a). We examine the sensitivity of two-phase equilibrium to variations in abundance, depletion, incident radiation and the like, in order to extract from the calculations those aspects which can be related to existing absorption line data on damped Lyman-$\alpha$ systems. In Sect. 3 we compare the results of these calculations with observed hydrogen and carbon column densities: we show that the observations are consistent with an origin in largely cool gas for the damped Lyman-$\alpha$ systems at lower z (z < 2.3) and in warm gas at higher redshift. In Sect.  4 we describe calculations of the ${\rm H}_2$ abundance in both warm and cool diffuse gas, under conditions of varying metallicity, etc., employing (slow) gas-phase processes to explore the minimum expected amounts of molecular gas, and the more usual grain-catalysis (as in Liszt & Lucas 2000) for cooler regions of higher molecular fraction. We also compute the variation of the molecular hydrogen fraction in small gas clots of constant density. From this, it follows that the low metallicities of damped Lyman-$\alpha$ systems are by themselves sufficient to cause decreases in the molecular fraction by many orders of magnitude, even if cool neutral clouds are present.