4 The formation of molecular hydrogen

Levshakov et al. (2002) provide a summary of the molecular fractions observed toward 14 sources (their Table 2) including most of those discussed here. The lowest molecular fraction is seen toward 0000-262 at z = 3.39, the highest toward 0013-004 at z = 1.97 (see Tables 1-2 here). The latter is the only direction for which the molecular fraction exceeds 1/3000, and most are below 10^-5. In this section we address the existence of damped Lyman-$\alpha$ systems with large hydrogen column densities, substantial amounts of cool gas, but very small fractions of molecular hydrogen. The models of ${\rm H}_2$-formation presented here are very similar to those of Liszt & Lucas (2000), who considered the formation of CO in local diffuse gas, given certain other conditions like the presence of HCO⁺, but there are a few differences which we now remark.

4.1 H $_\mathsfsl{2}$-formation in the gas phase

The present calculations include X-ray heating and ionization, because they are integral to the question of two-phase equilibrium. X-rays have little effect on the large neutral gas columns which harbor appreciable molecular column densities nearby in the Milky Way but are also included now because we are interested in understanding the minimum molecular fractions which can be expected, and these are presumably set by the slow gas-phase processes (not involving grains) which formed the first stars, and which occur in low-density regions of (unshielded) free space.

 

Table 2: Hydrogen and carbon at highest redshift.

Source	1337+113	0528-250	0347-382	0000-262
z	2.80	2.81	3.03	3.39
[Zn/H]	-1.00	-0.91	-1.23	-2.07
N(H I)	8.0E20	2.2E21	2.52E20	2.6E21
N(${\rm H}_2$)	<5.0E16	6.0E16	8.2E14	1.1E14
N(C I)	<1.6E13	<5.9E12	<4.0E11
N(C II)	2.0E17	1.7E17	5.1E15
N(C II*)		3.6E14	1.9E13
N(C II)$_{\rm m}$	2.8E17	9.6E16	3.1E15	7.9E15
C II/C I	>12658	>28862	>12700
C II*/C II		0.00211	0.00389

References:
1337+113: Lanzetta et al. (1989).
0528-250: Ge et al. (1997); Srianand & Petitjean (1998).
0347-382: Levshakov et al. (2002).
0000-262: Prochaska & Wolfe (1999); Levshakov et al. (2000).

Gas phase ${\rm H}_2$-formation occurs via the exothermic reaction pathways H + e^- $\rightarrow$ H^- + $h\nu$, H^- + H $\rightarrow$ ${\rm H}_2$ + e^- and H⁺ + H $\rightarrow$ ${\rm H}_2^+$ + $h\nu$, ${\rm H}_2^+$ + H $\rightarrow$ ${\rm H}_2$ + H⁺. Many of the basic reactions are cited by Puy et al. (1993) and most by Haiman et al. (1996) (all can straightforwardly be located in the UMIST reaction database) but the discussion of early-universe conditions must be modified to include relevant values for the photodissociation of H^-and ${\rm H}_2$ and the cosmic-ray ionization of ${\rm H}_2$. For the latter we assumed $\zeta = 2\times^{-17}\mbox{s$^{-1}$ }$ per H and 1.08 $\zeta$ per ${\rm H}_2$), and for the photodissociation rate of ${\rm H}_2$ in free space we followed Lee et al. (1996). In order to formulate a treatment of the variation of the photodissociation of H^- with extinction, we integrated the cross-sections of Wishart (1979) over the local ISRF, finding an unshielded rate of $1.5\times 10^{-7}$ s^-1, just over half of which arises at wavelengths beyond 800 nm; the photo-dissociation is, therefore, not strongly attenuated under the conditions discussed here and we elected to ignore extinction in this regard (the rate quoted in the UMIST database is $2.4\times 10^{-7}$ s^-1 but we used the smaller value).

Current values of the reaction rates for all important processes are given in the UMIST reaction database, whose values we employed unless otherwise noted. Species followed during modelling of the chemistry included H I, He⁺, H⁺ and e^- - all given by the calculations of two-phase equilibrium -as well as H^-, ${\rm H}_2$ and ${\rm H}_2^+$.

Figure 7: Molecular fraction as a function of fractional distance into cool clouds of density $n({\rm H}) = 32~{\rm cm}^{-3}$ and various central column density N(H) as indicated. The uppermost curve in each panel is for local conditions. The labelled curves are for varying metallicity and a dust/metal ratio D/Z = 0.6 (60% of the local value). The bottom curve represents formation solely in the gas phase, in the limit of zero metallicity parameter; see Sect. 5 of the text.

Figure 6 shows the free-space abundance of ${\rm H}_2$ arising solely from gas-phase processes in the two-phase models under conditions of varying metallicity and ISRF (the two left-hand panels of Fig. 1). Unlike grain formation scenarios, which take advantage of high N(H) to boost the molecular fraction at high density, free-space gas-phase formation of ${\rm H}_2$ does not seem to distinguish between warm and cool conditions. The molecular fractions calculated in Fig. 6 correspond well with the smallest values in the local ISM, or, for that matter, in damped Lyman-$\alpha$ systems. Unfortunately, this minimum is not diagnostic of the host gas conditions.

4.2 The H $_\mathsfsl{2}$ abundance in cool clouds

${\rm H}_2$ formation in cool neutral gas clouds is illustrated in Fig. 7, which indeed shows why even damped Lyman-$\alpha$ systems with appreciable cool gas still may lack molecular hydrogen. To create this diagram, we considered (following Liszt & Lucas 2000) a spherical clot of gas of constant density, immersed in isotropic radiation fields (X-ray, cosmic-ray, optical/uv, etc.). This was computationally divided up into 128 radial zones, in each of which we derived the temperature/ionization structure and ${\rm H}_2$ abundance. The latter requires iteration, because the maintenance of ${\rm H}_2$ is a sharply non-linear process dependent on the ${\rm H}_2$ column and extinction between any point and free-space (Lee et al. 1996). We adopted a fairly straightforward relaxation method which converged with gratifying rapidity.

Calculation of the abundance of molecular hydrogen is typically made feasible by employing a set of shielding factors which account in an average way for the many very complicated effects of line-overlap and radiation transport in the dissociation process. We used the shielding factors of (Lee et al. 1996) which were calculated for local gas. The justification for this is that the dominant effect requiring consideration here is the order of magnitude change in the number of grains at very low metallicity, not the factor of two differences in individual grain properties between local grains and those seen, for instance in the Magellanic clouds. The parametrization of Pei (1992) for the Milky Way, LMC and SMC shows that, for a given amount of B-band extinction, the grain distribution provides successively somewhat more extinction at (say) Ly-$\alpha$ as the metallicity declines; the inference is that graphite grains disappear and silicates do not. But this effect is dominated by the overall diminution of the extinction with lowered metallicity.

Figure 7 shows the radial variation of the fraction of H-nuclei in molecular form over spherical gas clots of constant density $n({\rm H}) = 32~{\rm cm}^{-3}$ for different column densities N(H) through the center of the clot. The mean line of sight averaged over the circular face of such a body intersects it at an impact parameter of 2/3 of the radius (at a value 0.33 along the horizontal axis in Fig. 7), where the column density is 3/4 of that through the center.

In each panel of the figure there are 8 vertically-separated curves. At top, shaded, is the result which would apply in the Milky Way, where we have taken the dust/metal ratio as observed locally (the reference model of the two-phase calculations) and depleted carbon and oxygen in the gas phase by a factor of 2.4. The bottom curve, also shaded, is the result when the metallicity goes to zero and only the gas-phase formation of ${\rm H}_2$ is included; a modest amount of self-shielding occurs and the molecular fraction is slightly higher than in free space (Fig. 6). The intermediate curves assume a dust/gas ratio 0.6 relative to the reference model for the Milky Way following Vladilo (1998) (a small effect at higher N(H) but of real importance to the thinnest model) and no depletion of carbon and oxygen. These curves are labelled by their varying metallicity as in the previous diagrams.

The cloud with $N({\rm H}) = 4 \times 10^{20}~{\rm cm}^{-2}$ would be a compact (4 pc), cool (130 K) Spitzer (1978) "standard'' cloud in the Solar neighborhood. It would also be very substantially molecular if found in the Solar vicinity, but a factor 4 decline in the metallicity suffices to reduce the molecular fraction by some four orders of magnitude. Even the model having a four times higher column density (compare with the entries in Tables 1-2) cannot sustain an appreciable molecular fraction when the metallicity is reduced by a factor 10, which is hardly extreme for one of the damped Lyman-$\alpha$ systems.

4.3 The role of geometry

The role of geometry can also be inferred from Fig. 7. At any given metallicity, a cloud with lower N(H) produces much less than one-fourth as much ${\rm H}_2$ as that illustrated in the next-lower panel. This is another reason why the molecular fraction may vary widely between two lines of sight with similar N(H), N(C II)/N(C I), and/or metallicity (for example). Molecular hydrogen will readily populate a region when the circumstances are propitious, but can easily be prevented from forming by the vagaries of local source structure.

4.4 PKS0528-250

This source (Table 2) has an overall molecular fraction $5.5\times10^{-5}$ despite the lack of evidence (in carbon) for any appreciable amounts of cool gas; earlier we asserted that (very roughly) no more than a few percent of the gas could be cool. Carilli et al. (1996b) did not detect 21cm H I absorption, placing a $2-\sigma$ upper limit N(H I)  $\le 2 \times 10^{18}~{\rm cm}^{-2}~\Tsub sp$. So, the molecule-bearing gas must be cool, occupying roughly 1% of the total gas column for $\Tsub sp\la 100$ K. In the context of our models the gas must also be fairly dense, $n({\rm H}) \ga 100~{\rm cm}^{-3}$, occurring over only a very small fraction of the path length (1 kpc or more) occupied by the gas as a whole.