Astron. Astrophys. 324, 1-10 (1997)

6. Conclusions

We have obtained regimes of thermal and thermo-reactive instability for a range of initial redshifts and fluxes (see Sect.5 and Table 1). In each case, the boundaries of the instability regime as defined by independent criteria (enhancement, density contrast, etc.) were mutually consistent and only moderately dependent on the instability thresholds defined. As seen in Sect. 5, it is remarkable that even in the presence of flux there is a lower temperature bound to the instability regime that is almost independent of the density as long as the lower bound is exceeded. The minimum (initial) number density for instability to occur corresponds to requiring that the flux be almost entirely shielded.

In order to estimate the quality of the approximation and averaging procedure in Eq. (10), we have performed a spatially dependent analysis for selected critical cases. This analysis shows that the qualitative picture does not change and the error is far below one order of magnitude. Moderate quantitative changes would be expected chiefly with respect to the characteristic sizes obtained. The advantages of the simplifying assumption made at the end of Sect.  2are that they allow one to obtain insight into the qualitative behavior of optically thick systems and that the analysis, which involves an exhaustive search of parameter space, can be provided for reasonable computing times.

The above characterization of instability regimes favorable for star formation is of particular use in the context of hydrodynamical numerical simulations of large-scale structure with galaxy formation (YKKK; Steinmetz, 1996). In such codes, smoothed estimates of the gas density and temperature are produced at every timestep. Consider for the present discussion a medium with primordial abundances; suppose that some estimate of the ultraviolet flux is available (see e.g. Mücket et al. (1996)). Now, the evolution of thermal and thermo-reactive instabilities involves the ionization states. These in turn depend in principle not only on the current local values of the temperature, density, and ultraviolet radiation flux, but also on the past history. However, considering the computational expense of explicitly simulating the time evolution of all relevant ionization species in every cell, it is useful to study what can be inferred from a knowledge of the current pressure, density, and ionizing flux alone. The present paper shows that it should generally be possible to estimate the instability regimes for any desired redshift/flux combinations by making use of look-up tables compiled "off-line" at each redshift as a function of temperature and pressure. The trends found here at least roughly characterize conditions favorable for formation of a multiphase medium as a function of redshift and the ambient temperature and density.

Such a characterization of instability regimes would be an extremely useful tool in studying the influence of photoheating on star formation, and hence on galaxy evolution. The suppression of thermal instability with increasing UV flux also illustrates how an antibiasing mechanism might arise if one were to take into account explicitly the emission and transport of UV radiation by massive galaxies and quasars (Efstathiou, 1992; Haardt & Madau, 1995; Ferrara & Giallongo, 1996).

Referring to Figs. 6 and 12, we identify the thermo-reactive instability seen in the evolution of the cold perturbation as a possible mechanism for at least some Lyman limit systems: During the course of the evolution, the optical depth of the perturbation increases dramatically according to our model. For a region of size 50 kpc, column densities ranging from 10¹⁷ to about 10²⁰ cm^-2 or neutral hydrogen densities of up to cm^-3 would result. (Note that the full hydrogen density also increases with respect to its initial value.)

Realistically, the evolution of the thermo-reactive instability leading to Lyman limit systems will need to be modeled more precisely taking properly the spatial variation of the optical depth and the onset of Jeans instability into account. One would expect that the thermo-reactive instability would have a tendency to propagate out from an initial core. Once the instability has affected a region exceeding the Jeans length, as occurs in Fig. 6, gravitational instability would set in.

The conditions would thus be favorable for protogalaxy formation leading to star formation and hence some degree of enrichment of the medium with heavy elements at a later stage. Thus, one would expect Lyman limit systems that had previously formed from primordial gas according to our scenario to contain heavy elements when observed now. Indeed, evidence for heavy elements in Lyman limit systems has been observed (Petitjean & Bergeron, 1990; Petitjean et al., 1994).

In Fig. 12 one also finds regimes for which the cold component is decoupled () but still optically thin. These regions could be of interest in more detailed models of the formation of Ly- clouds.

Since the main goals of this paper were to characterize the energy budget of primordial gas in terms of parameters (density, temperature) available from numerical simulations, the question of bifurcations was not addressed here in full mathematical detail. However, in view of the potential importance of bifurcations for a qualitative understanding of observations (such as Lyman limit systems), the general mathematical structure of ionization problems in a diffuse gas is worthy of further study in its own right.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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