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

1. Introduction

An understanding of processes occurring in diffuse gas interacting with a photoionizing flux is of direct importance in modeling a number of astrophysical phenomena, such as quasar absorption clouds (Doroshkevich, Mücket, and Müller, 1990; Ferrara & Giallongo, 1996) and the structure of neutral hydrogen clouds in galactic halos (Ferrara & Field, 1994). The effects of heating due to photoionization in galaxy formation have been considered by Efstathiou (1992) in the context of mechanisms for suppression of dwarf galaxy formation and by Navarro and Steinmetz (1996) in their study of the "overcooling" problem. These authors found that photoionization alone could not provide the heating mechanism required, i.e., supernova feedback should also be included. Cosmological simulations of galaxy formation including star formation and supernova feedback have been carried out by Yepes, Kates, Klypin, & Khokhlov (1996; hereafter YKKK). A reasonable hypothesis is that photoheating indirectly influences star formation by regulating the conditions for thermal instability and thus the formation of a multiphase (cloudy) medium.

Since many of the observable phenomena involving photoionization require a prediction of statistical properties of objects, it is important to integrate the local dynamics of diffuse gas (i.e., heating, cooling, ionization, formation of neutral hydrogen) into an overall approach including hydrodynamics and the evolution of large-scale structure. As these processes involve a large dynamical range of scales, some of them will inevitably occur below the limits of resolution of a numerical simulation. Due to the presence of nonlinearities, the effects of fluctuations below a typical cell size will require special attention, especially when instabilities are involved.

It has long been appreciated (Field 1965; Defouw 1970; Balbus 1986; Fall & Rees; 1985; Ibañez & Parravano 1982) that gas subject to cooling (and perhaps heating) processes in a cosmological setting may become thermally unstable, leading to enhanced cooling and the formation of a multiphase medium including cool clouds (Begelman & McKee, 1990). Cool clouds play an important role in theories of the interstellar medium (McKee & Ostriker, 1977) and galaxy dynamics, due to their effects on the energy budget (enhanced cooling), star formation and "supernova feedback" (YKKK).

Corbelli & Ferrara (1995) have demonstrated the existence of so-called "thermo-reactive" instability modes for gas containing metals in the presence of ionizing radiation. Here we will see that instabilities of thermo-reactive type are also possible in gas of primordial composition. In particular, our results imply that thermo-reactive instabilities could play an important role in the formation of at least some population of observed Lyman limit systems.

The principal goal of this paper is to study the dynamics of fluctuations in a gas of primordial composition which may already have been compressed to high ambient density (compared to the background density of the universe) and heated to high temperature as a consequence of large-scale structure formation. The intention is to characterize instability regimes approximately as a function of the ambient temperature and density of the gas, with local effects of gravitation excluded. Such a characterization is expected to be useful for an understanding of conditions for star formation in the context of hydrodynamical numerical simulations. For this application, linear instability analysis about ionization equilibrium solutions does not tell the whole story: For one thing, even if present, some linear instability modes are too slow to be of importance on a dynamical timescale. Moreover, some of the instabilities which occur involve nonlinearity in an essential way and could not be detected by linear analysis. (Because opacity is involved, the strength of the nonlinear effects increases with the size of the perturbed region; see also Ferrara and Field (1994).)

Apart from its importance for the intended applications, the nonlinear system studied in this paper is quite interesting from a purely mathematical point of view: As seen here [and as previously pointed out for example by Petitjean, Bergeron, and Puget (1992)], the equations of ionization equilibrium exhibit multivalued solutions in certain regimes of parameter space. We propose that these solutions are best understood in terms of the theory of bifurcations (or catastrophes): Bifurcations are typical in the equilibria of nonlinear systems, and the mathematical theory (see for example Arnol'd, 1979; Chow & Hale, 1985) gives strong hints and indications of the general behavior to be expected. Near bifurcations, a system can depart rapidly from equilibrium even if the control parameters vary slowly. (In the present case, the "slowly-varying" control parameters are any two thermodynamic variables, say temperature and - on a still longer time scale - pressure.) Hence, even a qualitative understanding of system evolution near bifurcations requires a time - dependent treatment of the coupled system of ionization, heating, and cooling equations. The importance of simulating the dynamical equations for evolution of ionization states was previously discussed for example by Cen (1992), and a dynamical treatment has been incorporated into the hydrodynamic code used in a number of papers (Cen et al. 1990,1992,1993,1994). However, for numerical efficiency it would be useful to know under what circumstances ionization equilibrium is in fact a good approximation.

The organization of this paper is as follows: Sect. 2 gives the model system of equations describing the dynamics of the three species , , and for a gas of primordial composition. Sect. 3 shows that the solution manifold of the ionization equilibrium equations can exhibit the properties of a mathematical bifurcation surface (Golubitsky & Schaeffer, 1985). Of special interest are the solution trajectories of constant pressure, for which one sees that evolution through equilibrium solutions is not always possible. Sect. 4 considers selected solutions of the full time-dependent equations exhibiting some typical behavior. Sect. 5 characterizes the regimes of temperature and density in which instabilities relevant to numerical simulations (i.e., those with sufficiently short timescales) are most likely to occur. Sect. 6 summarizes and describes in particular some potentially interesting implications for Lyman limit systems.

© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998

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