Astron. Astrophys. 334, 1047-1055 (1998)

2. Chemical models and strategy

The large gas-phase chemical network emphasized in this paper - the "new neutral-neutral model" of Bettens et al. (1995) - includes 409 species and almost 4,000 chemical reactions. The model contains a large selection of rapid neutral-neutral reactions based on an extrapolation of limited low-temperature experimental results (Sims & Smith 1995). The new neutral-neutral model fails to produce sufficiently large abundances of complex molecules at early time to explain observed results in the well-studied source TMC1. A second model of Bettens et al. (1995), the "new standard model", contains far fewer rapid neutral-neutral reactions and produces much larger abundances of complex molecules at early time. We adopt the new neutral-neutral model as the standard model here, since it is physically the more reasonable and since large abundances of complex molecules are not produced in either network with most of the elemental abundances discussed below. Some analogous calculations have been undertaken with the new standard model; differences between the two models relating to bistability are mentioned in the discussion section.

In the use of both networks here, we neglect photodestruction of molecules caused by external photons (), although we include photodissociation and photoionization induced by secondary UV photons produced by cosmic rays (Gredel 1990). A small but significant change to the networks is the use of the latest storage ring results (Larsson et al. 1993; Sundström et al. 1994) for the temperature dependence and product branching fraction of the dissociative recombination rate coefficient of with electrons; the rate is larger at low temperatures than the rate used by Bettens et al (1995). Specifically, the formula used for the overall rate coefficient k is 1.15 10^-7 cm³ s^-1 with a product fraction of 0.75 for the 3H channel and 0.25 for the H₂ + H channel. A large rate ( 10^-8 cm³ s^-1) is well known to be critical for bistability (Le Bourlot et al. 1995a).

For a large number of gas-phase elemental abundances, the region of bistability was elucidated by running two types of models under homogeneous physical conditions at a temperature of 10 K. At Meudon, steady-state calculations were run, while at Columbus, pseudo-time-dependent calculations were run until steady-state conditions were reached. The two methods are in excellent agreement. The time-dependent results were obtained by solving a system of ordinary but non-linear differential equations with the Gear method. The region of bistability (if any) for each set of elemental abundances was determined by plotting the ionization fraction vs the parameter . For each set of elemental abundances, two types of initial conditions (or, in the case of the steady-state calculations, initial guesses) were utilized. With the "atomic" initial conditions, all of the elements other than hydrogen are completely in atomic form. This set corresponds to the typical initial conditions in pseudo-time-dependent models (e.g. Lee et al. 1996). With the "molecular" conditions, the carbon, nitrogen, and oxygen start off in the molecules CO, N₂, and O₂. In the region of bistability, as mentioned previously, the atomic initial conditions lead to the HIP solution and the molecular initial conditions lead to the LIP solution. Outside of the region of bistability, the two sets of initial abundances lead to the same steady-state abundances. Fig. 1 contains a typical plot of fractional ionization vs in which a bistable region exists; the effect would be just as noticeable if we plotted other results, such as the abundance ratio between C and CO, vs . The vertical lines at the two critical points are to elucidate the bistable region only.

Fig. 1. The phenomenon of bistability at steady state shown by plotting the ionization fraction (with respect to ) vs. . The arrows depict the domain of where two solutions occur. The upper abscissa scale of density corresponds to s-1. The particular calculation makes use of "low metal" abundances except for a fractional sulfur abundance of 2 10-6.

A wide variety of elemental abundances has been used through the years in gas-phase models of dense interstellar clouds. Virtually all have, to a greater or a lesser extent, contained the assumption of depletion of heavy elements to form the dust particles. In diffuse clouds, optical measurements have been used for some time to determine gas-phase elemental abundances. Early measurements (Morton 1975) showed the elements C, N, and O to be depleted by factors of 6, 2.5, and 4, respectively, from solar values. More recent measurements (Cardelli et al. 1996; Savage & Sembach 1996; Grevesse et al. 1996; Meyer 1997) show somewhat more modest depletions of these elements of 3, 1-2, and 2 respectively, as well as hardly any depletion for S. In addition, Li & Greenberg (1997) have recently used indirect evidence and a so-called unified dust model to discuss elemental depletions in diffuse clouds. In dense clouds, rather modest additional elemental depletions from the gas for C (2.4%), and O(13%), have been derived by Shalabiea & Greenberg (1995) to account for the presence in grain mantles of water and CO.

In this global study, we have used a variety of elemental abundances, which can be regarded as variations of three primary sets of abundances. Two of these sets are the well-known "low metal" and "high metal" abundances (e.g. Leung et al. 1984; Graedel et al. 1982) in which the elements C, N, and O are depleted according to Morton (1975). The "high metal" and "low metal" results differ in abundances for the elements S, Si, Na, Mg, and Fe, with the former containing a modest depletion of 2 for sulfur and stronger depletions of 50, 10, 60, 110, respectively for the other elements while the "low metal" values contain additional depletions of 100 for each. Although the "high metal" abundances give a reasonable description of diffuse cloud gas-phase measurements, the "low metal" results typically yield substantially better agreement with observation in cold dense interstellar clouds (Graedel et al. 1982), without, it would appear, violating any obvious elemental constraints based on gas-phase observations. In addition to these two sets of primary abundances, we have also started with a set of abundances we refer to as the "dense core" abundances. These are derived by Flower et al. (1995) from the solar values of Anders & Grevesse (1989) via an analysis based on a review article by van Dishoeck et al. (1993). Here, near solar abundances are used for N, O, and S, while C is depleted from its solar value by a factor of 2-3. As opposed to these non-existent to small depletions, the "dense core" abundances contain only a small amount of true metals (Na, Mg, Fe) so that they resemble the "low metal" abundances except for a large amount of sulfur. The gas-phase abundance of sulfur appears to be particularly poorly constrained in dense clouds based on observations to date, which is unfortunate given the primary role of this element in bistability (see below). The three primary sets of abundances are all listed in Table 1 along with solar abundances. Since true metals lead to relatively high levels of ionization and the element sulfur can be regarded as a "soft ionizer", leading to moderate levels of ionization, the three sets of abundances will lead at moderate to high densities ( 5000 cm^-3 with "standard" ) to fractional ionizations that are relatively low ("low metal" abundances), moderate ("dense core" abundances), and high ("high metal" abundances). After studying the bistability phenomenon using these primary sets of abundances, we then undertook variations, mainly in the element S, to determine their effect on bistability. It should be reiterated that the Meudon group has also used a variety of other elemental abundances in their bistability calculations (Le Bourlot et al. 1993, 1995a).

Table 1. Solar and primary elemental abundances with respect to total hydrogen.
Note: the C, N, and O abundances in the "dense core" column are based on the earlier solar values of Anders & Grevesse (1989) which are slightly different from those of Savage & Sembach (1996).

© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998

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