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

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3. Results

We found that bistability occurs for certain ranges of [FORMULA] for a wide assortment of elemental abundances. However, certain elemental abundances do not lead to bistability. The most prominent are abundances with large amounts of actual metals (Na, Mg, Fe). Starting from the "high metal" abundances in Table 1, we varied the S abundance but found bistability under no circumstances. Some relevant results from the steady-state calculations are shown in Fig. 2a, in which fractional ionization is plotted vs both [FORMULA] and [FORMULA] (assuming [FORMULA] s-1.) The other sets of primary abundances in Table 1 do show bistability, as do variations from them, especially in the S abundance. The region of bistability obtained using the "dense core" abundances and variations lies at high [FORMULA] (low densities) for a wide variety of sulfur depletions. With an enhancement in S, however, the region of bistability for the "low metal" abundances lies at surprisingly small [FORMULA] (large densities). Steady-state results for these two cases are shown in Figs. 2b and 2c, respectively.

[FIGURE] Fig. 2. a The ionization fraction plotted vs. [FORMULA] and vs. [FORMULA] ([FORMULA] s-1) for "high metal" abundances and different sulfur depletions. No bistability can be seen. b The ionization fraction plotted vs. [FORMULA] and vs. [FORMULA] ([FORMULA] s-1) for "dense core" abundances and different sulfur depletions. Bistability can be seen but only at rather large values of [FORMULA] (low densities). c The ionization fraction plotted vs. [FORMULA] and vs. [FORMULA] ([FORMULA] s-1) for "low metal" abundances and different sulfur depletions. Bistability can be seen over a wide range of [FORMULA] values, depending strongly on the sulfur depletion.

A global view of our results can be seen in Fig. 3 where we have synthesized many individual results of the type shown in Figs. 1 and 2, with ordinate the fractional ionization and abscissa [FORMULA]. Note that for a "typical" value for [FORMULA] of [FORMULA] s-1, the densities on the plot range from a low near 10 cm-3 to a high near 105 cm-3 with increasing density going from right to left. In Fig. 3a, the individual results can be seen, while in Fig. 3b the outer borders of the region where bistability occurs are depicted. We have undertaken such a synthesis in the hope that, like the HR diagram, a global figure of this type would be instructive. To make Fig. 3a, we plotted actual points from bistable regions only. Calculations for elemental abundances in which bistability occurred for no range of [FORMULA] as well as the results outside of the bistability range were not plotted. Bistable solutions based on the "low metal" elemental abundances (Table 1) and variations as well as bistable solutions based on the "dense core" abundances and variations are included. It can be seen that for all solutions plotted, bistability occurs in a vertical band of fractional ionization which appears to widen somewhat as one goes from small to large [FORMULA] (large to small gas density). The amount of widening is uncertain since we may expect that at low densities, UV penetration leads to an increase in temperature, an effect that hinders bistability but, at the same time, increases the abundances of [FORMULA] and [FORMULA], which may have the opposite effect.

[FIGURE] Fig. 3. a A "global" diagram for bistability obtained by superimposing points from the bistability regions of different calculations on a plot with the fractional ionization as the ordinate and [FORMULA] as the abscissa. Both LIP and HIP results for elemental abundances based on the primary "low metal" and "dense core" abundances in Table 1 are depicted. The new neutral-neutral model network has been used. b The results of a are depicted so that only the outer borders of the region where bistable solutions occur are shown. Above the bistable region at high densities lie the single-phase "high metal" dense cloud solutions while below the band at these densities lie the "low metal", low sulfur solutions.

Within the band of intermediate ionization depicted in Fig.  3b, it is by no means guaranteed that bistability will occur for a given set of abundances, since calculations with individual elemental abundances only go bistable for limited ranges of [FORMULA]. Outside of the band, however, we have not been able to find bistability. Above the band at relatively high densities lie the single-phase "high metal" dense cloud solutions whereas below the band at these densities lie the "low metal", low sulfur solutions. These latter solutions pertain to the typical models used at early-times to explain molecular abundances in sources such as TMC-1 (note that the new neutral-neutral model used here fails in this task). The bistable solutions in the band at high density ([FORMULA] cm-3 with [FORMULA] s-1) are for "low metal" elemental abundances with, however, an increased sulfur abundance which results in higher ionization levels. The different points also derive from variations in the C/O elemental abundance ratio. These high density bistable solutions are new ; previous solutions exist mainly at densities lower than standard dense cloud values unless a high value of [FORMULA] is assumed.

We present detailed steady-state results for one such high density bistable solution - obtained with an elemental sulfur fractional abundance of 2 10-6 and the other elements held to their "low metal" values. The region of bistability occurs within a range for [FORMULA] of 2.7-6.2 10-22 cm3 s-1, which, with a "standard" value of [FORMULA] = 1.3 10-17 s-1, corresponds to a density range of 2.1-4.8 104 cm-3. Results for both phases at a density of 2.6 104 cm-3 are listed in Table 2 for species with fractional abundances greater than 10-12 in at least one phase; these are discussed in the next section.


[TABLE]

Table 2. Fractional LIP and HIP abundances for selected species with respect to H2 at steady state with [FORMULA] = [FORMULA] cm-3 and T = 10 K



[TABLE]

Table 2. (contitnued)


A similar plot to Fig. 3 with the abundance ratio between C and CO replacing the fractional ionization is not as useful since many "high-metal"-based solutions without bistability lie in the same region as the bistable results.

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© European Southern Observatory (ESO) 1998

Online publication: June 2, 1998

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