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Astron. Astrophys. 334, 678-684 (1998)

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2. Fractional ionization as a function of [FORMULA], [FORMULA], [FORMULA], [FORMULA] and [FORMULA].

Fig.  1 contains the results of steady state equilibrium chemical models for [FORMULA] as a function of [FORMULA], the visual extinction to the near edge of a semi-infinite plane-parallel cloud. Equilibrium results were obtained through the integration of rate equations from a time when all elements more massive than hydrogen were primarily in neutral and ionic atomic form but most of the hydrogen was in H2. The standard interstellar radiation background was assumed to be incident on the near side of the cloud, and it is assumed that a cosmic ray induced radiation field is present throughout the cloud (Prasad & Tarafdar 1983). In addition to an extensive network of gas phase reactions for which rate coefficients were taken primarily from the UMIST compilation (Millar et al. 1997), we included modifications to the gas phase chemistry due to the presence of grains by assuming that all material striking grains is immediately returned to the gas phase, but that many of the species are processed in such a way that much of the material is returned in the form of saturated species like H2, NH3, H2 S and CH4. The detection of NH in interstellar clouds (Meyer & Roth 1991; Crawford & Williams 1997) supports this view, at least for clouds with values of [FORMULA] [FORMULA] 1 (see Williams 1993). The rate at which particles of neutral species X were assumed to strike grain surfaces is 3.5 [FORMULA] 10-18 [FORMULA] ([FORMULA] /cm [FORMULA], where [FORMULA] is the number density of hydrogen nuclei; charged particles were assumed to strike grains (mostly negatively charged) at a rate that is 18 times larger (following Rawlings et al 1992). All ions striking grains were assumed to return to the gas phase as neutral atoms and molecules. The cosmic ray ionization rate was taken to be 1.3 [FORMULA] 10-17 s-1. Molecular hydrogen is self-shielding, and CO was assumed to be sufficiently self-shielded and shielded by H2 that only photons produced as a consequence of cosmic ray induced ionization are important for its photodissociation.

[FIGURE] Fig. 1. Steady state fractional ionization as a function of [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA]

The results in Fig. 1 are for the two sets of assumed gas phase elemental abundances given in Table 1 (see Shalabiea & Greenberg 1995). Set A is similar to that required in many cases to reproduce chemical abundances measured for dark cores while Set B, which has higher fractional abundances of low ionization potential elements, is selected because diffuse clouds have much higher gas phase fractional abundances of those species than dark cores. The issue of the mechanisms that control the depletion of the low ionization elements is unclear at present (hence the parametrised approach adopted here), but is one of major importance for the chemistry of star formation, as the results in this paper indicate.


Table 1. Fractional elemental abundances relative to [FORMULA]

Inspection of Fig. 1 shows that for [FORMULA] = 5 [FORMULA] 102 cm-3 (typical of the RMC clumps) the gradient -d(log [FORMULA])/d [FORMULA] reaches a maximum in the range [FORMULA] [FORMULA] for Set B depletions and a local maximum for the same range of [FORMULA] for Set A depletions. For Set A depletions and [FORMULA] = 500 cm-3 a much higher value of -d(log [FORMULA])/d [FORMULA] exists for a very small range of high values of [FORMULA] where a chemical phase transition occurs, but the high value of -d(log [FORMULA])/d [FORMULA] in the range of [FORMULA] [FORMULA] Set A as well as Set B depletions, is no doubt important for the ion-neutral damping of turbulence supporting RMC-like clumps. Taking (1 [FORMULA] 1016 cm-2 /2) to be half the critical measured value of the 13 CO column density from the edge of an RMC clump to its centre, [FORMULA] CO)/ [FORMULA] = 1 [FORMULA] 10-6 (Williams et al. 1995), and the standard conversion of the hydrogen nuclei column density to [FORMULA] (Savage & Mathis 1979), we find that the critical visual extinction from the centre of an RMC clump to its edge is 2.5, in agreement with the regime of maximum -d(log [FORMULA])/d [FORMULA] for [FORMULA] = 500 cm-3 identified in our calculations.

Results for [FORMULA] are also given in Fig. 1 for a range of values of [FORMULA]. As stated above, the ion-neutral damping rate of a wave of sufficiently low frequency is proportional to the square of the wave frequency divided by [FORMULA]. A comparison of the Set A graphs for [FORMULA] = 5 [FORMULA] 102 cm-3 and [FORMULA] = 5 [FORMULA] 104 cm-3 shows that at [FORMULA] = 3, the values of [FORMULA] for the two model clumps are roughly equal. Therefore the ion-neutral wave damping rate for a given frequency has roughly the same functional dependence on [FORMULA], implying that even for dense cores the [FORMULA] dependence of [FORMULA] must be taken into account in any consideration of wave propagation.

The chemical calculation yielding [FORMULA] also provides information about the depth dependence of many other chemical abundances. Figs. 2 and 3 give results for the fractional abundances of a number of species for a [FORMULA] = 1 [FORMULA] 103 cm-3, Set A clump and a [FORMULA] = 5 [FORMULA] 104 cm-3, Set A dense core. The model results indicate that other than CO emissions only CH and OH emissions are likely to be detectable towards RMC clumps with [FORMULA] = 1000 cm-3, and also that the CH, OH and CO fractional abundances are not sensitive to [FORMULA], [FORMULA] and [FORMULA]. Unfortunately, values of CH and OH abundances are difficult to infer observationally. A comparison of Fig. 3 with Fig. 2 shows that [FORMULA] is the dominant ion throughout a dense core, while [FORMULA] is the dominant ion over a large range of [FORMULA] in [FORMULA] = 1000 cm-3 gas. Unlike RMC-type clumps with [FORMULA] = 3, a dense core with [FORMULA] = 3 and [FORMULA] = 5 [FORMULA] 104 cm-3 is likely to produce observable emissions in a number of molecular species other than CH, OH and CO, including [FORMULA], N2 [FORMULA], H2 CO, CN, NH3, CS and possibly HNC and HCN.

[FIGURE] Fig. 2. Steady state fractional abundances of species as functions of [FORMULA] for [FORMULA] = 103 cm-3 and case A depletions (see text for description)

[FIGURE] Fig. 3. Steady state fractional abundances of species as functions of [FORMULA] for [FORMULA] = 5 [FORMULA] 104 cm-3 and case A depletions (see text for description)

Though our results are more extensive than those of previous authors, they are in harmony with those of Le Bourlot et al. (1993b), Flower et al. (1994), and Shalabiea & Greenberg (1995) in regions where comparisons can be made.

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© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998