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

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3. Collapse from [FORMULA] = 3

We propose that [FORMULA] [FORMULA] 2.5 to 3.0 represents a critical range for the stability of RMC-type clumps. Above this critical range, the level of ionization and hence the level of turbulent support both fall. We do not know what mechanism is most effective in increasing the extinction of a clump so that it is in or above this critical range, but various possibilities exist. These include compression by clump-clump collisions and increases in the interclump pressure caused by stellar winds or a supernova (e.g. Hartquist & Dyson 1997). In any case, in our study of the time dependent dynamics and chemical evolution of the centre of a collapsing clump, initial chemical conditions like those obtaining at [FORMULA] = 3 in the [FORMULA] = 1000 cm-3 Set A model of the previous section may be reasonably realistic. Thus, we have adopted such initial conditions for two models of collapsing clumps, and in this section we describe the chemical evolution of a parcel of gas during the collapse phase, in a more realistic but simple version of cloud collapse. At this time our contention that a value between 2.5 and 3 for [FORMULA] is a critical one may be viewed as being based on a plausibility argument requiring a more rigorous demonstration through numerical MHD studies of the responses of turbulent clumps to perturbations. In any case, the RMC clumps are translucent and dense cores form in the collapse of such objects. Thus, the results of this section are relevant to clump collapse and core formation and evolution and should be considered to be for initial conditions that not only differ from those used previously in investigations of chemistry during collapse but are also more representative of the chemical state in a typical RMC clump.

In each model, collapse was initially taken to be one-dimensional so that [FORMULA] remains at a constant value of 3. Such a collapse at constant [FORMULA] is an approximation to the type of collapse found to occur initially along the magnetic field in numerical simulations of the evolution of a cloud in which the magnetic pressure initially is by far the highest pressure (Fiedler & Mouschovias 1993). The density was taken to be that which results from the collapse from rest of a plane-parallel cloud with a column density of hydrogen nuclei of 1022 cm-2 and an initial value of [FORMULA] of 103 cm-3, until [FORMULA] reaches a critical value, [FORMULA]. For the results in Fig. 4, [FORMULA] = 2 [FORMULA] 104 cm-3 (model 1); for model 2 (see Fig. 5), we set [FORMULA] = 6 [FORMULA] 104 cm-3. After [FORMULA] = [FORMULA], collapse was taken to occur at a rate slower than the spherical free-fall timescale by a factor of 6.95([FORMULA] /10-3 cm-3)([FORMULA], in harmony with the object being magnetically subcritical and subsequent collapse taking place at a rate controlled by ambipolar diffusion (Mouschovias 1987). For [FORMULA] [FORMULA] [FORMULA] we set


in our models, where an interclump medium was assumed to be responsible for the constant contribution to the [FORMULA]. For the purposes of our calculation we assume that the collapse halts at a density of 3 [FORMULA] 106 cm-3, after which the cloud is treated to be static. The models use the grain-modified gas phase chemistry as described in the previous section. Note that this implies that freeze-out is off-set by chemically driven desorption. Results are given for both models in Table 2, where [FORMULA] and [FORMULA] are the times since collapse began in the cases [FORMULA] = 2 [FORMULA] 104 cm-3 (model 1) and the [FORMULA] = 6 [FORMULA] 104 cm-3 (model 2), respectively. Here, [FORMULA] (X) and [FORMULA] (X) denote the fractional abundance of species X relative to [FORMULA] in the two models, where i represents ions. These exploratory calculations of chemical evolution are the first in which the initial collapse is plane parallel along the field lines, setting up conditions for multidimensional collapse modulated realistically by ambipolar diffusion.

[FIGURE] Fig. 4. Fractional abundances as functions of time since the onset of collapse for model 1 ([FORMULA] = 2 [FORMULA] 104 cm-3). The dashed curves give ([FORMULA] /1012 cm-3)

[FIGURE] Fig. 5. Fractional abundances as functions of time since the onset of collapse for model 2 ([FORMULA] = 6 [FORMULA] 104 cm-3). The dashed curves give ([FORMULA] /1012 cm-3)


Table 2. Fractional abundances as functions of [FORMULA] for two collapse models

Note that the chemically driven desorption of unsaturated species are here assumed to eject all molecules arriving at grain surfaces (see Williams & Taylor 1996), so that the abundances in Figs. 4 and 5 are not constrained by freeze-out. Consequently, the late-time peaks in hydrocarbons and related species (Ruffle et al. 1997) are not evident. In fact, the results in Figs. 4 and 5 are similar in character to those of Howe et al. (1996) who showed that reasonable fits to the chemistry in cores A-D of TMC-1 could be obtained if a low effective freeze-out rate is assumed.

In Table 2, the calculated fractional abundances when the collapse has attained a density of [FORMULA] = 2 [FORMULA] 104 cm-3 of all species are the same for both models. At this point the two models have yet to diverge. After this point, the fractional ionization in model 1, in which the plane-parallel collapse at constant [FORMULA] occurs until [FORMULA] = 2 [FORMULA] 104 cm-3, is lower than in model 2 where [FORMULA] = 6 [FORMULA] 104 cm-3. This is a consequence of the visual extinction being lower in model 2, leading to the background interstellar photons continuing to be important for longer than in the model 1. This leads to a suppression by photodissociation of the early-time peaks of some species in model 2, which do exhibit an early-time peak in model 1. In addition, greater amounts of C, [FORMULA] and [FORMULA] occur in model 2 than in model 1 at [FORMULA] = 7 [FORMULA] 104 cm-3. However, one may account for the timescale difference between the two models, in that model 2 reaches the density of [FORMULA] = 7 [FORMULA] 104 cm-3 earlier than model 1 and then reaches the following displayed density of [FORMULA] = 2 [FORMULA] 105 cm-3 at a later time than model 1.

The late-time abundances of early-time species are reduced in model 2, as a result of the lower visual extinction at [FORMULA] = 2 [FORMULA] 105 cm-3, and the longer time that model 2 takes to reach that density. However, NH3, SO and CS are seen to still be increasing in abundance in model 2, whilst in model 1 their abundances have already peaked.

Recently Ruffle et al. (1997) have argued that the abundances of HC3 N and C2 H will rise at late times in dense cores as depletion occurs; Caselli et al. (1998) have even used the measured fractional abundances of HC3 N in numerous cores to infer the depletions, relative to solar abundances, of C, N and O on the assumption that they did not vary from element to element. Ultimately to establish reliably that depletion is usually the cause of high abundances of such putative early-time molecules, considerable data about the abundances of species with abundances that are sensitive to the relative depletions of trace elements must be gathered. Our current results show that for cores in which high fractional abundances of HC3 N and C2 H exist but are not due to high depletions, the ratio of the abundances of those two species will be significant for the inference of how [FORMULA] varied with [FORMULA] during collapse. In their work, Caselli et al. (1998) drew on published results for HC3 N in many cores; at the time of the submission of the present paper a comparably large set of data for C2 H do not exist in the literature.

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© European Southern Observatory (ESO) 1998

Online publication: May 15, 1998