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

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1. Introduction

In their work on the clumpy structure of the Rosette Molecular Cloud (RMC), Williams, Blitz and Stark (1995) identified seven star forming clumps, none of which has an estimated 13 CO column density, N(13 CO), of less than 1016 cm-2. From their Fig. 21, it appears that eight or nine other clumps have N(13 CO) [FORMULA] 1016 cm-2 but contain no stars. The estimated values of N(13 CO) for the star forming clumps, with one exception, vary by only a factor of about 2, and the estimated masses vary by a factor of about 7. While one could argue that the Williams et al. (1995) data point to the existence of a minimum mass required for star formation to occur in the RMC, we suggest that they indicate, at least as strongly, the existence of a minimum column density, N(13 CO) [FORMULA] 1016 cm-2, for stars to be born.

Hartquist et al. (1993) suggested that there should be a maximum value of the visual extinction, [FORMULA], of a clump supported against gravitational collapse along its large-scale magnetic field by internal Alfvén waves thought to comprise clump turbulence (Arons & Max 1975; Caselli & Myers 1995; Mouschovias & Psaltis 1995). Their idea was that above this critical visual extinction the damping rate of waves by ion-neutral friction (Kulsrud & Pierce 1969) is too rapid for the waves to be maintained at sufficient amplitudes to support a clump. This damping rate of waves in which the ion-neutral motions are well-coupled declines with extinction because it increases as the inverse of the number density of ions, [FORMULA], (and proportionally to the square of the frequency). The value of [FORMULA] itself decreases with visual extinction. Recent numerical simulations of wave behaviour in self-gravitating clumps show that nonlinear magnetohydrodynamic waves can, in fact, support such clumps against collapse (Gammie & Ostriker 1996). In view of its relevance to wave damping in clumps it is now particularly timely to return to the issue of the behaviour of [FORMULA] [FORMULA] [FORMULA] / [FORMULA] (where [FORMULA] is the number density of hydrogen nuclei) as a function of [FORMULA].

The view taken by Hartquist et al. (1993) and by us in the current work, is, thus, that much of the molecular material in giant molecular clouds is translucent to radiation and that photoionization affects its fractional ionization, which in turn plays a role in determining the rate at which material collapses. Our view is one that has much in common with that adopted by McKee (1989) who argued that star formation regulates itself because the births of stars lead to the production of radiation which raises the fractional ionization in translucent material and, consequently, lowers the rate of ambipolar diffusion. There are differences between McKee's and our points of view. We stress the role of turbulent support of clumps like those identified by Williams et al. (1995) (see also Bertoldi & McKee 1992); these are much more tenuous objects than the magnetically subcritical dense cores in which turbulent support is likely to be much less important than the support provided by the large-scale magnetic field and in which the collapse timescale is established by ambipolar diffusion. Consequently, we feel that McKee's (1989) considerations are likely to be of more relevance to more evolved objects formed through the collapse of the sorts of clumps Williams, Blitz & Stark (1995) found in their CO studies. In this paper we do present results for the fractional ionization for a wide variety of conditions; so many of our ionization calculations are of direct relevance for the application of McKee's (1989) model to self-regulation of later stages of the formation of low-mass stars.

Another reason for returning to this issue is that in the last few years a revision of ideas about the fractional ionization in some molecular cloud environments has occurred. Pineau des Forêts et al. (1992) and Le Bourlot et al. (1993a) discovered, for a range of assumed [FORMULA] dissociative recombination rate coefficients, a class of dark cloud gas phase chemical equilibrium solutions in which the [FORMULA] abundance is much lower and [FORMULA] much higher than in the solutions of the previously known class. Shalabiea & Greenberg (1995) have studied the effects on the existence of solutions belonging to the two classes of the assumed gas phase elemental fractional abundances of sulphur, [FORMULA], and low ionization potential metals, [FORMULA], such as sodium and magnesium, and of the assumed nature of grain surface chemistry modifications to the dark cloud gas phase chemistry. Though Le Bourlot et al. (1993b) and Flower et al. (1994) gave the fractional abundance of [FORMULA], x ([FORMULA]), relative to hydrogen nuclei as a function of [FORMULA] for one cloud model with an assumed radiation field like that of the typical interstellar background field, there has been no exploration of the simultaneous dependence of the fractional ionization [FORMULA] on [FORMULA], [FORMULA] (the number density of hydrogen nuclei), [FORMULA], [FORMULA] and [FORMULA]. The results of Le Bourlot et al. (1993b) and of Flower et al. (1994) indicate a transition at translucent depths for a model cloud in which [FORMULA] = 103 cm-3 from a solution of the high ionization class to one of the low ionization class.

In Sect. 2 of this paper we report the results of such an exploration and identify a variety of situations under which -d(log [FORMULA])/d [FORMULA] is large. In Sect. 3 we give results for the chemical evolution of a parcel of gas collapsing from [FORMULA] = 103 cm-3 with an initial [FORMULA] within the range in which -d(log [FORMULA])/d [FORMULA] is large. In our description of the collapse dynamics we include a plane parallel collapse phase, representing collapse along the field lines, followed by a phase during which the collapse takes place both across and along the large scale magnetic field and is regulated by ambipolar diffusion, the rate of which depends on the fractional ionization which is taken from our calculations.

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Online publication: May 15, 1998