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Astron. Astrophys. 362, 697-710 (2000)

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2. An illustrative, one-dimensional model

The formation of stars occurs in dense condensations within interstellar molecular clouds, which collapse under the influence of their own gravity. A widely used theoretical description of this process, constructed by Shu (1977), starts with the singular isothermal sphere,


Here, [FORMULA] is the density as function of radius r and time t, a is the isothermal sound speed, and G is the gravitational constant.

At [FORMULA] collapse starts at the center ([FORMULA]). After a time t, all regions [FORMULA] are collapsing, with speed [FORMULA] increasing from 0 at [FORMULA] to free-fall, [FORMULA], well within this `collapse expansion wave' ([FORMULA]). Shu (1977) constructed a solution for the density and velocity field of the collapsing core which is self-similar in the spatial coordinate [FORMULA]. The density follows a power-law behaviour as function of radius, with [FORMULA] for [FORMULA], [FORMULA] just inside [FORMULA], and the undisturbed [FORMULA] outside [FORMULA] (Fig. 1).

[FIGURE] Fig. 1a-d. Density (top left) and velocity (bottom left) structure of the spherically-symmetric inside-out collapse model of Shu (1977) used to illustrate the radiative transfer and molecular excitation problem (Sect. 2). The excitation of HCO+ (top right; solid line) ranges from LTE in the dense, central regions to sub-thermal in the lower density outer regions. Compared to the optically thin excitation of H13CO+ (top right; dashed line), line trapping significantly influences the HCO+ excitation. The distribution of the kinetic temperature is shown with the thick line for comparison. The lower right panel shows the emergent HCO+ and H13CO+ J=4-3 line profiles in a [FORMULA] beam for a source at 140 pc. The asymmetric profile of the optically thick HCO+ 4-3 line is characteristic of infall.

Many authors have tested this model against observations of cloud cores and envelopes around young stellar objects (YSOs), e.g. Zhou et al. (1993), Choi et al. (1995), Ward-Thompson et al. (1996) and Hogerheijde & Sandell (2000). Especially the spectral-line signature of collapse (Fig. 1d) has received much attention as a probe of ongoing collapse, although this signature is shared by all collapse models and is not unique to the particular model described here. The exact line shape, however, depends quantitatively on the adopted model.

The interpretation of this signature needs non-LTE radiative transfer. Both collisional and radiative processes can excite molecules, and for each transition a critical density can be defined where the two are of equal importance. At lower densities radiation dominates, while at higher densities collisions drive the level populations to thermodynamic equilibrium. The large range of densities of star forming cores ensures that many molecules and transitions will go through the entire range of excitation conditions, while line emission will have a significant impact on the excitation at the intensities and opacities expected for typical abundances of many species, not only locally but throughout the envelope (Fig. 1c).

In the following we will use this model to illustrate our method of solving the coupled problem of radiative transfer and excitation. In particular, we will consider emission lines of HCO+ and H13CO+, which are readily observed and often used as tracers of dense gas. The strong [FORMULA], [FORMULA] and [FORMULA] lines at millimetre wavelengths have critical densities of [FORMULA], [FORMULA], and [FORMULA] cm-3, using the molecular data in Table 1. We assume an abundance of HCO+/H2 = [FORMULA] and an isotopic ratio of 1:65 for H13CO+: HCO+. The sound speed of the adopted model is [FORMULA] km s-1, its age [FORMULA] yr, and its outer radius 8000 AU. The total mass of the model is 0.73 [FORMULA]. The kinetic temperature follows [FORMULA], appropriate for a centrally heated envelope at a luminosity of [FORMULA] [FORMULA] (Adams et al., 1987, e.g.). The turbulent line width of 0.2 km s-1 is smaller than the systematic velocities except in the outermost part (Fig. 1b).


Table 1. Molecular data used in this paper.
a) Calculation presented in Appendix B.
b) Calculation presented in Sects. 4.1 and 4.2.
c) Based on results of Green & Chapman (1978) for CS.
d) Levels up to [FORMULA] in both the [FORMULA] and [FORMULA] states.
e) See http://www.giss.nasa.gov/data/mcrates . labelt:rates

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© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000