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, is the density as function of radius r and time t, a is the isothermal sound speed, and G is the gravitational constant.
At collapse starts at the center (). After a time t, all regions are collapsing, with speed increasing from 0 at to free-fall, , well within this `collapse expansion wave' (). Shu (1977) constructed a solution for the density and velocity field of the collapsing core which is self-similar in the spatial coordinate . The density follows a power-law behaviour as function of radius, with for , just inside , and the undisturbed outside (Fig. 1).
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 , and lines at millimetre wavelengths have critical densities of , , and cm-3, using the molecular data in Table 1. We assume an abundance of HCO+/H2 = and an isotopic ratio of 1:65 for H13CO+: HCO+. The sound speed of the adopted model is km s-1, its age yr, and its outer radius 8000 AU. The total mass of the model is 0.73 . The kinetic temperature follows , appropriate for a centrally heated envelope at a luminosity of (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.
© European Southern Observatory (ESO) 2000
Online publication: October 24, 2000