This linear approximation is able to treat configurations with moderate velocity gradients where the basic assumption of the Sobolev approximation, taking constant physical parameters within the radiative interaction region, is no longer justified.
In systematic comparisons it turned out that the Sobolev approximation is amazingly accurate even far beyond the limits of its strict applicability. Maximum deviations in systems with a smooth, monotonic density and velocity structure reach a factor 2.5. In clouds where the density profile within the radiative interaction region can be approximated by a linear behaviour, the maximum error falls below 50\,\%. In spherical homogeneous flows, it is further reduced to 20\,\% independent of the model parameters.
For situations requiring high accuracies of line intensity computations, a simple way for the improvement of results obtained by the ordinary Sobolev approximation is demonstrated.
Keywords: Line: formation, Radiative transfer, Methods: numerical, Radio lines: ISM, ISM: jets and outflows}
Submitted to Astronomy & Astrophysics, Main Journal, Sect. 13.
For particles composed of amorphous carbon, the enhancement of the extinction at 1 mm wavelength with respect to the extinction of compact spheres of the same mass is in between 2.1 and 13.6. For silicate particles, the enhancement of the extinction at 1 mm wavelength is in the range 1.5-2.4.
Keywords: Interstellar medium: dust
Submitted to Astronomy and Astrophysics
Unfortunately, it has been overseen that the concepts valid for the line formation in stellar atmospheres do not necessarily hold in case of molecular clouds. Especially the assumption of complete redistribution or the use of a redistribution function which are both good approximations for stellar atmospheres (treated as a sequence of plane-parallel layers) is not justified for many other spatial configurations.
The underlying theoretical problem is the treatment of an ensemble of molecules at a given point r moving with randomly distributed velocities relative to an external coordinate system. In molecular clouds where direct collisions are negligible and gas collisions always lead to an excitation/deexcitation of rotational levels, there is no coupling between the molecules besides the radiation. Consequently, molecules with different velocities v can have quite different level populations n_j depending on whether the Doppler shift of the frequency of the exciting radiation makes a resonant absorption possible or not.
An exact treatment has to solve the balance equations for these molecules separately, and an integration over the full velocity space is necessary to determine the local opacity and the source function. Due to the complexity of this problem, two simpler approaches have been widely used. The ``standard'' approximation is the assumption of complete redistribution. Here, only one local set of balance equations is solved, and the distribution of random velocities is used both for the absorption and the emission profile. In the more sophisticated concept of redistribution functions, the integration of a special redistribution function allows to adapt the profile of the source function for each direction and frequency.
We have studied the behaviour of all three approaches for some special geometries in molecular clouds, where the exact approach was numerically tractable. These were the radiative interaction of separated clumps with relative motion, expanding/infalling shells, spheres with large velocity gradients, and highly collimated outflows.
Using parameters which should be typical of molecular clouds, we found that both the width of the velocity distribution as well as the relative velocity inferred from the line profiles may be wrong by a factor 2 when the complete redistribution approximation is used. In many cases, the results obtained from the redistribution function were still worse. The error appears only for clouds with systematic velocities which are at least of the order of the local line width and geometrical constellations with an nonnegligible anisotropy in the radiative transfer.
Numerically expensive studies have to be carried out to get a detailed estimate of the error made if complete redistribution or a redistribution function is applied to model the radiative transfer in a given object of interest.
Astronomischen Gesellschaft, Abstract Series 11, Positions, Motions, and Cosmic Evolution, Bonn, 1995