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

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1. Introduction

The dense and cool material in the interstellar medium of galaxies plays an important role in the life cycle of stars, from the earliest phases of star formation to the shells around evolved stars and the gas and dust tori around active galactic nuclei. Line emission from atoms and molecules, and continuum emission from dust particles, at radio, (sub) millimetre and infrared wavelengths are indispensable tools in the study of a wide variety of astrophysical problems. This is illustrated by the large number of infrared and submillimetre observatories planned for the near future, such as the Smithsonian Millimeter Array (SMA), the Atacama Large Millimeter Array (ALMA), the Far-Infrared and Submillimetre Space Telescope (FIRST) and the Stratospheric Observatory for Infrared Astronomy (SOFIA).

An essential step in the interpretation of the data from these instruments is the comparison with predicted emission from models. This paper presents a numerical method to solve the radiative transfer and molecular excitation in spherically symmetric and cylindrically symmetric source models. At the comparatively low densities of interstellar gas, the excitation of many molecules is out of local thermodynamic equilibrium (LTE), and the transfer of line (and continuum) radiation plays a significant role in determining the molecular excitation (Leung & Liszt, 1976; Black, 2000). Geometry thus becomes an important element, and the high spatial resolution of current and future instruments often demands that at least two-dimensional (axisymmetric) source structures are considered. In the implementation of our method discussed in this paper, we have limited the source structure to spherical and cylindrical symmetries. The large and often multidimensional parameter space further requires a fast and reliable method, which needs to be easily applicable to many different astrophysical problems.

This need for reliable and flexible tools calls for the use of Monte Carlo techniques, where the directions of integration are chosen randomly. This approach was first explored by Bernes (1979) for non-LTE molecular excitation; later, Choi et al. (1995), Juvela (1997) and Park & Hong (1998) augmented it and expanded it to multiple dimensions. However, Monte Carlo methods can be quite slow, especially at large optical depths ([FORMULA]), which has limited their use so far. We will show that this problem can be overcome by using a technique inspired on Accelerated Lambda Iteration: the local radiation field and excitation are solved self-consistently and separated from the overall radiative transfer problem (see Sect. 3.4). The greatest virtue of our code is its ability to deal with a wide variety of source models for many atomic and molecular species, with or without a dust continuum. Although for any individual problem a somewhat more efficient method could be constructed (Sect. 3.5), the Monte Carlo approach frees the user from having to fine-tune such a method and allows the user to focus on the astrophysics of the problem at hand.

This paper does not discuss the influence of radiative transfer on the source structure, through the thermal balance, ionization and chemistry (Takahashi et al., 1983; Ceccarelli et al., 1996; Doty & Neufeld, 1997, for example). However, our code is suited to become part of an iterative scheme to obtain self-consistent solutions for the source structure including radiative transfer and molecular excitation.

Throughout this paper, examples from studies of star formation will serve to illustrate the various topics - and to stress the link with the analysis of observations. Sect. 2 introduces a simple, spherically symmetric model of the core of an interstellar cloud, collapsing to form a star. Sect. 3 then discusses the coupled problem of radiative transfer and molecular excitation. It introduces the two most commonly used solution methods, and shows that these are closely related. This opens the possibility of constructing a hybrid method which combines the benefits of both; the implementation of this combined approach in our code is deferred to the Appendix. The paper continues by exploring the capabilities of our code through a number of astrophysically relevant examples, based on extensions of the simple one-dimensional model of Sect. 2. A brief summary concludes the paper in Sect. 5.

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© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000