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Astron. Astrophys. 337, 85-95 (1998)

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3. The radiative transfer code

For the calculation of the continuum radiative transfer in 1D (spherical symmetry) and 2D (flared disk geometry) dust configurations, we used the code developed by Manske et al. (1997) which is based on the method given by Men'shchikov & Henning (1997). The main approximation used in this code is that even for the disk geometry, the density depends on the radial coordinate only. In addition, mean intensities and temperatures are self-consistently calculated for points in the disk's midplane and at its upper and lower conical surfaces only. The disk itself is essentially a part of a sphere with two removed polar cones (Fig. 6). A more detailed descriptions of the strategy for the solution of the radiative transfer problem can be found in our earlier papers mentioned above.

[FIGURE] Fig. 6. Geometry of the model disk with the opening angle [FORMULA]

3.1. Implementation of quantum heating

In the code, a ray tracing technique is used to solve the radiative transfer equation. This means that this equation is reduced to a 1-dimensional equation (Eq. (14)), which has to be solved along rays [FORMULA], with corresponding impact parameters p.


Here r is the radial position, [FORMULA] is the direction of the ray, and [FORMULA]. Assuming isotropic scattering and that the grains are in thermal equilibrium, the source function is defined by


where [FORMULA] is the mean intensity, [FORMULA] is the Planck function and [FORMULA] is the temperature of the dust grains of the chemically distinct dust component i and the size bin k. The quantities [FORMULA] and [FORMULA] are the absorption and scattering coefficients of the dust component [FORMULA],[FORMULA], and [FORMULA] is the total extinction coefficient, defined by:


If the dust model contains small dust grains, so that the quantum heating method must be used to calculate the emission, the source function has to be modified in order to account for this effect:


Here "small" means grains with radii [FORMULA] Å and [FORMULA] denotes the probabilities obtained by the quantum heating algorithm from Fig. 1.

To illustrate the difficulties of considering quantum heated particles in radiative transfer codes, note that it may take up to 25 seconds CPU time (on a DEC-Alpha 3000/500 workstation) to apply the quantum heating algorithm (Fig. 1) for one grain species and at one spatial grid point for one iteration. However, due to the basic approximations in our radiative transfer code, the numerical effort in flared disk geometry is only twice as high as for spherical symmetry (1D codes). For the models presented in this paper, the typical CPU time on a DEC-Alpha 3000/500 was about 6 hours.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998