Astron. Astrophys. 337, 832-846 (1998)

3. The numerical model

3.1. Structure of the underlying models

The underlying numerical models were calculated on five multiply nested grids, each with 62 x 62 grid cells (see Yorke & Kaisig 1995, Paper I, and Paper II). The spatial resolution of the finest grid was  cm (R is the distance to the symmetry axis, Z to the equatorial plane). Axial symmetry and mirror symmetry with respect to the equatorial plane were assumed for the models. The simulations were performed within a volume  cm until a quasi-steady state was reached.

For the diagnostic radiation transfer calculations discussed here we use the final states of five simulations described in Paper II. Some of the relevant parameters of these simulations are given in Table 2. Fig. 2 and Fig. 3 display the density and ionization structure as well as the velocity field of the selected models. Models A and C are the results of simulations with the same moderate stellar wind and the same radiation source. But in the simulation leading to model A the diffuse UV radiation field originating from scattering on dust grains was completely neglected. For that reason we got a higher photoevaporation rate for model C. In Fig. 2 this is recognizable by the greater overall density in the ionized regions and by the higher velocity in the "shadow" regions of the disk in the case of model C. In order to investigate the variation of spectral characteristics with the stellar wind velocity we chose the models with the largest wind velocities G2, G3 and G4. Fig. 3 shows the increasing opening angle of the cone of freely expanding wind with increasing wind velocity.

Table 2. Scattering coefficient as well as parameters for the stellar wind (mass loss rate and velocity ) and the ionizing source (stellar photon rate and temperature ) used in the calculations. The evaporation time scale is calculated from with .

Fig. 2. Density, velocity and ionization structure of model A and C. Gray scale and black contour lines display the density structure. These contour lines vary from to in increments of . The white contour lines mark the position of the ionization front and the arrows show the velocity field. The normalization is given at the upper right corner.

Fig. 3. Density, velocity and ionization structure of model G2, G3 and G4. Symbols and lines have the same meaning as in Fig. 2 except the black density contour lines, which are drawn down to .

3.2. Strategy of solution

We use the model data to calculate the ionization structure and the level population. From the level populations we determine the emissivities of each line transition and the continuum emission at each point within the volume of the hydrodynamic model. For each viewing angle considered, we solve the time independent equation of radiation transfer in a non-relativistic moving medium along a grid of lines of sight (LOS) through the domain, neglecting the effects of scattering:

where the optical depth is defined as . Integrations were performed for a given set of frequencies, whereby the effects of Doppler shifts for the line emissivities were taken into account. The resulting intensities are used to determine SEDs, intensity maps and line profiles. Spectra are obtained from the spatial intensity distributions by integration, taking into account that each LOS has an associated "area". Depending on the symmetry of the configurations could be utilized to minimize the computational effort (see Fig. 4). For the pole-on view (), for example, only a one dimensional LOS array need be considered. For the edge-on view () lines of sight either through a single quadrant (continuum transfer) or through two quadrants (line transfer) are necessary. The resolution of the central regions is enhanced by overlaying a finer LOS grid in accordance with the multiple nested grid strategy used in the hydrodynamic calculations.

Fig. 4. Choice of lines of sight (LOS) and their associated areas for different viewing angles . Filled dots indicate the LOS used for the continuum calculations. Empty dots refer to the additional Lines of Sight necessary for the line profile calculations.

Each point in Fig. 4 corresponds to an LOS trajectory through the model. Mapping such a trajectory onto the (R,Z) model grid yields hyperbolic curves as displayed in Fig. 5. Beginning with a starting intensity (), the solution of Eq. (16) is obtained by subdividing the LOS into finite intervals and analytically integrating over each interval assuming a sub-grid model (see below).

Fig. 5. Projection of a typical LOS trajectory (curved dashed line) onto the model data grid (solid lines). Temperature, density, degree of ionization and velocity are defined at cell centers. The small circles divide the LOS into subintervals; the source function is evaluated at the location of the circles, chosen to lie on the intersections of the LOS with lines connecting the grid cell centers.

3.3. Continuum radiation transfer

If no discontinuities are present within the subinterval under consideration, we assume varies linearly with , i.e.

where i and denote the starting and end points of the interval, respectively, and is a mean optical depth over the interval:

With this formulation the solution of Eq. (16) over the entire interval is given by (see Yorke 1988):

For the cases considered here we choose as the starting LOS intensity. For "proplyd"-type models (considered in a subsequent paper of this series) a non-negligible background intensity should be specified.

3.4. Radiation transfer in emission lines

For the transitions considered here the radiation field can be considered "diffuse" and the contribution of spontaneous emission dominates over line absorption and stimulated emission processes. After separating the source function and the intensity of Eq. (16) into the contributions of the continuum and the line, we obtain

where and . Here is the Doppler-shifted frequency of the transition, the Doppler width and the net source function integrated over the line. Assuming that is linear in over the whole interval yields the analytical solution to Eq. (20):

where is the error function and a dimensionless frequency shift.

The net source function is calculated according to the algorithm suggested by Yorke (1988):

with . The line source functions and are calculated from the total line emission coefficient as defined in Eq. (9) at the boundaries of the evaluated interval and from the continuum absorption coefficient: .

If , i.e. there is negligible Doppler shift within the subinterval, the solution of Eq. (20) with is used:

3.5. Treatment of ionization fronts

The numerical models considered contain unresolved ionization fronts due to the coarseness of the hydrodynamic grid. At these positions jumps occur in the physical parameters and the solutions given by Eq. (19) and Eq. (21/23) are poor approximations. The exact location of the fronts within a grid cell are unknown; we assume they lie at the center of the corresponding interval. Our criterion for the presence of an ionization front is a change in the degree of ionization between two evaluation points.

For the continuum calculations Eq. (19) is applied to each half interval with . For the first half kept constant and for the second half is held constant. For the line calculations Eq. (23) is used with () and () for the first (second) half interval.

3.6. Treatment of the central radiation source

The central source is modeled by a black body radiator of temperature and radius . The integration along the line of sight through the center is started at the position of the source with the initial intensity

with A is the area associated with the central LOS.

© European Southern Observatory (ESO) 1998

Online publication: August 27, 1998
helpdesk.link@springer.de