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Astron. Astrophys. 340, 508-520 (1998)

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2. Numerical methods

We investigate the evolution of disks exposed to an external radiation field as shown in Fig. 1, using a modified version of the 2D radiation hydrodynamics code described in Yorke & Welz (1996, Paper I) and Richling & Yorke (1997, Paper II). In addition to the usual hydrodynamics with heating and cooling this code calculates the direct hydrogen-ionizing stellar radiation field as well as the diffuse radiation fields resulting from recombination of hydrogen into the ground state and from scattering on dust grains. The diffuse radiation fields are treated in the flux-limited diffusion (FLD) approximation (Levermore & Pomraning 1981) as described and implemented by Yorke & Kaisig (1995) for multiply nested grids. We also solve the Poisson equation for the gas' self-gravity and the FLD equations for grey (IR) continuum radiation transfer in order to determine the dust temperature (see Yorke et al. 1995). The code described in Paper I and II uses cylindrical coordinates [FORMULA] and solves the equations in the quadrant with [FORMULA] and [FORMULA] assuming axial symmetry and symmetry with respect to the equatorial plane. This code design is well suited for previous work where the ionizing source was located at the center of the disk.

[FIGURE] Fig. 1. Sketch of the astrophysical problem: A star-disk system is illuminated by the hydrogen-ionizing radiation emerging from a massive star. The dashed line marks the computational domain when using the 2D radiation hydrodynamics code.

Moving the radiation source along the rotation axis out of the computational domain breaks the symmetry with respect to the equatorial plane, necessitating several modifications to the code. The numerical grid was extended to include the quadrant with [FORMULA] and [FORMULA]. The nested grids are now distributed as shown in Fig. 2 for an example of two [FORMULA] grids. This arrangement maintains the enhanced resolution for the inner regions of the disk where most of the mass is concentrated. We also changed the boundary conditions at the lower boundary of the grid. For the coarsest grid we now assume semi-permeable outflow boundary conditions. All inner grids get their boundary conditions from the next coarser grid. Almost all parts of the code were affected by this procedure. A detailed description of the method of explicit nested grids and the treatment of grid interaction can be found in Yorke & Kaisig (1995).

[FIGURE] Fig. 2. Grid architecture and lines of sight arrangement. The lines of sight across the grids are determined as described in the text. The magnification on the right shows the location of the physical variables: degree of ionization x, gas temperature T, extinction coefficient [FORMULA] and optical depth [FORMULA]. The ray segments are denoted with [FORMULA]. The volume of integration is given by [FORMULA] and [FORMULA].

Major changes were necessary for the definition of the lines of sight across the numerical grid for the integration of the direct EUV radiation. In Papers I and II these lines of sight extended radially from the central source located at [FORMULA] to the centers of the outermost R and Z boundary cells. Here, the lines of sight are centered at the massive star outside the grid as illustrated in Fig. 2: The grids with radius [FORMULA] and height [FORMULA] are extended to imaginary quadratic grids (marked with dashed lines) until the radiation source at [FORMULA] is located at the upper left corner of these grids. Starting with the finest grid the lines of sight are drawn - with increasing angle between the line of sight and the rotation axis - through the center of the boundary cells of the quadratic grids (marked with little solid circles). When the angle between the line of sight and the rotation axis is greater than the corresponding angle of the line which touches the corner [FORMULA] of the original grid (marked with thick solid lines) we switch to the next coarser grid. This algorithm insures that every grid cell is traversed at least once, regardless of the location of the external source along the symmetry axis outside the outermost grid. Had we adopted the simpler procedure of non-diverging lines of sight, we would have necessarily restricted ourselves to the case of distant ([FORMULA] length scale of outermost grid) sources.

The coupling between the grid and the lines of sight is treated in the same way as in the original code. The magnification on the right of Fig. 2 displays the position of the physical variables on the grid. Scalar variables are located at the center of a cell, but the optical depth is defined at the intersection of the lines of sight with the boundaries of the cells. Therefore a mean optical depth [FORMULA] is calculated from


where we use the ray segments [FORMULA] as weight factors. The summation extends over all lines of sight crossing the cell. From [FORMULA] we get the photon density


within the cell. [FORMULA] is the stellar EUV photon rate and r the distance of the center of the cell from the star. As described in Paper I and II we simultaneously determine the degree of ionization x and the gas temperature T for the current time step at the center of the current cell and thus obtain the extinction coefficient


which includes both extinction resulting from ionization of hydrogen and dust extinction. Here n is the particle density, [FORMULA] the mean ionization cross section of hydrogen and [FORMULA] the dust extinction coefficient. The optical depth of each line of sight exiting from the other side of a cell is then calculated from


Outside the computational domain we assume constant density ([FORMULA]). Furthermore, we assume that the gas is fully ionized outside the grid ([FORMULA]). Thus, only dust extinction is effective when calculating the optical depth of each line of sight at the upper boundary of the coarsest grid


where [FORMULA] is the ray segment between the star and the upper boundary and [FORMULA]. In our simulations the outer ray segments are of order [FORMULA], which leads to a slight reduction of the EUV photon flux at the upper boundary of the grid. This simple assumption for the upper boundary condition is an attempt to take at least some extinction outside the grid into account. A proper treatment would require a more realistic density distribution towards the ionizing star and the integration of the optical depth along [FORMULA]. At the interface to the next finer grid [FORMULA] is stored for use as the boundary condition when calculating on the finer grid.

Note that the more general situation - an off-axis location of the source - is only possible using a 3D code.

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© European Southern Observatory (ESO) 1998

Online publication: November 9, 1998