Construction of a model of an X-ray
illuminated cloud consists of the simultaneous solution of the local
balance equations. The radiative transfer equation is
solved for both the continuum and for the lines that escape the
region near the point of emission. The large number of ions in the
calculation results in many ionization edges that may affect the
radiation field. We solve the transfer equation on a frequency grid
that includes a total of 9999 continuum grid points with even
logarithmic spacing in energy from 0.1 eV to 20 keV resulting
in a limiting resolution of 0.12 , corresponding to, e.g. 8.6 eV at 7 keV.
We calculate the luminosities of
10000 spectral lines and
solve the continuum transfer equation individually for each of these.
The emissivity of each line at each point is the product of the
emissivity and the local escape fraction for
that line. The continuum opacity for each
line is the opacity calculated for the energy bin that contains the
line. This procedure is repeated for each successive shell with
increasing radius.
Calculation of the escape of the diffuse radiation
field depends on a knowledge of the
optical depths of the cloud from any point to both the inner and outer
boundaries. Since these are not known a priori we iteratively
calculate the cloud structure by stepping through the radial shells
at least 3 times. For the initial pass through the shells we assume
that the optical depths in the outward direction are zero. This
procedure is found to converge satisfactorily within 3-5 passes
for most problems of interest. This procedure is tantamount to the
“-iteration” procedure familiar from stellar atmospheres, and
must suffer from the same convergence problems when applied to problems
with large optical depths. These problems are reduced in our case
by the use of escape probabilities rather than a full integration
of the equation of transfer.