## 2. MALI code for clouds and H synthetic profilesThe version of the MALI code used in this work is described in Mein
N. et al. (1996). The cloud-like structure is represented by a
horizontal 1D slab irradiated from below by the solar incident
radiation. For a thorough description of the appropriate boundary
conditions see also Heinzel et al. (1999). The cloud can move as a
whole in any arbitrary direction and with a prescribed bulk velocity,
which affects the H line source
function via the Doppler brightening effect (Heinzel et al. 1999). The
present MALI code has been largely optimized in order to achieve the
high performance capability needed for our grid construction. We use
the diagonal ALO (approximate lambda operator) of Rybicki and Hummer
(1991), together with the acceleration technique of Ng (1974). A
typical model requires some 10 to 20 s of CPU time on a 400 MHz-type
workstation. As an initial guess for the population of the different
atomic levels, one can use the results obtained for the previous
nearest model. The iteration procedure is then stopped when all
populations at all depths reach a relative error of
10 The cloud slab is assumed to be isothermal with a given kinetic temperature. Other input parameters are constant electron density, microturbulent velocity, macroscopic bulk velocity (and direction of motion), geometrical thickness and height above the solar surface (see Table 1). The helium to hydrogen abundance ratio is 0.1. According to Heinzel (1995), a constant electron density allows us to use the much simpler complete redistribution (CRD) for Lyman lines, in contrast to partial redistribution (PRD) generally needed for these resonance lines. CRD requires fewer iterations than PRD, which also accelerates the construction of the grid. Moreover, since we do not know a priori the pressure scale-height in various magnetically-confined features (clouds), a constant electron density seems to be a reasonable first-order estimate.
We use a standard five level plus continuum hydrogen model atom. Since the electron density is known a priori, the preconditioned MALI equations can be expressed as linear functions of the populations of the atomic levels, which also simplifies the solution (Heinzel, 1995). The computed non-LTE populations, as functions of the H line-center optical depth, are tabulated and used to evaluate the H line source function according to the standard formula: where ,
stand for the statistical weights of
the respective atomic levels, and
are the number densities of hydrogen
atoms in the 2 Clearly, the H source function is,
for practical purposes, frequency/wavelength independent across
H because of the assumption of CRD in
this line. Note that from the computed number densities one can
evaluate the gas pressure , where
The formal solution of the radiation transfer equation for the cloud is then computed numerically as follows: where is the background intensity along the line of sight, , is the optical depth and is the optical thickness. When far-wing intensities are discarded, the optical depth can be adequately described by a Doppler-shifted Gaussian profile: where is the
H line center wavelength
(656.2808 nm), is the
Doppler-shift of the line due to the line-of-sight motion of the
cloud, and
© European Southern Observatory (ESO) 1999 Online publication: April 19, 1999 |