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Astron. Astrophys. 345, 618-628 (1999)

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2. MALI code for clouds and H[FORMULA] synthetic profiles

The 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[FORMULA] 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 10-3.

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.


[TABLE]

Table 1. Parameters used in the calculation of the grid. The total number of MALI models computed is [FORMULA]. Given that the source function calculated by the MALI code is symmetric for negative and positive velocities, we only compute 3 points (0, 2.5 and 5 km s-1) for the parameter V (see text).


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[FORMULA] line-center optical depth, are tabulated and used to evaluate the H[FORMULA] line source function according to the standard formula:

[EQUATION]

where [FORMULA], [FORMULA] stand for the statistical weights of the respective atomic levels, [FORMULA] and [FORMULA] are the number densities of hydrogen atoms in the 2nd and 3rd levels, c is the speed of light and h denotes Planck's constant. [FORMULA] and [FORMULA] are calculated by the MALI code as a function of optical depth at line center [FORMULA].

Clearly, the H[FORMULA] source function is, for practical purposes, frequency/wavelength independent across H[FORMULA] because of the assumption of CRD in this line. Note that from the computed number densities one can evaluate the gas pressure [FORMULA], where k is Boltzmann's constant, T denotes the kinetic temperature, and N is the total density of particles, including neutral hydrogen atoms and protons as well as electrons and neutral helium. Since we neglect the ionization of helium, electron and proton densities are equal. However, our assumption of CRD for Lyman lines gives a less realistic ground-level population for hydrogen (Heinzel 1995), which affects the value of [FORMULA] in the case of a low ionization degree.

The formal solution of the radiation transfer equation for the cloud is then computed numerically as follows:

[EQUATION]

where [FORMULA] is the background intensity along the line of sight, [FORMULA], [FORMULA] is the optical depth and [FORMULA] is the optical thickness. When far-wing intensities are discarded, the optical depth [FORMULA] can be adequately described by a Doppler-shifted Gaussian profile:

[EQUATION]

where [FORMULA] is the H[FORMULA] line center wavelength (656.2808 nm), [FORMULA] is the Doppler-shift of the line due to the line-of-sight motion of the cloud, and W the line Doppler width, which in turn can be expressed as:

[EQUATION]

m being the hydrogen atom rest mass and [FORMULA] the microturbulent velocity. The background intensity profile [FORMULA] has been taken from David (1961). The integral in Eq. (2) is carried out numerically by a spline quadrature algorithm (Fortin 1995).

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© European Southern Observatory (ESO) 1999

Online publication: April 19, 1999
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