2 Modelling

2.1 Numerical procedure

The first step in our calculations is to solve the ionization, hydrostatic and statistical equilibrium equations as well as the radiative transfer equations out of local thermodynamic equilibrium (non LTE) for hydrogen. Computational details can be found in Gouttebroze et al. (1993, hereafter GHV). We obtain electron and hydrogen level population densities from which we can determine the hydrogen spectrum emitted by our prominence. We thus derive the radiation inside the slab, taking into account the incident radiation and the principal transitions of the hydrogen atom. This defines the new physical conditions for the computation of the helium spectrum. The second step is then to solve independently the statistical equilibrium and the radiative transfer equations for the multilevel, multi-ion helium atom. The solution of the statistical equilibrium equations is determined by iterations and the radiative transfer equations are solved by the Feautrier method (finite-difference method, Feautrier 1964) with variable Eddington factors (Auer & Mihalas 1970). In all calculations, partial redistribution in frequency is considered for the formation of the resonance lines H I Ly$\alpha$ and Ly$\beta$(Heinzel et al. 1987), He I $\lambda$584 Å and He II $\lambda$304 Å.

2.2 Prominence model

Figure 1: Prominence model. T, P, and V are respectively the electron temperature, the gas pressure, and the microturbulent velocity. H is the height of observation above the solar surface and W the slab geometrical thickness.

The prominence model used here is the same as described in GHV - see also Heinzel et al. (1987, 1994). It consists of a plane, parallel slab standing vertically above the solar surface (see Fig. 1). It is a one-dimensional representation defined by the slab thickness W. On both sides of this symmetric model, the prominence is illuminated by an incident radiation field coming from the underlying photosphere and chromosphere, and the surrounding corona. This radiation field determines the boundary conditions for the resolution of the radiative transfer equations and it is crucial to consider it properly in order to study line formation. This radiation field is diluted according to the height H of the line-of-sight above the solar surface and eventually to the center-to-limb variations for the incident lines. In our computations the center-to-limb variations are taken into account for the incident hydrogen lines with upper level $n \leq 5$ (Gouttebroze & Labrosse 2000). The emerging spectrum is computed for three viewing angles: $\mu = \cos \theta = 0.2, 0.6$and 1. This last value corresponds to a line-of-sight perpendicular to the slab and will be the only one considered in this paper, for brevity. Inside the slab, three physical quantities have to be defined: the electron temperature, the gas pressure and the microturbulent velocity. In this work we use isobaric and isothermal models, so these three quantities are fixed constant throughout the slab. Consequently, each model is defined by 5 parameters, namely the electron temperature, the gas pressure, the microturbulent velocity, the slab thickness and the height above the solar surface.

Numerical codes for the hydrogen emerging spectrum from a quiescent prominence can be found on the MEDOC web site (Gouttebroze & Labrosse 2000).

2.3 Atomic model

Figure 2: Diagram of the neutral helium energy levels. The energy scale has been shortened between 0 and 20 eV. We indicate for the 11 first states the index of the level with increasing energy relative to the ground state. Solid lines represent some of the transitions often observed in quiescent prominences, and the dashed line represents the ionization continuum at 504 Å.

We use very detailed atomic models for both hydrogen and helium. The hydrogen atom is the same as in GHV (20 bound levels plus continuum).

For neutral helium we use the atomic model of Benjamin et al. (1999 hereafter BSS). Thus we consider in our calculations 29 energy levels up to n=5, divided into individual (L,S) states, which allows us to keep the distinction between singlet (S=0) and triplet (S=1) levels. Energy levels and statistical weights are given by Wiese et al. (1966). It is important to have this distinction in the model atom between neutral helium singlet and triplet levels because they are not populated through the same processes. Figure 2 represents schematically the neutral helium states included in the calculations up to n = 5. Note that the energy scale has been modified in order to lessen the large gap between the ground state $\rm 1s^2$ and the first excited level $\rm 1s2s~^3S$. The latter is a metastable state and is the lower state of the infrared He I $\lambda$10830 Å line. In Fig. 2 we show some line transitions which will be the subject of further investigations in Sects. 4 and 5, as well as the resonance continuum transition which occurs at 504 Å. The optically thick resonance lines lie in the singlet system. There are no permitted radiative transitions between the two systems, but they are coupled through collisions. Effective collision strengths, collisional ionization coefficients and spontaneous emission coefficients are from BSS. Collisions strengths not defined in BSS are taken in Benson & Kulander (1972). Coefficients for Stark broadening are from Dimitrijevic & Sahal-Brechot (1984), and from Griem (1974) for transitions not defined in Dimitrijevic & Sahal-Brechot. Photoionization cross sections are from TOPBASE (Fernley et al. 1987).

For ionized helium we use a simple 4 bound levels atomic model. Energy levels and statistical weights again are given by Wiese et al. (1966). Effective collision strengths are from Aggarwal et al. (1992) for collisional transitions up to n=3 and from Aggarwal et al. (1991) for transitions up to n=4. Collisional ionization coefficients are calculated as in Mihalas & Stone (1968). Spontaneous emission coefficients are those of Allen (1973). Photoionization cross sections are from TOPBASE.

The He III ion is represented by one level. With this He I-He II-He III system we treat in our computations 76 permitted radiative transitions and 438 collisional transitions.