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Astron. Astrophys. 354, 714-724 (2000)

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2. Basic models

2.1. Physical model

An elementary magnetic flux tube filled by an hydrogenic plasma is considered to be affected by electron beam injection. The beam causes a hydrodynamic response of the ambient plasma calculated using the hydrodynamic models by Somov et al. (1981), developed by Zharkova & Brown (1995). The newer HD models have been calculated by Mariska et al. (1989), Li et al. (1989) and others. However, all these models have the differences which only occur at the upper coronal and chromospheric levels but do not affect the lower chromospheric levels. This allows to use the HD models by Somov et al. (1981) with temperature, density and macrovelocity variations in time and depth as shown in Fig. 1 in the paper by Zharkova & Kobylinsky (1993).

The main features of such a response can be briefly described as follows. The beam injection produces a strong downward shift of the transition region into deep atmospheric levels which, in turn, causes a sharp increase in temperature and a decrease in total density at the coronal levels. This is followed by evaporation of the chromospheric plasma into the corona and by formation of a dense cold condensation in the chromosphere which moves downwards to the photosphere as a shock wave. This paper deals with the chromospheric part of the HD simulations above, namely, with the cold and dense condensation in which the [FORMULA]-line emission originates. The HD models for temperature, density, macrovelocity were calculated for the first 10 sec after an electron beam onset.

2.2. Kinetic model

Electron beams are considered to be injected from the corona into a flaring atmosphere along the magnetic field direction with normal distributions in pitch-angles and time. In order to find an electron population at each depth, the time-dependent Boltzman equation was solved numerically for beam electrons with energy power law and normal distributions in pitch-angles and time, precipitating from the corona into deeper atmosphere in presence of the induced electric field of return current and converging magnetic field (Zharkova et al. 1995). The beam electrons are assumed to lose their energy in collisions with charged particles and neutral hydrogen atoms, in Ohmic heating and anisotropic scattering. While solving the kinetic equation in the phase space of time, depth, energy and pitch angle, the time and depth variables are considered to be independent whereas the energy and pitch-angle variables are to be dependent on the first two as they vary with depth and time (Emslie 1980). This, in turn, results in additional boundary conditions being imposed on the energy and pitch-angle dependence on depth and time (see for details Zharkova et al. 1995).

Some results of the simulations are shown in the paper by Zharkova et al. (1995) (Figs. 1 and 2) for the following beam parameters: the lower energy cutoff was equal to 15 keV, electron beam spectral index [FORMULA] and initial energy flux [FORMULA], upper energy cutoff [FORMULA] (Fig. 1) and [FORMULA], [FORMULA], [FORMULA] (Fig. 2). For the upper energy cutoff [FORMULA]keV, moderate spectral indices and initial fluxes on the top boundary, electron beams were shown to reach lower chromospheric and even photospheric levels as collimated beams with power law in energy, despite the disruptive effects of a return current at the transition region level and in the upper chromosphere.

[FIGURE] Fig. 1a-c. The [FORMULA]-line polarisation profiles (linear polarisation - parameter Q) plotted for the depths a - [FORMULA], b - [FORMULA], c - [FORMULA] and d - [FORMULA] for for a viewing angle [FORMULA], at the moment t= 4.39 s for beam parameters: solid line - spectral index [FORMULA], dashed line - [FORMULA]; [FORMULA]

[FIGURE] Fig. 2a and b. The [FORMULA]-line polarisation profiles (linear polarisation - parameter Q) plotted for the depth [FORMULA], at the moments t= 2.92s (a) and t= 5.94s (b) for 5 viewing angles [FORMULA] (from the top to the bottom in each subfigure, respectively). The peaks are correspondent to the allowed transitions as in the text; beam parameters are correspondent to the softer beam from Fig. 1.

There are significant variations in the energy loss mechanisms throughout this precipitation. At upper levels in the corona, where the ambient plasma is fully ionised, Coulomb collisions dominate the other energy losses; at the transition region and upper chromosphere, these are comparable with Ohmic losses, with the latter causing a disruption of the initial beam and the appearance of a `return current beam', returning to the source in the corona. This, in turn, produces a split in energy of the initial beam with the additional maximum at 30-35 keV.

In the lower chromosphere, where the ambient plasma is weakly ionised with ionisation degree of [FORMULA] (Zharkova & Kobylinskii 1989), inelastic collisions of beam electrons with neutral atoms prevail over any other energy losses. Since the cross-sections of such collisions are much lower than those of Coulomb collisions, beam electrons lose less their energy in these collisions and can precipitate downward to the photosphere, despite an increase in the total density. Electron beams, which are well directed along the magnetic field, still precipitate as collimated beams to the lower atmospheric levels, but they are transformed into softer beams with wider angular distributions (Zharkova et al. 1995).

The part of beam electrons which reaches the lower chromosphere can produce a dual effect on the atmosphere: first, the formation from the hydrodynamic response of a cold dense condensation, as described in Sect. 2.1; and, secondly, additional excitation, ionisation and polarisation of Hydrogen atoms by inelastic collisions, as considered below in Sect. 2.3.

2.3. Radiative model

For polarimetric simulations a 3 level Hydrogen model atom (total 9 sublevels: [FORMULA], [FORMULA] [FORMULA] and [FORMULA]) was considered with fine structure of the third and second levels caused by Zeeman splitting without level crossings. Only Zeeman coherence, like [FORMULA], will be taken into account whereas all other coherences with [FORMULA], [FORMULA] and [FORMULA] can be omitted following the estimation [FORMULA], where [FORMULA] is the atomic energy of the Bohr frequency set, B is the magnetic induction in G; x is about unity; [FORMULA] is the energy splitting from the spin-orbital interaction.

Steady state equations were solved for all these transitions (104 in total) using the density matrix with both collisional (thermal and non-thermal) and radiative tensors. Radiative transfer equations were not directly included into this solution because the opacities evaluated in the [FORMULA]-line fine structure transitions were appeared to be lower than unity in most cases, whereas their integrated opacity was well above unity. Therefore, the full radiative transfer problem solution can be omitted for these transitions by replacing it with an averaged radiative transfer solution applied to the [FORMULA]-transitions without the fine structure. This approach for [FORMULA]-line can be supported by conclusions of Bommier et al. (1991) where the radiative transfer problem was solved for a 3 level hydrogen atom with fine structure in optically thick media.

In order to calculate a diffusive mean intensity and a degree of ionisation in these dynamic events, we used a 5 levels plus continuum model atom without fine structure. The full non-LTE approach was applied for all the transitions at different depths and times, as described by Zharkova & Kobylinsky (1989, 1993) and Kobylinskii & Zharkova (1996). Collisions with thermal and beam electrons were considered along with radiative excitation/ionisation and deactivation by the external radiation in all the transitions considered.

The ionisation balance in a flaring atmosphere was found to be governed by Hydrogen atom ionisation. In the lower chromosphere, if thermal electrons only are considered, the hydrogen ionisation degree falls sharply down to a magnitude of [FORMULA]. Beam electrons reaching this depth mostly ([FORMULA]) lose their energy in inelastic collisions with neutral Hydrogen atoms (Aboudarham & Hénoux 1986, 1987; Zharkova & Kobylinsky 1989). These collisions increase the Hydrogen ionisation degree to about [FORMULA] (see Fig. 2 in Zharkova & Kobylinsky (1989). This, in turn, causes an enhancement in Lyman, Balmer, Pashen line wings and heads of continua.

2.3.1. Evaluation of elementary processes

Hydrogen atoms, embedded into a flaring atmosphere of a magnetic flux tube, are affected by depolarising collisions with thermal electrons of the ambient plasma and by polarising collisions with anisotropic beam electrons and external radiation. As the atom levels in a magnetic field are split into sublevels of the fine structure, it is important to evaluate the effect of each mechanism.

  1. Magnetic field effect

    According to Bommier & Landi Degl'Innocenti (1996), a weak magnetic field can partially destroy the coherence between magnetic sublevels, whereas Zeeman splitting should still not be taken into account (effect Hanle). In this case the following condition is valid:


    where [FORMULA] is the Larmor frequency and A is the Einstein spontaneous emission rate.

    In the case of an `intermediate' magnetic field the coherence between magnetic sublevels is absent and in comparison with the Doppler width, Zeeman splitting is noticeable. In this case the condition above can be rewritten as:


    here [FORMULA] is the Doppler width.

    The temperature in the model flaring atmospheres from Sect. 2.1 lies in a range of [FORMULA] oK which results in [FORMULA]. Magnetic field in the chromosphere can vary from 500 to 1500 G (Lozitskii & Baranovskii 1993; Silva et al. 1996), so [FORMULA]. Therefore, in the chromosphere, where the [FORMULA] emission originates, these two regimes may occur and both should be considered. It should also be noted that in the line formation region Hanle effect will prevail in the line core, whereas in the wings the `intermediate' magnetic field approximation is valid.

  2. Effects of collisions and radiation anisotropy

    Let evaluate a role of depolarising collisions with thermal electrons and of an anisotropy of incident radiation at these levels. The anisotropy of the incident radiation is described by the parameter [FORMULA] which is determined by the non-diagonal elements of the density matrix. The isotropic part of radiation is described by the parameter [FORMULA] which is determined by the diagonal elements. Therefore, spectral line polarisation, caused by anisotropy of radiation, is proportional to the ratio [FORMULA]. The effect of depolarising collisions can be described by the parameter D, which accordingly is proportional to the plasma density N:


    where [FORMULA] is a sum of the collisional cross-sections in the [FORMULA]-line transitions up and down, V is the relative velocity between atoms and colliding particles. If [FORMULA] is in units of [FORMULA] and V is in km/s, then:


    Radiation anisotropy [FORMULA] can be described by the formulae:


    where W is the dilution factor of the external chromospheric [FORMULA] radiation, varying from 0.3 to 0.5, [FORMULA] - the Einstein's absorption coefficient and J - the mean radiation intensity. For a strong optical line with the effective temperature of [FORMULA] K, [FORMULA]. The condition [FORMULA] leads to the following evaluation of the critical density where the collisional and radiative probabilities are comparable:


    Considering [FORMULA] as being of [FORMULA], for the temperature range relevant to a flaring atmosphere, the critical density is [FORMULA]. This corresponds to the temperature minimum region in the quiet atmosphere, while in flares this density can be reached at the lower chromospheric levels where the [FORMULA]-line wing emission occurs.

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© European Southern Observatory (ESO) 2000

Online publication: February 9, 2000