Astron. Astrophys. 360, 702-706 (2000)

2. Methodology

A four-levels plus continuum atomic model of hydrogen was used. Following the theory given in Paper I and Paper II, to a good approximation, for an electron beam bombardment, the non-thermal collisional excitation and ionization rates of hydrogen from its ground level can be obtained as

The non-thermal collisional excitation and ionization rates of CaII, as given in Paper I, can be expressed as

where is the ground level population. is the rate of energy deposit due to the excitation and ionization of hydrogen by an electron beam. The excitation and ionization from the higher levels are negligible. Neglecting return current effects in a dense atmosphere, the energy deposit rate is given by (Emslie 1978; Chambe & Hénoux 1979)

where x is the ionization degree. The particle flux is supposed to be proportional to , with a low cut-off energy . is the total energy input flux above . The meaning of other physical quantities can be found in the relavant references.

In most empirical flare models, the chromospheric flare is located at heights lower than the top of the chromosphere ( 2000 km) in the quiet-Sun model VAL-C given by Vernazza et al. (1981). If this property was valid for all flares, one would actually not see any stucture above the solar limb and no any limb flares. This is not the case, since limb flares are observed. Thus, in order to compute the profile of chromospheric lines in limb flares, we need a specific empirical model. Unfortunately no such model exists. Therefore, in this paper, the temperature distribution in the flaring atmosphere has been represented by the semi-empirical flare models F₁ and F₂ given by Machado et al. (1980). In these models, a plan parallel atmosphere has been used. Indeed we cannot, in such conditions, study in a satisfactory way the height dependance of the line profiles. However, this approach allows us to explore the role of the non-thermal effects of particle beams in the line profiles of limb flares. The heights given in the figures may not be correct; however, they correspond to specific values of density and temperature. The comparison between observed and computed profiles could allow us to estimate the height variation of the density and temperature for a given limb flare. We shall not go so far as to limit ourself to the demonstration that additional effects, due to non-thermal processes are required in order to explain the significant width of the observed line profiles.

The numerical code we used is similar to the one presented in Paper I and Paper II. That is, the non-thermal rates have been included in the statistical equilibrium equations; the statistical equilibrium equations and the transfer equations for hydrogen and CaII, coupled with the hydrostatic equilibrium and the particle conservation equations, have been solved iteratively. One hundred frequency-points have been used for each line concerned. Five broadening mechanisms, i.e. Doppler broadening, radiative damping, Van de Waals forces, linear and quadratic Stark effects, have been included in the calculation of the line profiles. The assumption of complete frequency redistribution was adopted for simplicity. This does not have a significant influence on the line wing intensities, where the non-thermal effects are the most pronounced. Using the flare atmospheric models F₁ and F₂, we computed the source function and the opacity at different heights of the atmosphere. Assuming that, for a limb flare, the horizontal distributions of the source function are constant for a given height, we can compute the line profiles at different heights by using

where . D is the thickness of the flare along the line-of-sight and is the absorption coefficient per unit length. We take D = 3000 km as typical for all the models. The value of D does not influence greatly the final results. The calculations have been made for an electron beam with an initial total energy flux = 5 10¹¹ ergs cm^-2 s^-1 and with a power index equal to 4, which are typical values for flares. The low cut-off energy was chosen to be 20 keV; the origin of the electron beam was taken at the top of the chromosphere, where the column mass = 0. In the upper and middle chromosphere, where the lines are formed, the mean hydrogen ionization degree and for an electron beam with an energy of several tens of keV. So and were adopted.

© European Southern Observatory (ESO) 2000

Online publication: August 17, 2000
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