Astron. Astrophys. 322, 674-678 (1997)

2. Numerical method and results

2.1. General procedure

In order to compute the Lc from an atmosphere, we must treat the full non-LTE problem, that is, to solve iteratively the equations describing the radiative transfer, statistical equilibrium, hydrostatic equilibrium, and particle conservation. Particularly, when the atmosphere is bombarded by an electron beam, the deposited energy causes nonthermal excitation and ionization of the hydrogen atoms. The rates of these nonthermal collisional transitions are to be added to the radiative and thermal collisional rates in the statistical equations. A four-level-plus-continuum atomic model for hydrogen is adopted here. The method is similar to that of Fang et al. (1993), where a detailed description about the energy deposition rate and the nonthermal collisional excitation and ionization rates is presented.

To study the Lc signatures under different flare processes, we modify the model atmosphere in a parametrized way. Based on the results of semiempirical flare models, it is known that their most significant modifications relative to the quiet-Sun model are both a temperature rise in the upper chromosphere and a downward shift of the transition region, which may result from the evaporation of chromospheric material into the corona (see e.g., Gan & Fang 1987). In what follows, we consider these two processes and the nonthermal effect independently to show what kinds of Lc emissions would be produced respectively. It should be mentioned that, solar flares certainly involve a drastic change of the coronal temperature structure, but it has little influence on the Lc emission as the hydrogen atoms are fully ionized there. Therefore, we will not discuss this issue here.

2.2. Effect of chromospheric temperature rise

We start from the quiet-Sun model VAL-C presented by Vernazza et al. (1981) and later modified by Avrett et al. (1984). The flare-induced temperature rise in the chromosphere is assumed to reach a maximum value at the top chromosphere, decrease downwards linearly with , and vanish at the temperature minimum region (TMR), i.e., , where and represent the column mass densities at the top chromosphere and the TMR, respectively. Under such circumstances, we have calculated the Lc intensities at the disk center for models with various values of , namely, 0, 1000, 2000, and 3000 K. The results are shown in Fig. 1 (solid lines). Note that is exactly the quiet-Sun case while K corresponds roughly to a strong chromospheric flare.

Fig. 1. Lyman continuum intensities at the disk center computed for various atmospheric models showing the effects of varying chromospheric temperatures and top column mass densities. From bottom to top, solid lines correspond to 0, 1000, 2000, and 3000 K, while dashed lines to 7.4 10-6, 3.0 10-5, 3.0 10-4, and 3.0 10-3 g cm-2. Notice the overlapping of the lowest solid and dashed lines (quiet-Sun case). The two dotted lines correspond to flare models F1 (the lower one) and F2 (the upper one). See text for details

From Fig. 1, it is easily found that the chromospheric temperature rise tends to increase the absolute Lc intensity, as expected. However, the effect of varying seems not to be large. Moreover, there is no obvious difference in the gradients of the four curves, i.e., the derivatives of the logarithmic intensity with respect to wavelength. This means that the color temperature of the spectrum is not changed significantly irrespective of the actual temperature rise in the chromosphere.

2.3. Effect of transition region shift

To show the effect of the downward shift of the transition region, we fix the T versus relation and only vary the values of . The results are superimposed in Fig. 1 by dashed lines for cases of 7.4 10^-6, 3.0 10^-5, 3.0 10^-4, and 3.0 10^-3 g cm^-2. Again, the first and last cases correspond to the quiet-Sun and a strong flare, respectively.

It is shown that a larger also increases the Lc intensity and its effect is more pronounced than the effect of a pure temperature rise. Besides this fact, the larger the value, the smaller the curve gradient, implying a higher color temperature of the spectrum.

A realistic flare model should incorporate the above two processes. For comparison, we also plot in Fig. 1 by dotted lines the results for flare models F1 and F2 from Machado et al. (1980), which clearly show a combination of both effects of the temperature rise and the transition region shift. Note that our computed Lc intensities for models F1 and F2 are consistent with the results given by Avrett et al. (1986), although different computation codes have been used.

2.4. Effect of nonthermal electron beam bombardment

Now we turn to the case when there exists a precipitating electron beam causing nonthermal excitation and ionization of the hydrogen atoms. We assume a power law distribution for the energy spectrum of the beam electrons with a lower energy cut-off at 20 keV. The Lc intensities are then computed for the quiet-Sun model imposed by electron beams with various energy fluxes 0, 10¹⁰, 10¹¹, and 10¹² ergs cm^-2 s^-1, and various power indices 3, 4, and 5. The zero-flux case just corresponds to the quiet-Sun without including the nonthermal effect. Fig. 2 displays the results.

Fig. 2. Lyman continuum intensities at disk center computed for the quiet-Sun model imposed by various electron beams showing the effects of varying their energy fluxes and power indices. Solid, dashed, and dotted lines represent cases of 3, 4, and 5 respectively. From bottom to top, each group of lines correspond to 0, 1010, 1011, and 1012 ergs cm-2 s-1. Notice the overlapping of the lowest three lines for 0 (quiet-Sun case without any nonthermal effect)

It is shown that the effect of nonthermal excitation and ionization caused by the electron beam greatly enhances the Lc. The most noticeable phenomenon is that the curve gradient tends to increase significantly with increasing energy fluxes. Therefore, a lower color temperature should be detected if there exist such nonthermal effects. For a quantitative comparison, we have further computed the color temperatures from the resulting spectra at Å, which are shown to be 8760, 7150, 6680, and 6470 K, corresponding to the four cases of 0, 10¹⁰, 10¹¹, and 10¹² ergs cm^-2 s^-1 with 4. Color temperatures for the spectra of models F1 and F2 without any nonthermal effect are about 11500 and 12220 K respectively.

One can also find that the energy flux is the main factor determining the enhancement of the Lc intensity. Comparatively, the effect of varying the power index is less pronounced though also very clear. We will further discuss this problem in Sect. 3.

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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