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Astron. Astrophys. 351, 368-372 (1999)

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3. Energetics in the lower atmosphere

The key question that remains to be solved is what mechanism can cause the unusually hot TMR in model A. Canonical heating processes, such as bombardment by energetic particles injected from the corona, irradiation by soft X-rays, or EUV irradiation, seem ineffective to heat the TMR (Machado et al. 1978; Machado & Hénoux 1982; Poland et al. 1988). The chromospheric backwarming effect operates if the chromosphere is hot and condensed enough (Machado et al. 1989), if there is a strong chromospheric condensation (Gan & Mauas 1994), or if the chromosphere is bombarded by an intense particle beam that causes non-thermal excitation and ionization of hydrogen atoms (Aboudarham & Hénoux 1987). All these requisites would also produce a significant Balmer jump in the continuum spectrum which is certainly not the case for the 1974 October 11 event. Therefore, the backwarming mechanism is not relevant either.

Based on these arguments, we suggest that the main energy source lies in the TMR itself, as in the events studied by Mauas et al. (1990) and Ding et al. (1994). Theoretically, magnetic reconnection can occur in lower atmospheric layers (e.g., Li et al. 1997). Although it is still unclear what is the direct consequence of such a reconnection, one may postulate that, high energy particles are accelerated, carrying a significant amount of energy. In the following, we assume that collisional heating by high energy particles, injected from lower layers, is the main heating source.

3.1. Thermal response of the lower atmosphere due to particle beam heating

To investigate how the very hot TMR is formed, we have to solve the full set of radiation hydrodynamics equations and follow the temporal evolution of the atmosphere. This is a very difficult task at present. In first approximation, we neglect hydrodynamic effects and heat conduction. Then, the energy equation can simply be written as


where [FORMULA] and [FORMULA] are the hydrogen level i number density and the corresponding excitation potential, [FORMULA] is the number density of ionized hydrogen, and [FORMULA] is the ionization potential. Eq. (1) is valid for a pure hydrogen atmosphere. The number of bound levels, N, is taken to be 4. H and L are respectively the collisional heating and radiative cooling rates. [FORMULA] is an extra heating term used to compensate for the radiative cooling in the pre-flare atmosphere, which is assumed not to change during the flare evolution.

The heating rate, due to bombardment by particle beams, can be computed according to formulae given by Emslie (1978). In particular, if the particles are injected from a low layer, we employ a method similar to that in Ding et al. (1998). Assuming a power law for the number flux distribution of the particle beam, we parameterized it by the total energy flux, [FORMULA], the power index, [FORMULA], and the low-energy cut-off, [FORMULA]. In addition, the particle production site is taken as the middle of the TMR ([FORMULA] g cm-2).

The radiative cooling rate is calculated through non-LTE computations. Here, we consider only the lines and continua of H and H-, which are the main cooling agents in the TMR. When high energy particles are present, the non-thermal excitation and ionization of the hydrogen atoms should be considered in the computations (e.g., Fang et al. 1993). These non-thermal effects raise greatly the local source function, and accordingly change the height distribution of the radiative cooling rate.

Eq. (1) can be solved iteratively together with non-LTE computations of the radiative cooling rate. A more detailed description of this procedure is given in Ding et al. (1999), where the heating due to non-thermal particles in lower layers is used to explain the continuum emission of WLFs. The procedure is summarized as follows. Starting from the pre-flare atmosphere, taken as model B, and injecting a particle beam described as above, the values of H and L are computed at each time step, upon which the atmospheric temperature at the next time step is obtained. This is an explicit procedure which requires that the time interval between successive steps is small enough ([FORMULA] s). The results indicate that, as long as the particle beam is steady, the whole atmosphere can reach after a certain time ([FORMULA] s) a quasi-stationary state (QSS) in which the radiative cooling is roughly compensated for by the particle beam heating; afterwards, the atmospheric temperature changes very slowly although the particle beam persists.

3.2. Limit of the beam flux due to return current instability

A beam of charged particles generates a return current which can excite instabilities if the drift velocity of the return current exceeds some critical value (Hoyng et al. 1978). For electron beams with a power law distribution, Aboudarham & Hénoux (1986) deduced a formula for the critical energy flux below which the return current is stable against excitation of ion-acoustic or electrostatic ion-cyclotron waves,


in the case that ambient electrons and protons have a same temperature. According to Eq. (2), Aboudarham & Hénoux (1986) found that in the coronal environment, electron beams with [FORMULA] and [FORMULA] keV are stable when [FORMULA] ergs cm- 2 s-1. Apparently, [FORMULA] is proportional to [FORMULA], the ambient electron density. In the TMR, [FORMULA] depends in turn on the energy flux of the particle beam owing to the non-thermal ionization effect. Thus, Eq. (2) becomes implicit and we can only solve it iteratively together with Eq. (1).

Without considering the non-thermal ionization effect, the electron density is only 1.3 1012 cm-3 at [FORMULA] g cm-2 in the pre-flare atmosphere. However, considering the non-thermal effect leads to a significant increase of the hydrogen ionization degree, which, in turn, raises the value of [FORMULA]. For an electron beam with [FORMULA] and [FORMULA] keV, [FORMULA] is computed to be as high as [FORMULA] 1015 ergs cm-2 s- 1 at [FORMULA] g cm-2. In fact, such a strong electron beam can result in an electron density at the particle injection site of [FORMULA] 1015 cm-3, three orders of magnitude larger than that in the case without non-thermal ionization effect.

3.3. Possible electron beams responsible for the very hot TMR

We have computed several cases for electron beam heating and obtained the respective QSSs. Fig. 3 plots the results for six cases. We see the effect of varying the low-energy cut-off in the top row and the effect of varying the energy flux in the bottom row, respectively. The average temperature around the TMR is found to increase with increasing [FORMULA]. This fact is conceivable. Particles of a higher energy deposit energy over a broader region; meanwhile, the line source function there is enhanced owing to the non-thermal excitation and ionization effect, leading to a stronger radiation field that helps more effectively to heat the particle-free region. The results show that, the QSS in the case where [FORMULA] ergs cm-2 s- 1, [FORMULA], and [FORMULA] keV roughly matches model A in the TMR, and that it leads to a Ca II K line profile and a continuum intensity comparable to observations (Fang et al. 1995). On the other hand, we can also raise the heating effect by purely increasing the energy flux. However, if a smaller [FORMULA] (say, 20 keV) is adopted, a very large [FORMULA], even approaching the critical value as discussed above, would be needed to yield a QSS comparable to model A. Therefore, such a case is less plausible.

[FIGURE] Fig. 3. Temperature distribution (dotted line) for the atmosphere at the quasi-stationary state in the presence of an electron beam heating. The six panels refer to six cases with beam parameters [FORMULA] (in units of ergs cm-2 s-1) and [FORMULA] labeled in each panel. In all cases, the power index [FORMULA] is fixed to be 4. Note that the temperature spikes at the electron production site are somewhat computational artifacts since we have neglected the thermal conduction and the hydrodynamic effect. Models A (solid line) and B (dashed line) are also plotted for reference. See text for details

To check further the energetics in the lower atmosphere, we plot in Fig. 4 the column mass distribution of the energy deposition rate, H, by an electron beam and that of the net radiative cooling rate, [FORMULA], at the very beginning of electron bombardment. An electron beam with [FORMULA] ergs cm-2 s- 1 and [FORMULA] keV is adopted which corresponds to the top-right panel in Fig. 3. As expected, there is initially a significant imbalance between the collisional heating and the radiative cooling, implying a net heating effect of the atmosphere. Just around the electron injection site, the energy deposited by beam electrons is obviously the dominant heating energy. However, this energy decreases rapidly with the column mass from the electron injection site. On the other hand, one can find from Fig. 4 that the radiative cooling rate becomes negative in the photosphere, which amounts to [FORMULA] 103 ergs cm-3 s- 1 at around [FORMULA] g cm-2. This means that radiative heating comes into effect and may be comparable to the effect of collisional heating in some particular layers. This negative radiative cooling is caused by an absorption of enhanced radiation from the beam heated layers, within which the non-thermal excitation and ionization effect by beam particles plays a chief role. With the atmosphere being heated, one may expect that the imbalance between the collisional heating and the radiative cooling becomes smaller and it nearly diminishes when the QSS is reached.

[FIGURE] Fig. 4. Rate of energy deposition by an electron beam (solid line), characterized by [FORMULA] ergs cm-2 s- 1 and [FORMULA] keV, and net radiative cooling rate (dashed line) computed at the very beginning of electron bombardment

Fig. 3 implies that a beam of hecta-keV electrons is a viable candidate responsible for the production of the very hot TMR. Moreover, a fairly large energy flux is needed if the electron beam is the only energy source. Of course, the presence of other heating sources will loose such a stringent requirement.

3.4. The role of proton beams

Protons of a given energy cross much shorter column depth than electrons of the same energy. According to Emslie et al. (1996), protons of an energy [FORMULA] times that of electrons are required in order to deposit energy over a difference in column mass the same as that for electrons. Therefore, one can expect that the heating results produced by 20, 50, and 100 keV electrons can be alternatively produced by 0.8, 2, and 4 MeV protons, respectively, if the total energy flux remains the same for both particles. For further confirmation, Fig. 5 plots the temperature distributions for the heated atmosphere in the three protons cases, showing a nice correspondence to the three electron cases plotted in the top row of Fig. 3.

[FIGURE] Fig. 5. Temperature distribution (dotted line) for the atmosphere at the quasi-stationary state in the presence of a proton beam heating. The three panels refer to three cases with beam parameters [FORMULA] (in units of ergs cm-2 s-1) and [FORMULA] labeled in each panel. In all cases, the power index [FORMULA] is fixed to be 4. Models A (solid line) and B (dashed line) are also plotted for reference

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Online publication: November 2, 1999