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Astron. Astrophys. 337, 887-896 (1998)

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7. Discussion and conclusions

The current oscillations in a loop result in the modulation of the magnetic field and the loop cross-sectional area. Thus both thermal and nonthermal emission from the loop should be modulated by these oscillations. It follows from Eq. (26) for the capacitance of an equivalent electric circuit, because [FORMULA], that the most important part in the capacitance is the coronal part of a magnetic loop. Supposing, for instance, that [FORMULA] and [FORMULA] we get


where [FORMULA], l, and S are the average plasma density, length, and cross-sectional area of coronal part of a loop, respectively, and I is the electric current along the loop axis. The total inductance of a slender [FORMULA] flux tube can be presented as (Alfvén & Carlqvist 1967)


From Eqs. (30) and (31) we obtain the period of eigen oscillations of an LRC-circuit


where [FORMULA] A. To find this last Equation we keep in mind the following average values of a magnetic loop (Bray et al.,1991): [FORMULA] cm-3, [FORMULA] cm, and [FORMULA] cm2. The current values inferred from Eq. (32) are given in Table 1 for the events under investigation. For these sixteen events the values of the total circuit energy [FORMULA] are also presented. These values are between [FORMULA] and [FORMULA] ergs. For two of the events, 22 June 1989 and 7 May 1991, we got a chance to compare the total energy stored in the circuit and the thermal energy of the flare. For the 22 June 1989 flare we obtained the current [FORMULA] A and [FORMULA] ergs. On the other hand the thermal energy of the evaporated plasma, inferred from mm-wave and soft X-ray data (Urpo et al. 1994), is equal to [FORMULA] ergs. For the flare of 7 May 1991, we got [FORMULA] ergs and the thermal energy of the flare was [FORMULA] ergs. The thermal energy of a hot evaporated chromospheric plasma filling the magnetic loop is about the total flare energy (Wu et al. 1986). Hence, at least for these two flares we can claim that only 5% of total energy of the electric current stored in the current-carrying magnetic loop was released during a single flare process. Such a situation seems quite reliable when the magnetic structure is stable trough the entire flare. One example of homologous flares having almost the same time behavior, demonstrated by the similarity in Pxx-X power spectral densities, is shown in Fig. 5. A rather good example has also been presented before by Urpo (1983). Fig. 6 shows the dependence of the maximum flux of a mm-wave burst vs. the total energy stored in the circuit, see also Table 1. The tendency is to have a decreasing mm-wave plasma flux with the increase of the total circuit energy. We interpret this tendency in terms of plasma beta [FORMULA]. The plasma beta decreases when the stored energy increases, because the magnetic field in the loop grows. In the case of a low beta plasma the plasma instabilities, for example the flute-instability which can act as a flare trigger (Zaitsev & Stepanov 1992), are stabilized (`hard' circuit) and the energy release drops.

[FIGURE] Fig. 6. Maximum fluxes of the observed mm-wave bursts (stars), and the plasma [FORMULA] (solid line), vs. the energy stored in a coronal loop

Note, that the Q-factor [FORMULA] of current oscillations is high enough. Indeed, taking Eqs. (30) and (31) into account for [FORMULA] CGS we get [FORMULA]. Under flare condition [FORMULA] and [FORMULA].

The self-consistent model of an equivalent circuit analog for the current-carrying magnetic loop that has been considered here, suggests relatively powerful long-duration Joule energy release in the loop footpoints in the photosphere. For the case of a steady-state flux tube, the kinetic energy of a solar plasma convective flow in the footpoints gives DC electric field that is due to the charge separation (Sen & White 1972). The power connected with the plasma flow is realized in the form of Joule heating


For the axially symmetric magnetic flux tube with [FORMULA] and partially ionized plasma flow [FORMULA], keeping in mind Eq. (5), we obtain for [FORMULA] that the Joule energy dissipation is independent of the magnetic field: [FORMULA]. Our estimates have shown that for the convection velocities 0.3-0.5 km/s that are usually observed (Bray et al. 1984; Simon & Leighton 1964), the Joule heating input is less as compared with the radiation losses, and the loop footpoints remain cool ([FORMULA] K). If the velocity of the convective flow grows up to extreme values, [FORMULA] 2 km/s, the Joule heating becomes more important than the radiation losses and the loop legs can be heated a lot. Such a heating may be one origin for the observed soft X-ray bright points (Bray et al.,1991).

Let us summarize the main results in our analysis. A self-consistent model of an equivalent LRC-circuit analog of the current-carrying magnetic loop reveals that both the resistance and the capacitance depend on the electric current along the loop axis. Quasi-periodic modulation of the mm-wave emission during a flare is the radiation signature of the eigen oscillations of an LRC-circuit and include the information on the current value in a magnetic loop. Spectral analysis of the sixteen mm-wave events presented in this paper give modulation time scales from 0.7 to 17 s, which, in turn, give current values [FORMULA] A and total circuit energies [FORMULA] ergs. We got the possibility to compare the total energy of the current stored in a loop and the flare energy for two of the events. In both cases the flare energy was less then 5% of the total circuit energy. It means that the magnetic structure of the flare loop or the loop system was stable and didn't change during the flares. There was a tendency of decrease in the maximal mm-wave emission flux (and most probably the total flare energy) with the increase of the energy stored in the magnetic loop in the sixteen flares that were investigated.

In conclusion we point out that the method of electric current diagnostics proposed here can be applied to the diagnostics of currents in stellar atmosphere. The loop size in the coronae of UV Cet-type stars is about an order of magnitude bigger as compared to the solar case. Taking for example the pulsation period observed on Hyades flare star H II 2411 by Rodono (1974), equal to 13 s, and using Eqs. (30) and (32) for the pulse period we have I [FORMULA] A and, consequently, the total energy stored in a stellar magnetic loop is [FORMULA] ergs.

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© European Southern Observatory (ESO) 1998

Online publication: August 27, 1998