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

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4. Results and discussion

The results of simulations of the [FORMULA]-line polarisation are presented in Figs. 1-4. In Fig. 1 the [FORMULA]-line linear polarisation profiles are plotted versus wavelengths at t= 4.39 s after an electron beam onset ([FORMULA]) for viewing angle [FORMULA] and different depths. In Fig. 2 the linear polarisation profiles are plotted for different viewing angles and two different times of the beam injection for softer beams ([FORMULA]). In Fig. 3 and Fig. 4 respectively the full (integrated in wavelength) linear polarisation and intensity caused by soft weak beam above ([FORMULA]) are plotted versus depth in a flaring atmosphere for viewing angle [FORMULA] at different times.

[FIGURE] Fig. 3a-c. The [FORMULA]-line full polarisation (linear polarisation - parameter Q), plotted versus depth in a flaring atmosphere at the moments t=0.01s (a), t=1.46s (b), t= 2.92s (c) and t= 5.94s (d) for different viewing angles [FORMULA] (from top to bottom in each subfigure). Beam parameters: [FORMULA], [FORMULA].

[FIGURE] Fig. 4. The [FORMULA]-line full intensity (parameter I), plotted versus depth in a flaring atmosphere for different transitions of the fine structure at the moment t= 4.39s (upper curve corresponds to [FORMULA], then in the descent order: to [FORMULA] and [FORMULA]. Beam parameters are the same as in Fig. 2.

The calculations show that impacts with beam electrons in a flaring atmosphere mainly lead to linear polarisation of the [FORMULA]-line emission whilst circular polarisation is negligible. This is in agreement with the most observations (Hénoux & Chambe 1990a; Firstova & Boulatov 1996). It could be understood in terms of radiation with circular polarisation to be produced by small angular deviations of higher energy electrons in collisions with the ambient plasma charged particles (Berestetskii et al. 1989). However, as the beam kinetics shows, in the lower chromosphere the number of such electrons is very small, and they can not produce a significant effect.

There is a wide range of absolute values of linear polarisation in the line profile varying from a few tenths of percent (t= 1.46 s, Fig. 1b) to [FORMULA] (t=4.39 s, Fig. 1d). The maximum ([FORMULA]) of the full [FORMULA]-line linear polarisation was found for softer weaker electron beams ([FORMULA], [FORMULA]= 7) whilst for harder beams the degree of linear polarisation is lower ([FORMULA]).

This behaviour is similar to those of hard X-ray bremsstrahlung polarisation (Zharkova et al. (1995), although the X-ray polarisation is slightly decreased with enhancement of initial flux during the injection whereas the [FORMULA] line polarisation steadyly increases with time. This occurs because of a difference in radiative cross-sections for radiation in these ranges: X-ray emission is a direct reproduction of electron beam distributions. However, the [FORMULA] quanta are slightly delayed by the line opacity caused by a radiative transfer and Doppler and Stark line broadening effects.

Therefore, at the higher chromosphere and transition region where the return current effect causes a decrease in a number of beam electrons precipitating downwards, it also decreases X-ray emission and polarisation. In the lower chromosphere where the [FORMULA]-emission originates, only a residual part of electrons which moved from higher to lower energy distributions can contribute to this emission. The number of these electrons increases with time and depth and their excitation and ionisation rates prevail over the thermal ones. Thus, beam electrons can excite or ionise more and more hydrogen atoms throughout their precipitation. This accumulative effect results in a steady increase of the [FORMULA]-line polarisation during the beam injection.

4.1. [FORMULA]-line polarisation profiles

Electron beam effects on the [FORMULA]-line profiles are rather asymmetric. The profiles are plotted with maximums corresponding to the main transitions with [FORMULA], and [FORMULA]). In order to demonstrate a pure polarisation effect, all other broadening factors such as Doppler and Stark effects are excluded from these profiles in Figs. 1-2, although the calculations were performed including these effects. As expected, the impact polarisation appears only in line cores whereas wings are fully depolarised by collisions with thermal electrons.

At the initial phase of beam injection, only the blue core ([FORMULA] from the central wavelength) in the [FORMULA]-line polarisation profile is affected by impacts with beam electrons, as it is seen in Fig. 2a. This effect varies from [FORMULA] for a viewing angle of [FORMULA] to [FORMULA] for a viewing angle of [FORMULA]. With further beam penetration into a flaring atmosphere the polarisation increases up to [FORMULA] in the blue core and to [FORMULA] in the red core (at [FORMULA] and more from the central wavelength). More powerful soft beams produce higher negative polarisation of about [FORMULA] at higher depths of [FORMULA] decreasing to a few percent at the lower depth of [FORMULA] (see Fig. 2b).

At the top depth where [FORMULA] radiation from the transitions [FORMULA], and [FORMULA]) have negative polarisation whereas in the transition [FORMULA] polarisation is positive reaching [FORMULA] (Figs. 1a and 2b) while the beam reaches a maximum energy flux. Positive polarisation shows a small decrease with depth down to [FORMULA] for a viewing angle of [FORMULA] and to [FORMULA] for a viewing angle of [FORMULA]. However, its contribution to the integrated polarisation is rather small and the full line polarisation stays negative.

These variations of [FORMULA]-line polarisation profiles are believed to reflect a pitch-angular anisotropy of beam electrons at these depths. For radiation in the transitions ([FORMULA], and [FORMULA])) more energetic beam electrons are required than for the lower energy transition ([FORMULA]. As the beam kinetics show, some part of the initial electron beam was transformed into a secondary beam with lower energy electrons returning to the corona. These electrons are scattered to pitch-angles between [FORMULA], according to the kinetic solutions by Zharkova et al. (1995).

It produces additional polarisation with the plane of polarisation being perpendicular to the direction of the secondary beam, but parallel the direction of the initial beam. For an observer at these levels it results in positive polarisation in this transition. At lower depths an anisotropic scattering effect decreases and coincides with a decrease of the number of electrons scattered to bigger pitch-angles as it was shown from the kinetics. This leads to a decrease to a few percent in this positive polarisation.

Therefore, full line polarisation will be a sum of polarisation degrees in each of these transitions and is dependent on the number of beam electrons in the initial and secondary beams.

4.2. Depth and time variations of [FORMULA]-line polarisation

Beam electron impacts become noticeable in the [FORMULA]-line polarisation from a column depth of [FORMULA] which coincides with those defined from the non-LTE simulations of Hydrogen emission without atomic fine structure (Zharkova & Kobylinskii 1989, 1993).

At lower chromospheric levels, from [FORMULA], anisotropic external radiation is a dominant mechanism in the [FORMULA]-line polarisation producing polarisation of about [FORMULA] (Fig. 3a). The mean intensity of this external [FORMULA]-radiation is higher by an order of magnitude than those of the diffusive [FORMULA]-line emission which is produced in absence of beam electrons.

The effect of beam impacts becomes noticeable at upper atmospheric levels (Fig. 3b) with increase in time and energy flux. At time t= 2.92s (Fig. 3c) being close to the beam's maximum energy flux, the impacts produce polarisation comparable and higher than that from the external radiation. At a column depth of [FORMULA], for instance, it reaches [FORMULA] and then sharply decreases with depth to 0.5-1%.

At a column depth of [FORMULA] there is a secondary increase of polarisation, caused by beam electrons, that has the same magnitude about [FORMULA] as for the external radiation.

This increase can be explained by the beam kinetics owing to the return current effect that at the upper latmospheric evels causes a split of the initial beam into two beams moving downwards with separate maximums in energy distributions (Zharkova et al. 1995). At depth of [FORMULA] the return current effect becomes negligible owing to the different reasons. These two beams merge again into a single directed beam with higher abundance of lower energy electrons appeared from Ohmic losses in addition to collisions. At this depth the electrons are responsible for increase in the [FORMULA]-line polarisation (Fig. 3) and intensity (Fig. 4).

When the beam reaches its maximum flux ([FORMULA]), at all levels impacts with beam electrons prevail over any other mechanisms including external radiation (Fig. 3d). For a viewing angle of [FORMULA], at the injection site the polarisation reaches [FORMULA], varying for other depths in a range of [FORMULA] with a secondary increase at the same level [FORMULA], as discussed above. Below this level the [FORMULA]-line polarisation falls to [FORMULA] which reflects the fact that the beam impacts become negligible, and polarisation is governed by the external chromospheric radiation.

The [FORMULA]-line impact polarisation is shown to be strongly affected by the viewing angle with a diversity of polarisation from [FORMULA] for [FORMULA] to [FORMULA] for [FORMULA] (see Fig. 3c-d). The bigger is the viewing angle the higher (in absolute value) is the negative polarisation, reaching its maximum at an angle of [FORMULA].

4.3. Interpretation of some observational features

Despite the fact that there are no [FORMULA]-line polarimetric observations available with high temporal resolution during the impulsive phase of flares when the current simulations are most applicable, an attempt is made to apply them to some observational features obtained during the first few minutes of a flare onset in the observations of Firstova & Boulatov (1996, thereafter FB). For comparison, the observations performed during the decay phase were also considered (Hénoux & Chambe 1990a; Hénoux et al. 1990b, thereafter HE).

The linear polarisation was found in about [FORMULA] of the observed [FORMULA]-spectra (FB). The absolute value of polarisation did not exceed [FORMULA] (HE), although in the observations of early phase of flare an average degree of polarisation was about [FORMULA] reaching [FORMULA] in the regions with weaker emission and in adjacent areas of the weakly perturbed chromosphere (FB). The [FORMULA]-line polarisation observed during the decay phase was positive in about [FORMULA] of observations with the plane of polarisation being parallel to the solar centre direction (HE). The plane of polarisation was close to the flare-to-solar centre direction for parts of a flare with higher emission whereas for weaker parts the plane of polarisation was perpendicular to this direction (FB).

Recently, an interpretation of the observed polarisation in [FORMULA]-line was attempted using the density matrix formalism for proton beams precipitating into a flaring atmosphere (Vogt et al. 1997). Collisions of hydrogen atoms with proton beams and the ambient plasma electrons were taken into account as well as the radiative transitions for incident and diffusive radiative fields in [FORMULA], [FORMULA] and [FORMULA] frequencies. However, the calculated [FORMULA]-line polarisation was found to be lower by up to an order of magnitude than the one observed in flares, thereby, fitting the observations only for a very weak emission at the very beginning of a flare onset (Vogt et al. 1997). In order to fit the observations, other agents of [FORMULA]- line polarisation in flares have to be considered.

In the present paper, the impacts with beam electrons during their precipitation into a flaring atmosphere were considered as a source of the [FORMULA]-line polarisation using the similar density matrix approach as it has been used for proton beams. The calculated magnitude of [FORMULA]-polarisation caused by these impacts with beam electrons varies from a few to [FORMULA] that is within the limits found in the observations. If hydrogen atoms can maintain a level of excitation and ionisation for a few minutes after the beam offset, for instance, because of lower recombination rates in comparison with the excitation and ionisation ones, then these figures for polarisation above can be valid during these few minutes and it can be observed in the early flash phase.

Another question is why for weaker parts of a flare polarisation is higher and for brighter parts lower. In the present study it was assumed that the [FORMULA]-line emission, originating in higher parts of the loop is likely to be less affected by beam electrons. It happens because they just started their precipitation from the top of the loop and at this depth the magnetic field is not high enough to give a strong splitting of atom levels into fine structure. However, this part of the loop could be the brightest one in the [FORMULA] emission owing to the number of neutral hydrogen atoms affected by beam electrons being high enough and their optical thickness being less than or about unity. Thus, [FORMULA] emission can escape completely from the volume that leads to observation of a higher intensity in emission. With further beam precipitation into the loop's feet themagnetic field increases as well as atom levels splitting into fine structure while the [FORMULA]-line optical opacity becomes much higher (Zharkova & Kobylinsky 1993; Kobylinsky & Zharkova 1996). This results in [FORMULA] radiation being trapped by radiative transfer effects and a lesser part of it being emitted while polarisation increases with depth owing to increasing Zeeman splitting.

The kinetics of beam precipitation into a flaring atmosphere (Sect. 2.2) can also explain why only [FORMULA] of flares expose linear polarisation. Only beams with a moderate intensity and spectral index can precipitate to the lower chromosphere as a directed beam, thereby, overcoming the effects of a return current at the transition region and the upper chromosphere. Very intense and hard beams are completely disrupted by this effect at the transition region whereas weak soft beams lose their energy in Coulomb collisions at the upper chromospheric levels. It is likely that [FORMULA] reflects the number of moderate intensity electron beams causing hydrogen emission and polarisation effects.

However, there is a key question to ask whether a calculated direction of plane of polarisation can meet the observational features of polarisation being positive in [FORMULA] of cases. The dependence of polarisation on a viewing angle seems to be rather important for the direction of polarisation (see Fig. 3 and Sect. 3.6 for the definitions) and it can be accounted for the plane of observed polarisation.

Beam electrons propagate along magnetic field lines which on top of the loop are nearly parallel to the solar surface and perpendicular to the observer looking from the top. This beam produces a higher intensity in the direction perpendicular to a beam propagation than in the parallel direction. Therefore, for the observer looking from the top, positive polarisation can be measured. On the other hand, in the loop's feet standing vertically, the direction of electron precipitation is parallel to the observer still looking from the top, and, hence, the resulting [FORMULA]-line polarisation is negative.

In real observations the observer can look onto the loop from the side or it can have inclinations to one or other side. Also, areas projected from the top of the loop on the horizontal plane are normally bigger by a factor 2/3 than those from the feet. Therefore, the fact that [FORMULA] of flares have positive polarisation in the observations by Firstova & Boulatov (1996) might indicate both a projection effect and a dominance of inclined loops in the observations.

However, these are speculations that need to be confirmed with observations with high spatial and temporal resolutions carried out during the impulsive and early flash phases of flares that, we believe, will become available in the future.

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© European Southern Observatory (ESO) 2000

Online publication: February 9, 2000