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Astron. Astrophys. 337, 294-298 (1998)

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2. A possible mechanism for emission in the H[FORMULA] far line wings

The rôle of the magnetic field in the formation of moustaches is still unclear. Kitai and Muller(1983) suggested that moustaches originate in elementary flux tubes. On the othe hand, Rust(1968) and Rust and Keil(1992) pointed out that Ebs are located at places where magnetic features of one polarity meet opposite polarity features. The location of Ebs at the interface between regions of opposite magnetic polarity suggest that they are associated with the presence of horizontal magnetic fields. Diver et al.(1996) proposed the Kelvin-Helmotz instability resulting from a laminar flow along a horizontal magnetic field as the origin of Ebs. It is also possible that magnetic reconnection between horizontal and non-aligned magnetic fields does occur in the lower atmosphere (Li et al.1997). If this is true, energetic protons accelerated there would propagate horizontally in opposite directions.

2.1. Proton-hydrogen charge exchange

When an proton beam precipitates into a neutral hydrogen atmosphere, a beam proton [FORMULA] may capture an electron from a target hydrogen atom, [FORMULA], becoming a superthermal one, [FORMULA], excited to level j :

[EQUATION]

The line intensity enhancement in a transition from upper level j to lower level i is [FORMULA] where the photon emission rate is (ZFH):

[EQUATION]

[EQUATION]

where C is the speed of light, [FORMULA] is the spontaneous radiative transition probability from level j to level i, [FORMULA] the number density of non-thermal protons of energy E excited to the level j in [FORMULA]-phase space, s is the distance along the trajectory of the protons of pitch angle [FORMULA]. The parameters [FORMULA] and [FORMULA] are dependent on the geometry and are given in ZFH. m and E are respectively the mass and the energy of a proton; [FORMULA] is the angle between the line of sight and the magnetic field direction.

In order to get a line profile symmetrical around [FORMULA], [FORMULA] must be equal to [FORMULA]. In that case, the maximum amplitude [FORMULA] of the Doppler shift associated with the variation of the component along the line of sight of the velocity of a recombined hydrogen atom is given by:

[EQUATION]

Since [FORMULA] is related to the heliocentric angle [FORMULA] and to the angle between the magnetic field direction and the plane defined by the line of sight and the local solar vertical, [FORMULA], by [FORMULA], this imposes to assume that the particles are moving around an horizontal magnetic field, either at disk center ([FORMULA]) with any orientation, or at the limb, perpendicular in that case to the line of sight ([FORMULA]). The Doppler shift is then given by [FORMULA] = 607 Å [FORMULA].

2.2. Non-thermal emission

Three bound levels plus an ionized state were used to represent the hydrogen atom. The same procedure as in ZFH was used to compute the number density [FORMULA].

The computations were restrained to the hydrogen H[FORMULA] emission line profile at the center of the solar disk, assuming that the proton beam was accelerated in the lower chromosphere at a column mass [FORMULA]. At the site of acceleration, the energy distribution of the flux, [FORMULA], of energetic protons of energy [FORMULA] was represented by a power law, [FORMULA], above an energy cut-off [FORMULA]. After crossing an horizontal distance s, with column number density N, the energy distribution of the proton beam flux is given by (ZFH):

[EQUATION]

where [FORMULA] is the total energy flux above the low energy cutoff [FORMULA]. [FORMULA] is the energy needed for a proton to cross a distance s, corresponding to a column number density N such that [FORMULA], with K=2 [FORMULA], [FORMULA], [FORMULA]; x is the ionization degree. As x is very small ([FORMULA]) in the lower chromosphere, [FORMULA], where [FORMULA] represents the effect of inelastic collisions on neutral hydrogen atoms. We took [FORMULA], typical value for a proton with an energy of 1 MeV.

By using Eqs. (2) and (3), for different values of [FORMULA], [FORMULA] and [FORMULA], the intensity enhancements in Ly[FORMULA], Ly[FORMULA] and H[FORMULA] lines have been computed. A fixed hydrogen density, [FORMULA], equal to 1 [FORMULA] 1015 cm -3, and an ionization degree [FORMULA], were used. For very weak ionization, changing x does not influence significantly the results.

Fig. 1 gives the computed non-thermal emission profiles of the Ly[FORMULA], Ly[FORMULA] and H[FORMULA] lines for different values of the total input energy flux [FORMULA]. The intensity plotted in this figure, as well as in Figs. 2 and 3, is not the intensity at the solar surface but rather the energy emitted per steradiant, per second and per Å[FORMULA] by a horizontal beam of protons of Sect. 1 cm2. It cannot be compared directly to the quiet sun intensity. However, Figs. 1, 2 and 3 give the wavelength dependence of the hydrogen emission lines and show that intensities comparable to the quiet sun intensity can be reached by assuming a vertical extension of the moustaches of a few ten of km. The intensities of the Ly[FORMULA] and Ly[FORMULA] line wings may increase by two to three orders of magnitude relatively to the quiet-Sun line profile intensities (see ZFH), while the intensity of the central part ([FORMULA] [FORMULA] 5 Å) of the H[FORMULA] line also increases and reaches about 10 [FORMULA]20 [FORMULA] of the quiet-Sun continuum. This is quite different to the case where a proton beam bombards the chromosphere from the corona producing a non-thermal H[FORMULA] emission three or four orders of magnitude weaker than the continuum background (see ZFH). The reason is simply that the proton beam loses all its energy locally in the chromosphere (see Fang et al., 1995).

[FIGURE] Fig. 1. Computed non-thermal emission profiles of the Ly[FORMULA], Ly[FORMULA] and H[FORMULA] lines, at disk center ([FORMULA]), for a total input energy flux [FORMULA] = 1 [FORMULA] (solid line), 5 [FORMULA] 1011 (dashed line) and 1 [FORMULA] (dotted line) erg cm-2 s-1 above a low energy cut-off [FORMULA] = 300 KeV and a power index [FORMULA] = 5 for the quiet-Sun model C (Vernazza et al. 1981). In all cases the pitch angle [FORMULA] is taken to be [FORMULA]

[FIGURE] Fig. 2. [FORMULA]-dependence of the non-thermal emission profiles of the Ly[FORMULA], Ly[FORMULA] and H[FORMULA] lines for [FORMULA] = 150 keV (solid line), 300 keV (dashed line) and 600 keV (dotted line) and for the values [FORMULA] = 5 [FORMULA] 1011 erg cm-2 s-1, [FORMULA] = 5, [FORMULA] and [FORMULA] = [FORMULA]

[FIGURE] Fig. 3. [FORMULA]-dependence of the non-thermal emission profiles for [FORMULA] = 5 (solid line), 4 (dashed line) and 3 (dotted line). The values of the parameters are the same as in Fig. 2, but [FORMULA] = [FORMULA]

Fig. 2 shows the [FORMULA]-dependence of the non-thermal profiles for [FORMULA] = 5 [FORMULA] 1011 erg cm-2 s-1, [FORMULA] = 5 and [FORMULA] = [FORMULA]. An interesting point to be noticed is that the highest intensity corresponds to [FORMULA] = 300 KeV. When [FORMULA] increases, the non-thermal emission decreases. This is probably due to the non linear variation of the charge-exchange cross section with the particle energies, and to the fact that the superthermal hydrogen atoms with higher energy produce non-thermal emission at wavelengths further away from the line center, so that the profile becomes broader and flatter.

Fig. 3 gives the [FORMULA]-dependence of the non-thermal emission profiles for the parameter values [FORMULA] = 5 [FORMULA] 1011 erg cm-2 s-1, [FORMULA] = 300 keV and [FORMULA] = [FORMULA]. It is worth to notice that the line wing intensities decrease with increasing values of [FORMULA], while the intensities in the line center part increase. This is due to the fixed value of [FORMULA], so that the number density of protons with lower energy is higher than that of protons with higher energy when [FORMULA] increases. Another point that should be mentioned is that with [FORMULA] increasing, the line intensity decreases. This is especially obvious for the H[FORMULA] line.

2.3. Transfer of H[FORMULA] radiation, net excess H[FORMULA] emission

The non-thermal H[FORMULA] photons emitted in the low chromosphere will be absorbed by the ambient atmosphere. The non-thermal H[FORMULA] emission line profile resulting from photon propagation through the atmosphere has been computed. Considering that the intensity of the non-thermal emission is at most only about 10-20 [FORMULA] of the background continuum emitted deeper in the atmosphere, we assumed simply that the atmosphere was playing only an absorbing role. Thus, the emerging emission is given by

[EQUATION]

where [FORMULA] is the opacity of the upper atmosphere at a distance [FORMULA] from H[FORMULA] line center. It was computed from the quiet-Sun model C (Vernazza et al. 1981) used as basic model representative of the solar atmosphere including the non-thermal excitation and ionization caused by the proton beam (see Hénoux et al. 1993). Fig. 4 gives the computed H[FORMULA] excess profile, [FORMULA], where [FORMULA] is the background continuum emission, for the case of [FORMULA] = 5 [FORMULA] 1011 erg cm-2 s-1, [FORMULA] = 300 keV, [FORMULA] = 5 and [FORMULA] = [FORMULA]. This profile is convolved with a Doppler profile with a macro-velocity. It can be seen that only near the line center ([FORMULA] [FORMULA]1 Å), there is an obvious absorption. It is favourable to the reproduction of the observed H[FORMULA] line profiles of EBs, because, as indicated by many authors, the latter just have a strong central reversal.

[FIGURE] Fig. 4. Computed H[FORMULA] excess profile for the case of [FORMULA] =5 [FORMULA] 1011 erg cm-2 s-1, [FORMULA] = 300 keV, [FORMULA] = 5 [FORMULA] and [FORMULA] = [FORMULA]. Convolution with a line profile with a macro-velocity of 10 km s-1 is made.

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© European Southern Observatory (ESO) 1998

Online publication: August 6, 1998
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