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Astron. Astrophys. 357, 351-358 (2000)

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3. Synthetic Stokes profiles emerging from configurations with discontinuous spatial variations of magnetic field and velocity

One-lobe Stokes V profiles comprise about 3% of the total number of measured profiles in our data set. Thus they cannot be dismissed as extremely rare exceptions. On the other hand, neither simplified models of vertical magnetic flux tubes (e.g., Grossmann-Doerth et al. 1989) nor numerical MHD simulations of magnetic flux slabs (e.g. Steiner et al. 1998) yield such extremely asymmetric, let alone one-lobe Stokes V profiles as averages over the whole magnetic structure. We have therefore considered several simplified configurations in order to investigate under which conditions highly asymmetric Stokes V profiles could be formed. Having learned that gradual variations of magnetic field and velocity do not give rise to very large asymmetries we restricted our analysis to models with discontinuous variations of both these quantities.

3.1. Magnetic canopy

This model consists of a quiet sun atmosphere divided into an upper part in which a magnetic field is present but no bulk velocity and a lower part with no magnetic field but a finite velocity. Both magnetic field strength and velocity are uniform in their respective regions. We computed the Stokes parameters of the spectral lines FeI  5250.2 and FeI  6302.5 for rays emerging from this atmosphere with a magnetic field strength of 1500 G and a velocity of 3 km s-1 (positive velocity corresponds to down-flow). These values were chosen in order to achieve approximate equality of Doppler- and Zeeman shift, thereby maximizing the asymmetry (Grossmann-Doerth et al. 1988). The area asymmetry of Stokes V was determined as function of position of the interface. The results are shown in Fig. 2 ; solid and dashed lines represent magnetic field inclinations with respect to the vertical of 30o and 60o, respectively. We use the common definition of the Stokes V asymmetry, i.e. [FORMULA] where [FORMULA] is the (absolute) value of the (blue or red) Stokes V amplitude or lobe area depending on whether amplitude or area asymmetry is meant. Note that the behaviour of both lines is quite similar and the inclination angle of the field has almost no influence on the asymmetry.

[FIGURE] Fig. 2. Stokes V area asymmetry of the spectral lines FeI  5250.2 and FeI  6302.5 emerging from a magnetic canopy configuration as function of interface optical depth for 1500 G and 3 km s-1. Solid and dashed lines represent 30o and 60o field inclination with respect to the vertical, respectively.

The Stokes V area asymmetry increases with interface height up to values around [FORMULA] where it reaches about 71 % for FeI 5250 and 67 % for FeI 6302 ([FORMULA] is the optical depth of the continuous radiation at 5000 Å). Fig. 3 shows the four Stokes parameters of the FeI 5250.2 line for the case of maximum asymmetry.

[FIGURE] Fig. 3. Stokes parameters of FeI 5250.2 emerging from a quiet sun magnetic canopy configuration with 1500 G, 30o, 3.3 km s-1 and the interface at [FORMULA]. The Stokes V area asymmetry is 71.3 %.

In order to understand the rapid rise of the Stokes V asymmetry with height of canopy interface we analyzed the Unno-Rachkovsky equations for a canopy configuration with vertical magnetic field because then the equations decouple and allow a simple solution for the right- and left-hand circularly polarized radiation and thus for Stokes V. This analysis, valid for a standard quiet solar atmosphere, is presented in the Appendix. It shows that, when growing, the asymmetry is increasingly governed by the intensity difference of the (unpolarized) radiation entering the magnetic layer at the wavelength positions of the centers of both Zeeman shifted absorption coefficients. The difference is growing with interface height since the wavelength position of the Doppler shifted absorption line formed in the non-magnetic layer coincides with one of the Zeeman components and, therefore, the asymmetry increases with decreasing line intensity. This also explains why the limiting asymmetry of FeI 6302 is smaller than that of FeI 5250: the former line is not as strong as the latter, so in the case of FeI 6302 the intensity difference is smaller than in the case of the other line. It should be mentioned though, that - owing to the decreasing thickness of the magnetic layer - the amplitude of the remaining lobe of Stokes V decreases from about 40 % to a few percent in the same range in which the asymmetry increases from a few percent to more than 70 %.

So far we have considered Stokes V profiles for single rays corresponding to infinite spatial resolution. In a real observation features much smaller than about 1" cannot be resolved. Hence, our computational results should be spatially averaged in order to make them comparable with observations. We assume the (inclined) canopy interface to run parallel to the magnetic field. Fig. 4 shows the average Stokes V profile of FeI  5250.2, i.e. the arithmetic mean of 30 rays placed equidistantly in [FORMULA] from -4 to -0.5 of the canopy interface, corresponding to a horizontal interval of about 300 km. Note that the amplitude of the averaged Stokes V profile no longer is small but neither is the asymmetry still very high.

[FIGURE] Fig. 4. Spatially averaged Stokes parameters of FeI 5250.2 emerging from a quiet sun magnetic canopy configuration with 1500 G, 30o, [FORMULA]. The interface position varied linearly from -4 to -0.5 of [FORMULA]. The Stokes V area asymmetry is 40 %.

We also investigated a similar configuration, consisting of a low magnetic layer superposed by a non-magnetic region. We found the properties of Stokes V to be similar to those of the magnetic canopy: The asymmetry grows with decreasing thickness of the magnetic layer , i.e. with increasing depth of the interface position; it reaches values of about 80 % in the limit when the magnetic layer, this time deep in the atmosphere, becomes very thin. This configuration may represent the initial phase of magnetic flux emergence. However, in view of the short duration of this phase we do not believe that this configuration is frequently observed so for the sake of brevity we omit the details of this analysis.

3.2. Embedded magnetic flux tube

Next let us consider a magnetic flux tube of finite width embedded in a quiet solar atmosphere and inclined with respect to the vertical. We assume a velocity to be present within the tube and no flow in the external atmosphere. The flow is necessarily along the magnetic field, for us only the (vertical) line-of-sight component is relevant. Both magnetic field strength and velocity within the flux tube are supposed to be uniform. It should be noted, though, that since we are dealing with 2D-configurations this flux tube is, strictly speaking, a flux sheet . The configuration differs from those previously considered by the fact that three regions contribute to the resulting Stokes parameters: The region below the sheet, the sheet itself and the region above the sheet. We computed the profiles of the Stokes parameters of a single ray emerging from the atmosphere for a set of flux tube positions. Fig. 5 shows both area asymmetry and amplitude of Stokes V of FeI  5250.2 for 1500 G, 30o and 3 km s-1 as function of optical depth of the center of the tube, whose thickness was chosen 0.5 in [FORMULA]. It will be noted that unlike in the previously studied configurations the area asymmetry is fairly independent of the tube position and stays at high values. The amplitude of the intact (blue) lobe of Stokes V, on the other hand, varies greatly with tube position: its value is small if the tube is high up or deep down in the atmosphere and it assumes large values if the tube is located between -1 and -2 of [FORMULA]. This behaviour is easily understood. If high up or deep down the flux tube corresponds to the configurations discussed before, i.e. thin canopy or deeply embedded magnetic slab. In both cases the amplitude is small because these regions contribute only marginally to the spectral line. The amplitude is fairly large if the flux tube is located in a height range where the contribution function is large.

[FIGURE] Fig. 5. Stokes V area asymmetry and amplitude of FeI  5250.2 for a ray penetrating an embedded magnetic flux tube as function of optical depth of flux tube center. Quiet sun atmosphere with 1500 G, 30o, 3 km s-1 and tube width 0.5 in [FORMULA].

Again we have to average spatially in order to get a result that could be compared with observations. In Fig. 6 the arithmetic mean of the Stokes parameters of 30 rays are plotted which penetrate the flux tube in the range -4.5 to -0.5 of [FORMULA], equally spaced in [FORMULA] of the sheet's upper interface and corresponding to a horizotal interval of about 300 km. Note that, unlike the canopy configuration, the embedded flux tube, even when spatially averaged, gives rise to highly asymmetric Stokes V profiles with fairly large amplitudes.

[FIGURE] Fig. 6. Spatial averages of Stokes parameters of FeI  5250.2 of 30 rays penetrating an embedded magnetic flux tube with 1500 G, 30o, 3 km s-1 and tube width 0.5 in [FORMULA]. The intersection depth of the rays with the upper surface of the flux tube varied linearly from -0.5 to -4.5 of [FORMULA]. The Stokes V area asymmetry is [FORMULA] and the zero-crossing shift corresponds to 2.3 km s-1.

3.3. The role of the temperature

The results shown above indicate that extremely asymmetric Stokes V profiles arise in a configuration with superposed magnetic and non-magnetic layers provided the contribution of the magnetic layer to the line formation is small, which normally means that its optical thickness must be small. However, in the course of the analysis of magnetic slabs produced by a numerical MHD simulation, we found highly asymmetric Stokes V profiles arising from configurations which seem to contradict these ideas. An example is shown in Fig. 7 where Stokes I and Stokes V of FeI  5250.2 are plotted together with the run of flow velocity and magnetic field with optical depth. It is seen that we deal with a canopy situation with almost no flow velocity in the upper layer where a magnetic field of about 500 G is present and a field-free region with flow below. The fact that the flow direction is upward instead of downward as one would normally expect is of no relevance in the present context. So we would expect the Stokes V profile to be asymmetric but not as highly asymmetric as 75 % because the interface between both layers is located at about [FORMULA]. At that level Fig. 2 would predict an asymmetry of about 35 %, only half the actual value. The discrepancy may be resolved when considering the atmospheric temperature profile. The data of Fig. 2 were obtained with an atmospheric model of the quiet sun while the Stokes V profile of Fig. 7 was formed in a magnetic slab atmosphere produced by a numerical simulation. Fig. 8 shows the run of temperature with depth for both atmospheres. Note that unlike in the quiet sun the temperature in the magnetic slab remains almost constant down to about [FORMULA] where it begins to rise rapidly. Thus we would expect the spectral line to be already fully developed at this depth and therefore, according to the deliberations of Sect. 3.1, the maximum asymmetry is already reached for that interface position.

[FIGURE] Fig. 7. Stokes I and Stokes V of FeI  5250.2 of a ray penetrating the canopy area of a numerically simulated (non-stationary) magnetic flux slab at a chosen position and time (upper two panels). The lower two panels show magnetic field strength and velocity as function of [FORMULA]. The area asymmetry of Stokes V is [FORMULA]; the value is negative because the velocity below the canopy is negative (up-flow).

[FIGURE] Fig. 8. Run of temperature with [FORMULA] for a quiet sun model and for a ray penetrating the canopy region of a computer simulated (non-stationary) magnetic flux slab at a chosen position and time.

The validity of our explanation is demonstrated by Fig. 9 which shows in the upper panels Stokes I and Stokes V of FeI 5250.2 emerging from a canopy configuration identical to those analyzed in Sect. 3.1 with the interface located at [FORMULA]. According to Fig. 2 we should expect an asymmetry of about 50 %. Instead we obtained a true one-lobe Stokes V profile with 97 % area asymmetry solely because we replaced the quiet sun model atmosphere by the slab temperature profile shown in Fig. 8. This example demonstrates that an `abnormal' temperature profile may enhance the asymmetry of a Stokes V profile considerably allowing a large amplitude at the same time.

[FIGURE] Fig. 9. Stokes I and Stokes V of FeI  5250.2 emerging from a canopy configuration inclined by 30o and the interface at [FORMULA]. The lower panels show magnetic field strength and bulk velocity as function of [FORMULA]. The thermodynamic structure of the atmosphere is identical with that of the magnetic flux slab model shown in Fig. 7. The Stokes V area asymmetry is -97%.

Encouraged by this result we repeated the computation of the average Stokes parameters for a magnetic canopy whose results are shown in Fig. 4, replacing the quiet sun temperature profile by the abnormal profile shown in Fig. 8. The result is shown in Fig. 10. Comparing Fig. 4 with Fig. 10 it will be seen that a different temperature profile may more than double the area asymmetry while there is no significant reduction of amplitude relative to the local continuum intensity.

[FIGURE] Fig. 10. Spatially averaged Stokes parameters of FeI  5250.2 emerging from a quiet sun magnetic canopy configuration. Same as Fig. 4 except for the temperature: the quiet sun temperature profile was replaced by the abnormal profile shown in Fig. 8. The Stokes V area asymmetry is 86 %.

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© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000
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