ON THE STOKES V AMPLITUDE RATIO AS AN INDICATOR OF THE FIELD STRENGTH IN THE SOLAR INTERNETWORK

Transcription

1 The Astrophysical Journal, 659:1726Y1735, 2007 April 20 # The American Astronomical Society. All rights reserved. Printed in U.S.A. A ON THE STOKES V AMPLITUDE RATIO AS AN INDICATOR OF THE FIELD STRENGTH IN THE SOLAR INTERNETWORK E. Khomenko 1, 2 and M. Collados 1 Received 2006 October 20; accepted 2006 December 21 ABSTRACT The results of the determination of magnetic field strength from weak polarimetric signals in solar internetwork regions are contradictory. We investigate the origin of this contradiction with the help of MHD simulations. It is shown that the Stokes V amplitude ratio of the Fe i kk15652y15648 lines is a good indicator of kg magnetic field concentrations, even for magnetic fields with a complex internal structure like those in MHD simulations. The Stokes Vamplitude ratio of the Fe i kk5247y5250 lines also shows a good correlation with magnetic field strength. However, in simulations with a flux level appropriate for the internetwork, it gives values corresponding to sub-kg fields. The reason is the rapid decrease of the field strength with height in kg magnetic field concentrations. These lines sample high regions of the atmosphere, where the field is already below kg levels. We also find that the Stokes V amplitude ratio of the Fe i kk6301y6302 lines shows no correlation with the magnetic field strength. The reason lies in the large difference in the heights of formation of these two lines. The value of the magnetic field strength obtained from the Fe i kk6301 and 6302 lines depends crucially on the treatment of gradients of the magnetic field, line-of-sight velocity, and temperature, even at a numerical spatial resolution of 20 km. Subject headinggs: MHD polarization Sun: infrared Sun: magnetic fields Online material: color figures 1. INTRODUCTION This paper is a contribution to resolving the long-standing problem of determining the magnetic field strength from weak spectropolarimetric signals measured in internetwork solar regions. The mixed-polarity weak-flux magnetic elements in the internetwork are proved to have a broad distribution of field strengths (see, e.g., Sánchez Almeida & Lites 2000; Socas- Navarro & Sánchez Almeida 2003; Khomenko et al. 2003; Trujillo Bueno et al. 2004; Domínguez Cerdeña et al. 2006a). However, the fraction of the strong kg fields in this distribution is still unknown, and as a consequence it is not clear whether the characteristic field strength in the internetwork lies in the kg or sub-kg range. The answer to this question is important, since kg fields may determine the amount of magnetic energy contained in quiet solar regions. If this energy turns out to be large, it may contribute to the heating of the upper solar atmosphere. There is no agreement between the magnetic field strength distribution measured with the help of different spectral lines. The infrared Fe i lines at 1.56 m reveal mostly weak fields with an exponential distribution (Lin 1995; Lin & Rimmele 1999; Khomenko et al. 2003; Martínez González et al. 2006; Domínguez Cerdeña et al. 2006b). The dominance of the weak fields in the internetwork is also suggested by observations of Stokes V of the Mn i lines, which are sensitive to hyperfine structure effects (López Ariste et al. 2006, 2007). On the contrary, the visible Fe i kk6301 and 6302 lines suggest that the characteristic field strength is in the kg range (Lites 2002; Socas-Navarro & Sánchez Almeida 2002; Sánchez Almeida et al. 2003; Domínguez Cerdeña et al. 2003, 2006b). Since Stenflo (1973), the Fe i kk5257, 5250 pair of lines has been used to measure the magnetic field strength of the network elements in the quiet Sun. These lines have practically the same 1 Instituto de Astrofísica de Canarias, Tenerife, Spain; khomenko@ iac.es, mcv@iac.es. 2 Main Astronomical Observatory, National Academy of Sciences, 03680, Kyiv, Ukraine oscillator strength and excitation potential of the lower level but differ in their effective Landé factor. The Stokes Vamplitude ratio (called the magnetic line ratio ) was used as an indicator of the intrinsic field strength of the unresolved network elements. Except for Keller et al. (1994), no observations of internetwork fields using these spectral lines have been reported yet. A common assumption used a priori in many observational works about internetwork fields is that the ratio of the Stokes V amplitudes of the Fe i kk6301, 6302 and Fe i kk15648, lines is directly related to the field strength contained in the resolution element, i.e., the same as for the Fe i kk5257, 5250 pair (see, e.g., Domínguez Cerdeña et al. 2003; Socas-Navarro & Sánchez Almeida 2003; Lites & Socas-Navarro 2004; Socas- Navarro et al. 2004). It has been proved that the line-ratio method gives the exact value of the longitudinal field component under the following conditions: (1) the two spectral lines are identical in their sensitivity to the thermodynamic parameters and form at the same height; (2) the magnetic field has only longitudinal components and is constant with height; and (3) the line formation is outside weak-field and strong-field limits, i.e., the field strength is intermediate (Sánchez Almeida et al. 1988). The presence of the velocity field does not affect the validity of the line-ratio method. Both lines should have a triplet Zeeman pattern; however, the presence of more than two -components affects the line ratio to a minor extent (Solanki 1993). If the two selected spectral lines are not sufficiently similar, the ratio of the Stokes V amplitudes becomes dependent on the line-of-sight ( LOS) velocity and thermodynamic parameters. As was recently shown by Martínez González et al. (2006), temperature gradients of reasonable magnitude can change the line ratio of the Fe i kk6301, 6302 lines in such a way that kg and sub-kg field strengths may be indistinguishable by this method. Results based on Stokes V magnetogram calibration, line ratios, and Milne-Eddington inversions depend on the assumption of the constancy of the LOS velocity and magnetic field with height. The weak profiles in internetwork solar regions are strongly asymmetric and have irregular shapes. The interpretation of such spectra in terms of simplified

2 STOKES V AMPLITUDE RATIO 1727 TABLE 1 Atomic Parameters of the Fe ilines k (8) EPL a (ev) log gf g eff a Excitation potential of the lower level of the transition. modelsmayleadtoconfusionandcontradictionsbetweendifferent measurements. The purpose of this paper is to check how well the Stokes V amplitude ratio constrains the magnetic field strength in situations with magnetic fields that have complex structures, such as those expected to exist in solar internetwork regions. We investigate the validity of the Stokes V line-ratio method for the Fe i kk5247y5250, kk6301y6302, and kk15652y15648 pairs of lines, as applied to the complex fields obtained via MHD simulations, assuming that these fields provide an adequate representation of the fields in solar internetwork regions. 2. MHD SIMULATIONS We use a snapshot of realistic three-dimensional simulations of solar magnetoconvection ( Vögler et al. 2005). The simulation has a bipolar magnetic field structure and an average unsigned magnetic field strength of 30 G at the level of optical depth log 5 ¼ 1 (see Vögler et al. 2005; Khomenko et al for details). The horizontal spatial resolution of the simulations is 20 km and the size of computational domain is 6000 ; 6000 ; 1400 km 3, with 600 km being above log 5 ¼ 0. A detailed description of this particular simulation run and analysis of Stokes spectra can be found in Khomenko et al. (2005). This snapshot, with a 30 G average magnetic field, is representative of a bipolar solar internetwork region. The Stokes spectra of the Fe i kk5247.1, , , , 15648, and lines formed at solar disk center ( ¼ 1) were calculated for every vertical column of the snapshot with the help of the SIR code (Ruiz Cobo & del Toro Iniesta 1992). The atomic parameters of the selected lines are given in Table 1. Note that neither micro- nor macroturbulent velocities were used for the synthesis. Thus, the profiles at the numerical resolution of 20 km are rather narrow. The broadening of the profiles occurs naturally due to the velocity field present in the simulations after averaging the spectra over the simulation box. It was shown that the average Stokes I spectra compares rather satisfactorily with the observed one (Khomenko et al. 2005). In order to decrease the spatial resolution and the continuum contrast of the synthetic spectra and to make these parameters close to typical observed values, we performed a convolution of the two-dimensional snapshot with an adequate point-spread function (Khomenko et al. 2005). In addition to the smearing, the smoothed images were binned 4 ; 4 pixels to decrease the pixel resolution. The resolution of the smoothed images is about Y0.7 00, and the contrast at the continuum wavelength is 3.5% in the infrared and 6%Y7% in the visible. These correspond to observations obtained under very good seeing conditions. 3. LINE-RATIO DEFINITION AND CALIBRATION Since Stenflo (1973), many modifications of the line ratio have been applied in the literature to derive the intrinsic magnetic Fig. 1. Calibration curves for the line ratio. Thin lines: V mic ¼ 0and V mac ¼ 0. Thick lines: V mic ¼ 0:3 kms 1 and V mac ¼ 0:7 kms 1. Solid lines: Fe i kk6301, Dashed lines: Fei kk5247, Dotted lines: Fei kk15648, lines. field strength of the magnetic elements in the Sun (see, e.g., Sánchez Almeida et al. 1988; Keller et al. 1990a; Domínguez Cerdeña et al. 2003). We chose the simplest one and calculated the ratio of the amplitudes of the Stokes V blue lobe peaks of each of the three pairs of lines. In order to have the same reference system, the line ratio is defined as the amplitude of the V-peak of the line with the smaller Landé factor divided by the amplitude of the line with the larger Landé factor (see Table 1). No scaling with the ratio of g ea values was done. Note that for the Fe i kk5247, 5250 lines, the line ratio defined in this way is the opposite of the one commonly used in the literature. Some fraction of the Stokes V profiles has irregular shape with three or more lobes (Khomenko et al. 2005). In the case of the original resolution, these profiles are due to the gradients along the line of sight and represent cases of extreme asymmetry. We do not treat these profiles differently from the rest of the asymmetric profiles. The maps of the line ratio shown below confirm a smooth variation of the line ratio along the snapshot even if the profiles have irregular shapes. In the case of the reduced-resolution profiles, most of the irregular profiles remain below the threshold that we apply. In order to calibrate the line ratio we performed radiativetransfer calculations of the Stokes spectra in the Harvard-Smithsonian Reference Atmosphere (HSRA) model atmosphere, varying the constant vertical magnetic field strength. The calibration curves are given in Figure 1. According to many investigations, the parameter that most influences the calibration of the Fe i kk5247, 5250 line ratio is the macroturbulent velocity (Solanki et al. 1987; Steiner & Pizzo 1989). For the original profiles, the calibration with null V mic and V mac was applied, since the profiles in the three-dimensional snapshot were synthesized in this way ( Fig. 1, thin curves). For the profiles with reduced resolution, we set V mic ¼ 0:3 and V mac ¼ 0:7 kms 1 (Fig. 1, thick curves). These numbers are the average numbers derived from the inversions of the spectra with reduced resolution done with the SIR inversion code. Both V mic and V mac obtained after the inversion show a spatial variation over the simulation snapshot; however, the distribution is rather narrow. It is evident from the figure that the calibration curve of the Fe i kk5247, 5250 lines is the one that is most affected by the change of the micro- and macroturbulent velocities. The calibration curves for the infrared lines and the Fe i kk5247, 5250 lines (for the original resolution profiles) enter into the saturation regime after about 1000 G. We applied a limit of

3 1728 KHOMENKO & COLLADOS Vol. 659 Fig. 2. Longitudinal magnetic field component obtained from the line ratio of the Fe i kk6301 and 6302 lines (top left), Fe i kk5247 and 5250 lines (top right) and Fe i kk15652 and lines (bottom left). Bottom right panel is the map of the longitudinal magnetic field component at log 5 ¼ 1 in the 30 G flux snapshot from the simulations of Vögler et al. (2005). [See the electronic edition of the Journal for a color version of this figure.] 1200 G in these cases, i.e., larger fields cannot be measured using the line-ratio technique and the only conclusion that can be derived is that the field strength is above the given value. 4. LINE RATIO WITH FULL NUMERICAL RESOLUTION Figure 2 shows maps of the longitudinal magnetic field component obtained after the calibration of the line ratio for the different pairs of lines. The profiles are taken at their original numerical resolution. No noise is added and no threshold is applied to the profiles. The original longitudinal magnetic field from the simulation snapshot is presented in the bottom right panel. It corresponds to the level where log 5 ¼ 1 (where 5 is the optical depth at ). The locations with strong fields in the original snapshot correspond rather well to locations with the maximum field derived from the line ratio of the Fe i kk5247, 5250 and kk15648, line pairs. However, in most of the locations the field strength derived from the Fe i kk5247, 5250 lines stays lower than in the simulations and does not reach kg values. In contrast, the IR lines reflect the kg field concentrations rather well but seem to give a worse representation for the weak fields. There are a lot of 100Y200 G field signals in between the strong field concentrations, while the true field strength in the original snapshot is close to zero. In the case of the Fe i kk6301, 6302 lines, the field strength derived from the line ratio is largest not where the field is largest, but rather in the canopy regions surrounding the magnetic field concentrations and in the granular-intergranular borders. The correlation between the true field strength and the one derived from the line ratio of the Fe i kk6301, 6302 lines is absent. 5. THE REASON FOR THE INCORRECT LINE RATIO OF THE Fe i kk6301, 6302 LINES To find the reason for the wrong behavior of the Fe i kk6301, 6302 lines, we performed several test calculations. The results of these calculations are displayed in Figure 3. The V mic and V mac parameters were set to zero in the calculations described below. In the first test (Fig. 3, top left), we removed all the horizontal variations of the thermodynamic parameters from the original snapshot but preserved the original values of the magnetic field and velocity. In all the pixels, the thermodynamics was represented by the one taken from HSRA model atmosphere. Then the radiative-transfer calculations were repeated. Compared to the case displayed in Figure 2 (top left), the spatial distribution of the line ratio has become slightly more homogeneous in granular regions. However, it still shows no correlation with the original magnetic field distribution. Thus, despite the results of Martínez González et al. (2006), the temperature gradients change the line ratio of these lines only slightly and are not the mechanism primarily responsible for leading to the kg line ratios of these lines observed in the simulations. In the second test (Fig. 3, top right), we retained the original variations of the thermodynamic parameters. The field strength

4 No. 2, 2007 STOKES V AMPLITUDE RATIO 1729 Fig. 3. Longitudinal magnetic field component from the line ratio of the Fe i kk6301 and 6302 lines. Top left : All the pixels have the same thermodynamics. Top right: B is constant with height in all the pixels. Bottom left: B and V LOS are constant with height. Bottom right: B and V LOS are constant with height and all the pixels have the same thermodynamics. [See the electronic edition of the Journal for a color version of this figure.] and orientation were made constant, however, in the vertical direction, varying from pixel to pixel according to their values at log 5 ¼ 1 in the original snapshot. Now the map of the line ratio has changed significantly. The strongest values are located in intergranular lanes, similar to the original field distribution. However, the line-ratio map suggests kg fields in all intergranular lanes, while originally only some intergranules contained strong fields. In addition, the amplitude of the spatial variations of the line ratio remains too large. There are a lot of spurious signals of 300Y400 G that can be seen in between the strong field concentrations. In the next test (Fig. 3, bottom left), both LOS velocity and magnetic field strength were kept constant with height (again using their corresponding values at log 5 ¼ 1 at each individual pixel). The removal of velocity gradients has led to an increase of the correlation between the line ratio and the original field strength. Now, the strong line-ratio locations correspond rather well to the locations of enhanced magnetic field. Thus, the velocity gradients can produce a false apparent increase of the line ratio in intergranular lanes where the field is, in reality, weak. However, the spatial variations of the line ratio are still larger than those for the Fe i kk5247, 5250 and Fe i kk15648, lines (Fig. 2). In the last calculation (Fig. 3, bottom right), we removed the horizontal variations of the thermodynamic parameters, as well as the vertical variations of the velocity and field strength. Only in this case the line ratio of the Fe i kk6301, 6302 lines reflects the true magnetic field distribution well. Thus, the vertical gradients of the magnetic field and velocity and variations of the thermodynamic parameters along the snapshot produce a distortion of the line ratio of these lines in the way that appears in Figure RESPONSE FUNCTIONS AND REGIONS OF FORMATION The reason for such line-ratio behavior becomes clear in terms of the region of formation of the considered set of lines. According to non-lte (NLTE) radiative-transfer calculations by Shchukina & Trujillo Bueno (2001), the difference in the average heights of formation of the Fe i kk6301, 6302 lines can reach about 100 km (see also Trujillo Bueno et al. 2007). This difference is not so large in the case of the infrared lines and is close to zero in the case of Fe i kk5247, 5250 lines. According to Sánchez Almeida et al. (1996), a single height of formation cannot be ascribed to a spectral line, since it samples very different layers of the atmosphere. One can use, however, the response functions (RFs; see Ruiz Cobo & del Toro Iniesta 1994) to obtain information about the line formation region. Following this idea, we have calculated the RFs of the intensity profile to temperature in one horizontal slice of the three-dimensional MHD model for each considered pair of lines with the help of the SIR code. The results of these calculations are given in Figures 4 and 5. The slice corresponds to the position Y 3:2 Mm and crosses a strong magnetic field concentration located at X 3:6 Mm.

5 1730 KHOMENKO & COLLADOS Vol. 659 Fig. 4. Heights of formation of the pairs of lines (see definition in the text) along the slice of the simulations around Y ¼ 3:2 Mm. The background image is magnetic field strength. Solid lines: Lines with a smaller Landé factor Fe i kk15652, 5247, Dashed lines: Lines with a larger Landé factor Fe i kk15648, 5250, 6302 ( from top to bottom). Dotted lines are the continuum level at [See the electronic edition of the Journal for a color version of this figure.] The RFs are calculated as a function of optical depth 5 and wavelength. The translation from 5 to geometrical height is done afterward. This translation is straightforward in our case, since the heights are available from the simulation snapshot. Due to the presence of gradients of the physical quantities, the shape of the RFs and the position of their maxima vary strongly along the slice. In the case of the visible lines, the region of formation covers a large portion of the upper photosphere. Despite the rather large regions of formation, we can define an average height of formation by calculating the centroid of the RF at the wavelength of the line core intensity. Figure 4 gives the height of formation defined in this way for the three pairs of lines as a function of horizontal position along the slice plotted over the magnetic field map. In each panel, the dashed (solid) line corresponds to the spectral line with a larger (smaller) Landé factor. As expected, the infrared lines see deeper than the visible ones, and their heights of formation vary around 0 km, i.e., close to the continuum formation level at (see the dotted line in Fig. 4). The Fe i k15648 line (dashed line in the top panel of Fig. 4) forms slightly higher than the Fe i k15652 line, with the difference being small and rather constant over the slice. Inside the magnetic field concentration, around the position X 3:6 Mm, all the lines form deeper due to the partial evacuation of the magnetic flux tube caused by the presence of the strong kg magnetic field. The Fe i kk5247, 5250 lines heights offormation are nearly the same. A small difference is observed only inside the strong magnetic field concentration. These lines form rather high in the atmosphere. Unlike the other lines, the Fe i kk5247, 5250 pair shows a strong variation of the formation height along the slice from 100 to 500 km due to the variations of temperature caused by granulation. It should be noted that the information about the magnetic field extracted from these lines can correspond to very different atmospheric heights, depending on the structure. If the profiles are smeared by atmospheric seeing, the information from different Fig. 5. Height variation of the RFs (at the wavelength of the line core intensity) in a selected point inside a strong magnetic field concentration marked by a vertical line in Fig. 4. From top to bottom: Fei kk15652, pair; Fe i kk5247, 5250 pair; Fe i kk6301, 6302 pair. Dashed (solid) lines are for the spectral lines with larger (smaller) Landé factors in each pair. The vertical lines are the centroids of the RFs. The bottom two panels represent the height variations of B z and V LOS at the selected point. heights will be mixed up and will correspond to some average magnetic field over a large portion of the atmosphere. The Fe i kk6301, 6302 lines form around 300 km with smooth variations of the formation heights along the slice. The Fe i k6301 line forms about 100 km higher than the other one, in agreement with Shchukina & Trujillo Bueno (2001). Thus, these two lines see markedly different atmospheric layers. Results of magnetic field measurements based on line ratios using the Fe i kk6301, 6302 lines should be taken with care, even at the numerical spatial resolution of 20 km. For example, in canopy regions surrounding strong magnetic field concentrations, the magnetic field increases with height along the LOS (see the background image in Fig. 4). This leads to an increase of the amplitude of the Fe i k6301 (with smaller Landé factor, but forming higher) relative to the Fe i k6302 line and produces a line-ratio value close to unity, which can be interpreted as a presence of kg fields, while in reality the field is sub-kg. This explains the presence of kg lineratio signals in the canopy regions in Figure 2 (top left).

6 No. 2, 2007 STOKES V AMPLITUDE RATIO 1731 Figure 5 gives an example of the height variation of the RFs (at the wavelength of the line core intensity) at the horizontal position inside the strong magnetic field concentration, marked by vertical lines in Figure 4. The same as before, dashed lines correspond to spectral lines with larger Landé factors and solid lines to those with smaller ones. In the top three panels, the vertical lines mark the position of the centroid of the RF. The bottom two panels give the magnetic field and LOS velocity as functions of height at the given point in the atmosphere. According to Figure 5, the RFs of the infrared lines (top) are rather narrow. This is a very useful property, since information about the magnetic field comes from a small portion of the solar atmosphere. The magnetic field does not change much within the formation region. At the same time, strong LOS velocity gradients are typical at these heights (see bottom panel of the figure). The velocity can change several km s 1 within the narrow formation region of the infrared lines. Due to the velocity gradients, the contributions to the absorption profiles are shifted in wavelength, producing asymmetric spectra. Potentially, this can lead to a distortion of the line ratio, leading to incorrect field strength even in this case. However, the direct calculations performed above show that, despite these gradients, the line ratio of the infrared lines gives rather satisfactory results. The situation is more complicated in the case of the visible lines. The formation regions of the Fe i kk5247 and 5250 lines are nearly the same, but the significant contribution comes from a very large portion of the solar atmosphere. Thus, the Stokes V ratio of these lines corresponds to some average value of the magnetic field strength over a wide formation region. The maximum of the contribution appears to come from upper layers, where the magnetic field is already small even inside the kg flux tube. For the Fe i kk6301, 6302 lines, the formation regions are also very extended. In addition, the shapes of the RFs are rather different. This explains why the gradients in physical quantities affect these lines in a different way, leading to Stokes V amplitude ratios that do not carry information about the magnetic field strength. An effective field strength measured by each of the lines may be estimated with the help of the RFs as a weighted mean over the formation region (Ruiz Cobo & del Toro Iniesta 1994): R RT ()B()d B ea ¼ R ; ð1þ RT ()d where R T () is the RF of the intensity to temperature taken at a given wavelength. The B ea values calculated from this equation are given in Figure 6 as a function of horizontal position along the slice. According to this estimation, in the weak-field case the lines of each pair see nearly the same field strength. The gradients of the magnetic field strength are not very large in the weakfield case. Integration over a large formation region results in average numbers that are not very different, even though the formation regions are not exactly the same, such as, for example, in the case of the Fe i kk6301, 6302 lines. In the strong-field case, the effective field strength obtained from equation (1) for the Fe i k6301 line is slightly larger in the canopies and smaller inside the flux tube, compared to that of the Fe i k6302 line. According to equation (1), the B ea value measured by the Fe i 6301 line always remains below kg levels. The field strength obtained from the line ratio of these lines does not agree with the one estimated from equation (1) because the gradients in the remaining physical quantities affect both lines in a different way. The latter does not happen for the Fe i kk5247, 5250 and the infrared lines. There, the field strength obtained from equation (1) Fig. 6. Effective magnetic field strength defined by eq. (1) as a function of the horizontal position along the slice. From top to bottom: Fe i kk15652, pair; Fe i kk5247, 5250 pair; Fe i kk6301, 6302 pair. Dashed (solid) lines are for the spectral lines with the larger (smaller) Landé factor in each pair. Symbols are the field strength obtained from the line ratio. and from the line ratio are similar. B ea does not reach kg values for the Fe i kk5247, 5250 lines, since these lines have maximum contribution at high atmospheric levels. This explains the absence of kg values in Figure 2 (top right)comparedtothemhd model at log 5 ¼ 1. Since the infrared lines measure deeper, the field strength obtained from them is larger. It reaches about 1.5 kg inside the strong magnetic field concentration and remains within 100Y200 G in the surroundings. Due to the saturation of the calibrationcurve(fig.1),wewereunabletomeasurethefield strength above 1200 G. Thus, the field strength from the line ratio given in Figure 6 remains smaller than the B ea. This, however, does not mean any limitation for the IR lines, since the field strength in the strong-field regime can be derived directly from the Stokes V splitting. 7. LINE RATIO WITH REDUCED SPATIAL RESOLUTION Figure 7 gives the magnetic field strength obtained from the calibration of the line ratio applied to the profiles at reduced resolution (see Fig. 2). Since now we aim at simulating observational conditions, only profiles with amplitudes above a threshold of

7 1732 KHOMENKO & COLLADOS Vol. 659 Fig. 7. Same as Fig. 2, but with reduced spatial resolution. [See the electronic edition of the Journal for a color version of this figure.] 5 ; 10 3 in units of I C were used. This threshold is somewhat higher than normally applied in observations (about 1 ; 10 3 ). This high threshold is due to the larger Stokes V amplitudes of the 30 G MHD snapshot than actually observed (see the comparison of the amplitudes between observations and MHD simulations in Khomenko et al. 2005). In observations, some 30%Y50% of the signals remain above threshold. Our intention is to maintain this fraction rather than the exact threshold level. To help the comparison, the bottom right panel of Figure 7 gives the map of the magnetic field strength from MHD simulations at log 5 ¼ 1, binned 4 ; 4 pixels to have the same pixel resolution of 80 km as the smoothed spectra. As expected, the profiles with the largest field strength are situated, in all cases, at pixels surrounding the strong magnetic field concentrations. The patches with enhanced line ratios are much bigger than the actual field concentrations. Note that these patches are much smaller in the infrared than in the visible. Possibly, this happens due to the presence of canopies, since visible lines form higher in the atmosphere. The best correlation between the actual magnetic field strength and that obtained from the line ratio is observed for the infrared lines (bottom left). There, the patches are narrow and have a central part with kg fields surrounded by weaker field regions. It is surprising that a simple line ratio of the infrared lines gives such satisfactory results, even for complicated field structures as those in the simulations. This good correlation is due to the narrow formation regions of these lines, which in addition, are nearly the same. The field strength obtained from the Fe i kk5247, 5250 line ratio does not reach kg values at any point in the simulation snapshot, staying within 600Y800 G at those pixels corresponding to kg fields in the original snapshot. According to x 6, such a low field strength is due to the larger formation height of these lines. In addition, since the formation regions are very extended and a large difference in formation heights exists between the different points (depending on their temperature; see Fig. 4), the smeared profiles carry information about a large portion of the solar atmosphere, both in vertical and horizontal directions. Thus, the field strength extracted from the line ratio of the Fe i kk5247, 5250 lines corresponds to some average value and stays below kg. The line ratio of the smoothed Fe i kk6301y6302 profiles corresponds to kg fields in many pixels of the simulation box. Should this line-ratio value have been obtained after observational data, one would have been tempted to erroneously infer that the whole observed area contains strong fields, which is not the case in our example. Thus, we conclude that, due to the large difference in their formation heights, the Fe i kk6301y6302 Stokes V ratio does not carry information about the field strength under either original or reduced resolution. A similar conclusion for the spectra with reduced resolution, was reached previously by Martínez González et al. (2006), based on different arguments that we will discuss below. 8. LINE RATIO IN THE IR FROM OBSERVATIONS According to the above results, the line ratio of the infrared lines should be a good indicator of the field strength, even under

8 No. 2, 2007 STOKES V AMPLITUDE RATIO 1733 Fig. 8. Line ratio in IR from observations (top) compared to the map of the longitudinal magnetic field derived from splitting of the IR lines (bottom). [See the electronic edition of the Journal for a color version of this figure.] reduced resolution. This conclusion was obtained for the ideal profiles from the MHD simulations that do not include noise or any other observational effect that might affect the quality of the spectropolarimetric data. In order to confirm this result, we have used spectra of the full Stokes vector recorded on 2000 July 29, using the Tenerife Infrared Polarimeter ( TIP; see Martínez Pillet et al. 1999) attached to the echelle spectrograph of the 70 cm German Vacuum Tower Telescope operating at the Spanish Observatorio del Teide ( Tenerife) of the Instituto de Astrofísica de Canarias. The same observations were used by Khomenko et al. (2003). Noise in the data has been reduced by a filtering process based on the principal component analysis method (see M. J. Martínez González et al. 2007, in preparation). After filtering, the noise at the continuum wavelength was 4 ; For the analysis, only profiles with a double-lobe shape were selected, with amplitudes above a threshold of 10 3 in units of I C. A parabolic fit was applied to the Stokes V profiles in order to calculate the amplitudes of the lobes. The line ratio of Fe i k15652 to k15648 is given in the top panel of Figure 8. We did not perform any calibration and the line-ratio scales from about 0.3 (weak field) up to 1.0 (strong field). For comparison, the bottom panel gives the longitudinal magnetic field component. It was measured from the splitting of Stokes V profiles applying a Gaussian fit (in order to derive the magnetic field strength from the positions of the -components) and the amplitude ratio between the amplitudes of Stokes Q, U, and V profiles (in order to obtain the magnetic field inclination), as discussed in Khomenko et al. (2003). According to this paper, the inclination of the magnetic field in the observed area is close to the vertical in most of the pixels (see their Fig. 10) and the magnetic field strength stays below 800 G in most of the field of view. For fields of such strength, the Stokes V amplitude ratio does not enter into the saturation regime. The amplitude of the Fe i k15648 line (with the larger Landé factor) already becomes independent of the magnetic field for fields as low as 500 G. However, the amplitude of the other Fe i k15652 line (with the smaller Landé factor) continues increasing with B. Thus, as follows from the calibration curves for the IR line ratio displayed in Figure 1, the saturation is not produced before some 800 G. Because of the above arguments, we expect that for fields of intermediate strength both methods should give similar results. Indeed, the comparison of both panels shows that there is a good correlation between the maps. The kg field patches correspond to the line ratio close to 0.8Y1, while weaker fields give a smaller line ratio. 9. DISCUSSION The purpose of this paper is to verify the common assumption that the Stokes Vamplitude ratio of the Fe i kk15648, 15652; Fe i kk5247, 5250; and Fe i kk6301, 6302 pairs of lines is directly related to the magnetic field strength contained in the resolution element of observations. The answer to this question is positive in the case of the Fe i kk15648, and Fe i kk5247, 5250 lines, whose formation regions are nearly the same for the lines of each pair. However, a somewhat surprising result is obtained for the Fe i kk5247, 5250 lines, since these are unable to reveal kg fields in internetwork. The line ratio of these lines, calculated in the simulation snapshot with 30 G average field strength and containing kg field concentrations, corresponds to sub-kg fields in all the cases. It is shown that this low field strength is due to the large formation height of the Fe i kk5247, 5250 lines. The maximum contribution to the observed profile comes from high layers, where the field strength is already below kg. At the same time, since Stenflo (1973) many observations have revealed kg line ratios using the Fe i kk5247, 5250 lines (Stenflo & Harvey 1985; Solanki et al. 1987; Sánchez Almeida et al. 1988; Keller et al. 1990b; Grossmann-Doerth et al. 1996), unlike the results presented above. However, it should be noted that all measurements with kg line ratios correspond to the solar network. There, the magnetic field concentrations are expected to be stronger. On one hand, this implies a larger field strength. On the other hand, the partial evacuation should be more pronounced, thus allowing the spectral lines to see deeper in layers where the field strength is still kg. In order to check whether the kg Fe i kk5247, 5250 line ratio can be obtained from simulations, we repeated the calculations given above for a simulation snapshot with an average field strength of 200 G (see Vögleretal. 2005). Figure 9 gives the magnetic field strength obtained from the line-ratio calibration for the profiles with original (left) and reduced resolution (middle) in the 200 G simulation snapshot. For comparison, the right panel represents the original magnetic field in MHD simulations at log 5 ¼ 1. Now, the strong kg field concentrations located in intergranular lanes in this simulation correspond to kg line ratio, both at the original and reduced resolution. It confirms that the presence of kg line ratios of the Fe i kk5247, 5250 lines in observations is closely related to the average magnetic flux in the observed area. According to our results, the treatment of gradients is crucial while inverting of the Fe i kk5247, 5250 lines. Since the region of formation of these lines covers almost the whole photosphere, a Milne-Eddington inversion (assuming a constant magnetic field) would reveal some average magnetic field strength, which would likely remain below kg levels. Allowing for gradients, the magnetic field strength should increase in the deeper layers. In general, strong lines forming through all the photosphere are not good for the inversion since contributions from the very different layers are mixed up. In the case of the Fe i kk5247, 5250 lines, not only is vertical mixing present, but the information from different heights is also overlapped due to the nearly 400 km difference in the formation heights depending on the temperature of the observed structure (see Fig. 4). Unlike the Fe i kk5247, 5250 and the infrared lines, the Stokes V amplitude ratio of the Fe i kk6301, 6302 pair turns out not to carry information about the magnetic field strength, even at the

9 1734 KHOMENKO & COLLADOS Vol. 659 Fig. 9. Magnetic field strength from the line ratio of the Fe i kk5247 and 5250 lines for the simulation snapshot with 200 G average magnetic field strength. Left: Full resolution. Middle : Reduced resolution. Right: Original magnetic field at log 5 ¼ 1 binned 4 ; 4 pixels. [See the electronic edition of the Journal for a color version of this figure.] numerical spatial resolution of 20 km. Under the reduced resolution, most of the pixels in the simulation reveal a line ratio corresponding to kg levels, while in reality the field strength can vary in a very wide range. Possibly, this could explain the large Stokes Vamplitude ratios observed recently by López Ariste et al. (2006) simultaneously with the Mn i lines. In their observations, in most of the locations the amplitude ratio of the Fe i kk6301, 6302 lines suggests kg fields, while the Mn i lines reveal weak fields. Thus, the amplitude ratio of the Fe i kk6301, 6302 lines cannot be considered a reliable diagnostic method for the field strength. More sophisticated methods should be applied, such as inversion techniques that allow fitting the line asymmetry. The conclusion that the Stokes V amplitude ratio of the Fe i kk6301, 6302 lines does not carry information about the magnetic field was previously reached by Martínez González et al. (2006). These authors argue that by changing the temperature gradient of the model atmosphere during the inversion, the amplitude ratio can reach values corresponding to any field strength from sub-kg to kg. In our work, we reach the conclusion that the temperature variations along the simulation snapshot affect the line ratio to a small extent, and the effects of the LOS magnetic field and velocity gradients are significantly larger. These results can be reconciled if one recalls that in the inversion performed in Martínez González et al. (2006) the LOS velocity and magnetic field were forced to be constant with height. Taking into account the difference in the formation heights of the Fe i kk6301, 6302 lines, the inversion code modifies the temperature gradient (as the only available parameter) in order to fit the line ratio. Since the Stokes V amplitude ratio of these lines can be modified by a number of different reasonable physical scenarios, the weak and noisy signals observed in the internetwork might not contain enough information to constrain the magnetic field strength. However, when in the future better resolution and signal-to-noise ratio are achieved in observations, fitting the line asymmetries could help in deriving a correct magnetic field strength from the Fe i kk6301, 6302 lines. 10. CONCLUSIONS Calculations of the Stokes Vamplitude ratio for complex magnetic fields generated by MHD simulations have been performed. We conclude that the Stokes V amplitude ratio of the Fe i kk15648y15652 line pair gives a reasonable estimate of the field strength even for fields with complex internal structure, such as those from MHD simulations. This good correspondence is due to the narrow formation regions of these lines, which are nearly the same. The Fe i kk5247y5250 lines also have very similar formation regions. However, these lines obtain significant contributions through the whole photosphere, being maximum at the upper layers close to the temperature minimum. The Stokes Vamplitude ratio of these lines gives an average value of the field strength over a large portion of the solar atmosphere, both in vertical and horizontal directions. If we assume that the simulations give a correct representation of the magnetic field in solar internetwork regions, kg line-ratio values are unlikely to be obtained from observations of the Fe i kk5247y5250 line pair. In contrast, the calculations of the Stokes V amplitude ratio of the Fe i kk6301 and 6302 lines for the complex magnetic fields of MHD simulations show that this is not a good parameter for the indication of kg magnetic field concentrations. The line ratio is affected by the vertical gradients of the magnetic field, velocity, and thermodynamic parameters due to the large difference in the heights of formation of the Fe i kk6301 and 6302 lines. The lineratio method cannot be applied for the interpretation of the of the Fe i kk6301 and 6302 spectra in order to derive the magnetic field strength. Such an analysis would lead to an excess of kg values. The treatment of the gradients is thus crucial in the analysis of these lines. The Fe i kk6301 and 6302 lines still contain information about the magnetic field strength via the line asymmetries, which should be taken into account in the interpretation of observations of the internetwork made with this line pair. 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