We discuss the results for the different line parameters and diagnostics individually. The infrared line is discussed separately in Sect. 3.7.
3.1. Profile shapes
The spectral shapes of Stokes Q, U and V resulting from multi-ray calculations as described in Sect. 2.2, can deviate significantly from profiles calculated in a plane parallel (i.e. 1-D) atmosphere. At these differences are minute and the Stokes V profiles are almost indistinguishable while at their shapes are totally different, with the multi-ray profile having narrow, widely separated lobes and often an inversion at the line core. Examples of multi-ray V profiles at , unbroadened by macroturbulence, are shown in Fig. 2a.
The V lobe separation of the multi-ray profiles, if taken at face value to represent Zeeman splitting, suggests a larger B at than at . Such a hypothetical observation could thus be misconstrued to indicate a field strength increasing with height, while exactly the opposite is the case in the underlying model.
To understand the multi-ray profile shape we first note that the inversion in the centre of the V profile is not due to magneto-optical effects. We confirmed this with calculations that had magneto-optical effects switched off. Next we consider the transfer equations for a Zeeman split spectral line in LTE,
Here is the Stokes vector at frequency , and are line absorption coefficients relative to the continuum absorption coefficient, and are the corresponding magneto-optical coefficients (which in general are considerably smaller than , , etc.), is the Planck function and is the continuum optical depth along the line of sight. Expressions for and are given by, e.g., Landi Degl'Innocenti (1976). Along most rays in the geometry considered here, light enters the flux tube from the field-free atmosphere, in which . For simplicity we assume that the magnetic filling factor of the unresolved flux-tubes is sufficiently small, so that always . This condition is fulfilled by most observations in the quiet sun and in solar plages ( amplitudes typically do not exceed a few percent). For large enough it is fulfilled along each ray individually, since then a large fraction of it passes through field-free atmosphere. In addition, for lines of sufficient strength, such as the ones we consider, . In this case Eqs. (1)-(4) can be simplified to
where and V, in turn. For a thin flux tube and a sufficiently large we can write as solution of Eq. (6):
(We have made use of the boundary condition at the first boundary of the flux tube pierced by the ray.) is the continuum optical depth step across the flux tube. The sign of depends on , i.e. whether the temperature decreases along the ray towards the observer or not. A temperature inversion, i.e. a flux tube that is hotter than its surroundings, gives and leads to an inversion in .
Our model flux tube is hotter than the non-magnetic atmosphere at equal geometrical height in the middle and upper photosphere. This is in agreement with stationary theoretical models incorporating radiative energy transport (e.g. Grossmann-Doerth et al. 1989; Steiner 1990; Steiner & Stenflo 1990; Knölker et al. 1990; Knölker & Schüssler 1992) and empirical models (e.g. Keller et al. 1990; Bruls & Solanki 1993; Briand & Solanki 1995). Within the flux tubes the spectral line cores, which are formed in these layers, thus sample a region in which , which drives them into emission. If the magnetic filling factor is sufficiently small, however, Stokes I gets its major contribution outside the flux tubes, where , so that a pure absorption profile results, as shown in Fig 2c. Stokes , on the other hand, obtain their constructive contributions only within the flux tube, with in the upper layers, which leads to the inversions seen in Figs. 2a and b. In summary, at small µ the sign of in the line core depends on the temperature of the flux tube relative to its surroundings for all but the weakest lines.
In either of the considered geometries another effect, first pointed out by Audic (1991), also plays an important role. The polarized signal in the line core is greatly reduced due to absorption in the gas between the flux tube (or the magnetic central layer) and the observer. There, the transfer equations for polarized light are simply:
The solution to Eq. (9) reads
Inserting from Eq. (7) we obtain
where and are the intensity and source function referring to the portion of the ray within the flux tube.
This effect tends to make Q, U, V largest in the line wing (where is small) and gives them flat cores (where is large), i.e. it decreases the - and (to a lesser extent) the -peaks and moves the -maxima away from the line core (Audic 1991). Again, this effect strongly depends on the thermal structure of the non-magnetic atmosphere, through the temperature dependence of (e.g. the of low excitation Fe I lines increases with decreasing temperature). Both effects, the inversion and the absorption at the line core increase the Stokes V lobe separation and therefore lead to an unexpected increase of that measure with decreasing µ.
3.2. Shapes of turbulence-broadened profiles
Consider briefly the influence of a macroturbulent velocity on the Stokes profiles. Such a source of line broadening is generally employed to reproduce the widths and shapes of Stokes profiles (e.g. Solanki 1986; Bünte et al. 1993). If we broaden the calculated profiles by 2 km s-1, a value typically required to reproduce observed profiles, then the Stokes profiles of Fig. 2 are transformed into the corresponding profiles plotted in Fig. 3. Note that only the extreme profile produced by model G still shows a significant central inversion after broadening. The Stokes V profiles show large variations from one model to the next, whereas the Q profiles of all the models are very similar. Consequently, the variation in flux tube diameter is only distinguishable in Stokes V, if at all.
It has recently been proposed by Sánchez Almeida et al. (1996) that magnetic elements with sizes even smaller than model G exist (so called MISMAs). Fig. 3a suggests that if these are significantly hotter than their surroundings at the solar limb they should produce V profiles with strong inversions in their cores. The Stokes V observations by Stenflo et al. (1987) do not exhibit any significant inversions, suggesting that either the magnetic elements have diameters larger than approximately 50 km or are at roughly the same temperature as the ambient medium. However, a much larger set of observations close to the limb is required to settle this point.
3.3. Stokes ratio and the magnetic inclination angle
The ratio , usually taken at a fixed wavelength, is a measure of the inclination angle of the magnetic vector relative to the line-of-sight. In the past has been determined from a plane-parallel, 1-D model with either a height dependent or homogeneous magnetic field (e.g. Ronan et al. 1987; Lites & Skumanich 1990; Lites et al. 1993; Bernasconi et al. 1995; Martínez Pillet et al. 1997). Here we test the accuracy of this approximation by comparing the results of a plane-parallel model with our multi-ray calculations. We note the value of produced for a given 2-D model and then search for the angle that gives the same ratio, using the plane-parallel 1-D model PLA. Here and signify the Q and V -amplitudes, respectively. Our tests are restricted in the sense that we consider only vertical 2-D flux tubes, but we nevertheless test at different values. Since the azimuthal angle , Stokes U is not required for this test (although, due to magneto-optical effects, it is generally non-zero).
The estimated error in introduced through approximating a flux tube by a plane-parallel atmosphere is on average only , with the largest error found by us being . We also find that for all models and spectral lines the errors are largest either closest to the limb , or closest to disc centre .
In summary, for the determination of the magnetic inclination only, it is quite sufficient to employ a plane parallel model for spatially unresolved observations of a rotationally symmetric magnetic flux tube. Note that, however, if either the spatial resolution element of the spectroscopic observation is much smaller than the flux tube diameter, or the magnetic flux is concentrated into a sheet-like structure (magnetic flux sheet) then the ratio is non-vanishing even at disc centre for a perfectly vertical, symmetric structure, thereby falsely suggesting a non-vanishing inclination.
3.4. Centre-to-limb variation of the Stokes V amplitude
For an optically thin spectral line formed in a homogeneous, vertical magnetic field the Stokes V amplitude scales with and thus to first order with µ. In a plane-parallel model of a flux tube (with the magnetic field decreasing with height as in our model PLA) there are two effects acting to produce departures from this behaviour. On the one hand the decrease of B with z lowers near the limb for incompletely split lines like 5247.1 Å and 5250.2 Å since they are formed higher in the atmosphere at small µ. On the other hand, due to the thermal structure of, e.g., the plage flux-tube model, these temperature sensitive lines, and thus their , are considerably enhanced with decreasing µ. The numerical results for 5250.2 Å, plotted as the dotted curve in Fig. 4a, show that for a plane-parallel model the latter effect is the dominant one (the dotted curve varies less rapidly than µ). The curves for the multi-ray models lie below the plane-parallel results for sufficiently small µ. Also, they strongly differ from each other at intermediate µ.
Van Ballegooijen (1985) first proposed that when observing at an angle to the axis of an isolated thin flux tube the V amplitude should be much lower than in the plane-parallel case. In his calculations Stokes V at a fixed wavelength drops by a factor of over 10 at for a vertical flux tube with a diameter of 15 km. We qualitatively confirm his result, but find a much gentler drop in with decreasing µ than he did. This is the case even for our model G with km, comparable with the of 7.5 km for Van Ballegooijen's model.
At large µ one important difference between his and our models is that our flux tubes expand with height, whereas he considered straight flux sheets. At small µ, however, the dominant reason for the discrepancy between Van Ballegooijen's and our results is that he considered only a single flux tube and not a whole array. In order to understand that, let us consider a bundle of rays equally distributed over a period of the flux tube array as shown in Fig. 1. Then, with increasing inclination (decreasing µ) an increasing number of rays passes at some height in the atmosphere through one or more flux tubes, so that, loosely speaking, the `effective' magnetic filling factor increases with inclination relative to the case of a single flux tube, although it does not increase in an absolute sense. We expect that at least in active region plages and the enhanced network a periodic collection of flux tubes, like we model here, comes closer to reality than in a single, isolated flux tube. Consequently, although the effect pointed out by Van Ballegooijen (1985) is present, it is expected to be of smaller magnitude in active regions and the network than his calculations suggested. The response of isolated flux tubes may well be more in accordance with the calculations of Van Ballegooijen (1985).
We attribute the enhanced of model E at relative to the plane-parallel case to the hot wall present below the external level in the 2-D MHS models, which becomes increasingly visible as one moves away from disc centre (e.g. Spruit 1976; Knölker & Schüssler 1988). Due to the large continuum intensity resulting from the hot wall the V profiles formed along rays piercing it are greatly enhanced in amplitude. On the other hand, below a certain µ rays piercing the flux tube wall mainly lie outside the flux tube, so that Stokes V is small along such rays (partly due to the effects described in Sect. 3.1.). Thus, we expect that there is an intermediate µ value at which the visibility of the wall is largest, in which case the rays passing through the hot wall just manage to stay within the flux tube.
In this picture the maximum hot-wall-induced enhancement happens at increasingly smaller µ for increasingly large flux tubes. A comparison of Fig. 1a with 1b shows that whereas all the rays passing through the hot wall in the thick flux tube lie within the flux tube for all , this is patently not the case for the thinner flux tube, although µ is the same in both cases. Hence we expect the enhancement of the V profiles due to the hot walls of the thinner flux tubes to be largest at , which µ region is unfortunately not resolved by our µ grid.
The application of a macroturbulence reduces relative to for all models. The , normalized to its disc centre value, is plotted in Fig. 4b for profiles broadened by a macroturbulence of 2 km s-1. The effect is larger for narrower flux tubes. Near the limb all of our 2-D models produce that are approximately a factor of 2 lower than the simple estimate. This effect should be kept in mind when determining fluxes from, e.g., full-disc magnetograms.
In Fig. 5 we plot of 5247.1 Å vs. µ for the same models as in Fig. 4a. Interestingly, (5247.1 Å) decreases more rapidly with decreasing µ than Å) for all models. The only difference between the two lines is their differing Landé factors. The larger Zeeman sensitivity of 5250.2 Å relative to 5247.1 Å leads to a greater separation between its -components. Consequently, the -components are less affected by absorption in the field-free part of the atmosphere and by the temperature enhancement in the flux tube (Sect. 3.1.), compared to the 5247.1 Å line. This weakens (5247.1 Å) relative to (5250.2 Å) at small µ, leading to significant and surprising consequences for the ratio between the V profiles of these two lines (see Sect. 3.5.). The above explanation for the difference between (5250) and (5247) vs. µ is confirmed by the behaviour of the extremely Zeeman sensitive line at 15648 Å (see Sect. 3.7.).
3.5. Magnetic line ratio
is widely accepted as one of the most powerful diagnostics of the intrinsic magnetic field strength of solar magnetic elements (Stenflo 1973; Stenflo & Harvey 1985) and is sometimes referred to simply as the magnetic line ratio, or MLR for short. For a simple model with a homogeneous magnetic field, as used traditionally to calibrate the MLR, we have MLR for a sufficiently weak field and MLR for stronger fields.
This diagnostic is well studied and understood for longitudinal fields and flux tubes located at . Its centre-to-limb variation, however, has been studied only for plane-parallel model atmospheres. These showed that for fields inclined to the line of sight the blending and saturation of the and components becomes important (Solanki et al. 1987), causing the line ratio to increase with increasing strength of the component. For vertical flux tubes this implies that the MLR increases towards the limb. An illustration of this effect is provided by the dotted curve in Fig. 6a, which shows the MLR as a function of µ calculated in the plane-parallel PLA atmosphere. A part of its increase towards the solar limb is also due to the decrease of B with height. The four other curves represent the MLR resulting from multi-ray calculations through models B, C, D and E. At small µ all multi-ray models give MLR values greater than unity. 2 Even slightly off disc centre at ) one may obtain a MLR of unity which could be misinterpreted as the measurement of an intrinsically weak magnetic field.
MLR values can be explained using the results of Sects. 3.1. and 3.4. The saturated line core in the field-free part of the atmosphere weakens the V profile in the core of a sufficiently strong line. As already discussed in Sect. 3.4., the V lobes of a more strongly Zeeman split line (e.g. 5250.2 Å) lie further from the line core than the V lobes of a less Zeeman split line (e.g. 5247.1 Å) and are consequently less weakened by transfer effects in the non-magnetic atmosphere. The MLR is thereby increased and exceeds unity in this case.
In order to test this explanation we consider four different, plane-parallel atmospheres with a vertical magnetic field confined to a specific optical depth range only, i.e., a 3-layered model as described in Sect. 3.1:
In addition, the temperature stratification in the range (i.e. in the central layer) may correspond to a different model atmosphere than that outside this optical depth range. The chosen model combination and the corresponding computed MLR for two different angles between the line of sight and the magnetic field are given in Table 2.
Table 2. MLR for various plane-parallel models
As far as line transfer is concerned such an atmosphere composed of 3 horizontal layers, with the (thin) central layer possessing a magnetic field (i.e. it corresponds to the flux tube) and the top and bottom layers being field-free (corresponding to its surroundings), is formally the same as a vertical (thin) flux tube pierced by an isolated inclined ray. In this simplified geometry we need only consider a vertical ray. In the 3-layer picture a hot flux tube embedded in cool surroundings produces a negative temperature gradient () at the lower boundary of the central layer, while the gradient remains positive at all other heights. Using Eqs. (5) and (7) it is then straightforward to deduce that for a sufficiently thin central layer Stokes I does not show an inversion in its core, whereas Stokes Q, U and V do.
Table 2 shows that the MLR is larger for a cooler atmosphere, i.e. for larger line saturation (test No. 2), as compared to the case of a hot atmosphere (test No. 1). Furthermore, it is largest when the magnetic portion is hot and the field-free layers are cool (test No. 3) due to Stokes V absorption in the field-free gas and the inversion of Stokes V in the line core (see Sect. 3.1). Note also that the MLR increases with . This effect is largest for a low temperature and correspondingly high saturation within the magnetic layer (tests 2 and 4) and it is largely due to blending of the - and -components.
Finally, the MLR becomes even larger if the line profiles are broadened by a macroturbulent velocity (Fig. 6b). In summary, for small magnetic flux tubes the MLR is not a reliable diagnostic for B too far outside solar disc centre, since it becomes excessively sensitive to the temperature, flux-tube size and turbulent velocity. At present it is not clear from observations whether MLR values greater than unity are present in solar data or not. The few MLR values determined by Stenflo et al. (1987) near the limb all lie below unity, but a larger number of observations are required for a definitive answer.
3.6. Centre-of-gravity method
The centre-of-gravity method (or c.g. method) for the longitudinal magnetic flux or filling factor was introduced by Semel (l967). The centre-of-gravity wavelengths of the positively and negatively polarized components of a spectral line are defined as:
According to the c.g. method the spatially averaged longitudinal magnetic field strength (in Gauss) is related to in a straightforward manner:
with the wavelengths given in Å. Here is the line centre wavelength and is the effective Landé factor.
For an optically thin line Eq. (14) can be shown to be exact for any µ or value. In plane-parallel atmospheres the accuracy is still high even for optically thick lines (errors 10%, Semel 1971) and the c.g. technique shows distinct advantages over other techniques for determining (e.g. Rees & Semel 1979; Grossmann-Doerth et al. 1987; Cauzzi et al. 1993).
This section may be considered to be an extension of the work of Rees & Semel (1979), who based their conclusions on 1.5-D radiative transfer through a vertical flux tube at . In Fig. 7a we plot derived from Eq. (14), as applied to Fe I 5250.2 Å (the plotted is normalized to its value at ). The dotted curve representing the plane-parallel PLA model lies very close to the expected linear dependence on µ, confirming the superiority of the c.g. method over the Stokes V amplitude in this type of model (compare with the dotted curve in Fig. 4a). Unfortunately, the curves obtained from the 2-D models using the c.g. method are less satisfactory. Thus, for flux tubes is underestimated by up to a factor of 2 near the limb when diagnosed by the centre-of-gravity method in conjunction with a plane-parallel model. Also, Fe I 5247.1 Å (Fig. 7b) gives consistently lower values near the limb than 5250.2 Å.
The reason for the difference between the resulting from plane-parallel and 1.5-D calculations lies once again in transfer effects introduced by the field-free medium into the polarized profiles. Note that the V profiles produced by the 1.5-D calculations not only have lower amplitudes, but also narrower V lobes, which reduces their additionally. The c.g. method may thus be an excellent technique for determining at , but its reliability is reduced at smaller µ for thin flux tubes. Inspite of these problems, the c.g. method still provides more reliable results than the V amplitude or magnetographic observations. Another advantage of the c.g. method lies therein that it is insensitive to macroturbulence or instrumental broadening. Thus Fig. 7 is also valid for broadened profiles.
3.7. The 15648 Å line
Fig. 8 shows V profiles of Fe I 15648 Å for models B, D, E and PLA at (Fig. 8a) and (Fig. 8b). Profiles due to the remaining models (C and G) are rather similar to those plotted and are not shown for clarity. The splitting at is 20-25% smaller than at , due to larger formation height near the limb, together with the vertical gradient of the field. The decrease of 15-25% between and 0.4 observed by Stenflo et al. (1987) is in rough agreement with the corresponding calculated decrease of 15-20% between and 0.4, which our models give.
According to Fig. 8 the Stokes V peak separation is almost independent of the flux-tube radius, , unlike the findings of Zayer et al. (1989). We traced the difference to an insufficient resolution of the flux tube boundary for the integration of the radiative transfer equation in the code employed by Zayer et al. (1989), which becomes most important for small flux tube radii. In the present work we have remedied the problem by using an adaptive mesh for that integration, as explained in Sect. 2.2.
We see little influence of the field-free atmosphere on the profile shapes of the 15648 Å line, in contrast to Fe I 5250.2 Å and 5247.1 Å; the inversion and flat portion in the cores of the visible Stokes profiles are absent in the infrared line (compare Figs. 2 and 3 with Fig. 8). This is partly due to the low formation height of the line. At this height the flux tube is cooler than its surroundings (i.e. in the flux tube, so that no temperature inversion is produced). Furthermore, the 15648 Å line is relatively weak, unsaturated and temperature insensitive, all of which substantially reduces the absorption (and saturation) in the field-free part of the atmosphere. Finally, due to this line's large Zeeman sensitivity, its -components peak outside the wavelength range within which it absorbs in the field-free atmosphere. The curves for the 2-D models (not shown) deviate correspondingly less from the plane-parallel case, compared to the visible lines. The counterparts of Figs. 4 and 7a for Fe I 15648 Å therefore also show correspondingly little scatter from one model to the next.
© European Southern Observatory (ESO) 1998
Online publication: April 20, 1998