3. Observational material and data reduction procedure
All the observations used here have been obtained with the ZIMPOL I polarimeter (cf. Povel 1995; Stenflo & Keller 1997) at a disk position of (corresponding to a limb distance of 5 arcsec). Three spectral regions that contained a particularly high concentration of polarization structures due to atomic (rather than molecular) transitions were selected: 4884-4888 Å , 4931-4936 Å , and 5682-5687 Å . In these three wavelength ranges 20 polarization features were identified as potentially useful candidates for Hanle diagnostics.
The recordings of these lines were done in 8 different solar regions (all at ) during three different observing runs with ZIMPOL I at the McMath-Pierce facility of NSO/Kitt Peak. While 6 of the regions were recorded in September 1996, one was recorded in April 1995 and one in February 1996. Most of the solar regions were selected at different position angles near the geographic north or south position of the solar limb, while one region was near the geographic east limb.
As the main objective of the present work is to explore the diagnostic potential of the differential Hanle effect and identify problem areas to be addressed by future work, rather than to present definite quantitative results on field strength and line parameters, we refrain here from listing all the technical details concerning the observations and the selected lines.
3.1. Evidence for varying turbulent magnetic fields
Fig. 1 provides examples of spectra for the 4886 and 4934 Å wavelength ranges. The top panels show the usual intensity spectra, which hardly vary from region to region on the Sun (as long as the limb distance is kept fixed), while the panels below show the "second solar spectrum" (linear polarization or ) for two solar regions that differed greatly in Hanle depolarization. The inversions that we will comment on later give turbulent field strengths of 4 and 30 G, respectively, for these two regions.
Regardless of whether one trusts these quantitative values or not, it is apparent from direct visual inspection of the spectra that the polarizations in the bottom panels are subject to much more Hanle depolarization than the spectra of the middle panels. One may therefore immediately draw the qualitative conclusion, without any modelling or special assumptions, that the turbulent fields of the solar region that is represented by the bottom panels are much stronger than those of the region that is represented by the middle panels. The inversions merely try to quantify this direct conclusion in terms of G units.
The different spectral regions could not be recorded simultaneously, and in some cases the time separation could be as large as 1-2 days for the "same" solar region. Due to solar rotation the heliographic location is then not identical. However, at at high latitudes and with our crude spatial resolution (including averaging along the 50 arcsec long slit), this difference may not be so important here. This could be seen in Fig. 5 of Stenflo & Keller (1997), where recordings separated in time by two days were plotted on top of each other.
It is interesting to note that although both the solar regions that are represented in Fig. 1 were located at the position angle of geographic north, which was at almost the same distance from the heliographic north pole in both cases (about , but on opposite sides of the heliographic pole), the turbulent field strength differs dramatically, by a factor of about 8 according to the inversions. This difference can be due to both spatial and temporal changes (since the observations were separated in time by years). Whatever the cause, it demonstrates that the turbulent field does not have an invariant field strength distribution, but its properties need to be mapped in space and time.
Let us here again stress that Zeeman-effect or magnetograph observations are blind to this type of field. Use of the Hanle effect is the only real avenue that we have to access this domain of magnetoturbulence.
3.2. Extraction of the line polarization
As has been noted above, the zero point of the polarization scale is unknown, so we have to rely on theoretical or indirect considerations to estimate its location. Here we will use heuristic arguments in connection with the definition of a procedure to extract a quantity that would represent the line polarization , as we will discuss next.
Previous surveys of the second solar spectrum (Stenflo et al 1983a,b) have indicated that there is a statistical relation between the shapes of the depolarizing, absorption-like profiles of lines with apparently no intrinsic polarization, and their intensity profiles . Here and are the polarization and intensity of the continuous spectrum. A simple and natural way to parametrize this relation in terms of one free parameter is to write
Here we have written instead of p to mark that the above relation is supposed to apply to non-scattering lines with no intrinsic polarization. If a line does contribute with some intrinsic scattering polarization, then this contribution would in the framework of this heuristic model be added on top of . Accordingly the intrinsic line contribution to the observed polarization p is given by
thus represents a kind of zero or reference level to be used for the extraction of the line contribution .
In reality the continuum and line polarizations are entangled in a non-linear way, which can only be properly modelled with radiative transfer. Our extraction procedure here in terms of Eqs. (8) and (9) with the introduction of an "intrinsic" line polarization is thus mainly formal, to allow us to more explicitly illustrate the diagnostic possibilities of the differential Hanle effect. In our ZIMPOL observations we do not find any purely depolarizing lines that could serve as good references. This, in combination with the unknown zero point of the polarization scale, makes the whole extraction procedure questionable and error prone, but without it we would have no quantity to which a Hanle depolarization factor could be applied (since so far nobody has the capabilities to do proper simulations of these lines with polarized radiative transfer).
Assuming that we have defined the value of and the position of the zero point of the polarization scale (see below), then the formal extraction procedure is straightforward. From the intensity spectrum we get I and , which gives us the profile via Eq. (8). This is then subtracted from the observed p () spectrum, which via Eq. (9) gives us .
While previous work (Stenflo et al. 1983a,b) has suggested that , implying a direct proportionality between the profile shapes, our present data indicate that a substantially smaller value of is needed. This can be seen in Fig. 2, where we have illustrated the situation for the third of our selected wavelength ranges, 5682-5687 Å. As usual the top panel shows the intensity spectrum, while the middle panel shows the observed Stokes for the same two solar regions that were chosen for Fig. 1. The 4 and 30 G regions are represented by the dotted and solid curves, respectively. A common straight line has been fitted through the continuum points to represent the level of the continuum polarization. This level is slanted, because the slow wavelength dependence of the instrumental polarization ( cross talk) makes the position of the zero point of the polarization scale vary with wavelength. The magnitude and sign of the gradient (slope) is consistent with theoretical calculations of the wavelength dependence of the instrumental polarization of the McMath-Pierce telescope and the compensating glass plate. The zero level in the diagram has been chosen to optimize the procedure for the extraction of (see below).
The bottom panel of Fig. 2 shows two versions of , both of which have been extracted from the solid curve with G in the middle panel. Thus the profile form parameter for the dotted curve, 0.55 for the solid curve. The model with is unable to make all the values of positive for any reasonable choice of the polarization zero level. One can see that a substantially smaller value of is needed, since the depolarizing profiles are generally broader than the corresponding intensity profiles, and a smaller is needed to mimic this change in profile shape. Assuming that there is no subtle contribution from the transverse Zeeman effect mixed in our data, we find that an as small as about 0.55 is needed to make the values of positive for all our spectral lines and all our solar regions.
Choosing a value for is however not the only thing that we have to do to define and , since the scale of the observed polarizations has an unknown zero point, which is common for all three quantities p, , and . If we use a theoretical value for in combination with the observed continuum level to determine the zero point, then we are unable to always get positive values for any value of . We are forced to choose another, considerably higher value for to make our heuristic concept of a self-consistent. This in turn suggests that current theoretical modelling of the continuum polarization is inadequate, but this is a separate problem that is outside the scope of the present paper.
Another uncertainty is our assumption that the line polarization must always be a positive quantity to be "physical". As has been shown by Faurobert-Scholl (1994), this is not always a valid assumption, since some special magnetic configurations may lead to negative polarizations in some spectral lines, in particular in chromospheric canopies. However, for turbulent fields in the photosphere negative polarizations should normally not occur.
Thus, by exploring the two-parameter - space, we have estimated the minimum value of and the maximum value of (0.55) that always give "physical" (positive) values. These values of and have then been used to fully define the procedure for extracting from the data. Since the continuum data points can be fitted with a straight line, the choice of a value for defines the zero point of the polarization scale.
This heuristic procedure is of course very unsatisfactory, but it is the best that we can do for the time being. Unfortunately it may also produce spurious polarization features, as suggested by Fig. 2 at places, where no obvious features are seen in , e.g. at Å . There is clearly an urgent need to explore the continuum polarization, both empirically and theoretically, to model the behavior of depolarizing lines, and to control the instrumental polarization.
© European Southern Observatory (ESO) 1998
Online publication: November 24, 1997