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Astron. Astrophys. 355, 781-788 (2000)

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2. Observed set of polarized line profiles

2.1. Effects from magnetic fields and scattered light

The polarization features of the second solar spectrum are expected to have their largest amplitudes in the absence of magnetic fields, since the modification by magnetic fields, via the Hanle effect, manifests itself primarily as depolarization and rotation of the plane of linear polarization. For a spherically symmetric sun and in the absence of magnetic fields the linear polarization increases steeply in amplitude as we approach the solar limb, and the orientation of the electric vector of the scattered radiation is parallel (and in rare cases perpendicular) to the nearest limb. The orientation of the "nearest solar limb" is always perpendicular to the radius vector that connects the disk center with the point of observation, so we could also say that the electric vector is perpendicular (and in rare cases parallel) to the radius vector. To minimize at the present stage the extra complication of magnetic effects we have carried out all our observations far from active regions, always with the slit oriented perpendicular to the radius vector.

Since the non-magnetic scattering polarization should only vary with center-to-limb distance (for a spherically symmetric sun), we usually average the spectra along the slit direction to enhance the polarimetric accuracy, after first having inspected the 2-D spectral images to make sure that significant magnetic-field effects are absent. When we move away from the limb, however, it gets increasingly difficult to avoid magnetic regions with significant Zeeman effect signals, due to the ubiquitous character of solar magnetic fields. Still it is possible to identify the slit portions that are most affected by the Zeeman effect and exclude these portions when determining the polarized line profiles.

The present analysis is based on observations with ZIMPOL I at the McMath-Pierce facility of the National Solar Observatory (Kitt Peak) on March 11 and 12, 1998. ZIMPOL I records two Stokes parameters, here I and Q, simultaneously with great precision ([FORMULA] in the degree of polarization). The spectrograph slit is always placed parallel to the nearest solar limb (perpendicular to the radius vector from disk center), and Stokes Q is defined as linear polarization with the electric vector oriented parallel to the slit. The McMath-Pierce telescope has large and varying instrumental polarization. Before each spectral recording the linear polarization of the telescope is compensated by a tilting glass plate down to a level of about 0.1 %, which is necessary to avoid effects due to the coupling of detector non-linearities with instrumental polarization (Keller 1996). The remaining instrumental polarization is dominated by [FORMULA] cross talk since I is always large, while [FORMULA] cross talk is practically non-existent in quiet regions, since U is always small there. The [FORMULA] cross talk divides out to become spectrally flat in the fractional polarization [FORMULA]. It therefore simply masquerades as a zero-line offset of the polarization scale. [FORMULA] cross talk is only significant where V, produced by the longitudinal Zeeman effect, is fairly large. Since the longitudinal Zeeman effect generates characteristic and nearly anti-symmetric V profiles and is spatially very structured (along the slit), one can rather easily identify the slit portions where such signatures are present and remove them from the analysis. In practice, if one avoids placing the slit over a limb facula, such contamination from the longitudinal Zeeman effect only becomes a problem rather far from the limb, for µ (the cosine of the heliocentric angle) [FORMULA].

To determine the center-to-limb variation of the Na I D2 and D1 [FORMULA] profiles that are due to coherent scattering on the Sun we selected the part of the solar disk that appeared to have the smallest facular activity and visible magnetic flux, so that the obtained polarized spectra would as closely as possible be representative of non-magnetic scattering. Since more or less "hidden" magnetic flux exists everywhere on the Sun, our results may well be affected by such magnetic fields via the Hanle effect. Since however the Hanle effect only operates in the Doppler cores of spectral lines but not in their wings, it is only the core peaks in the D2 and D1 lines that would be affected. For randomly oriented and spatially unresolved magnetic fields the Hanle effect should always depolarize the linear polarization, which means that the [FORMULA] amplitudes that we determine in the line cores should represent lower limits to the amplitudes that they would have in the ideal, non-magnetic case.

We did careful recordings with ZIMPOL I of the Na I D2 and D1 polarization at 11 different µ positions on the disk, from [FORMULA] to [FORMULA]. In addition we did recordings at the extreme limb ([FORMULA]) and a few arcsec outside the limb, but these recordings are not reliable because of the influence of wide-angle scattered light when we are outside the limb, so they have been discarded. The origin of these polarized scattered light effects has been identified, explained, and modeled by Keller & Sheeley (1999). Quantitative analysis shows that they become insignificant as soon as one is inside the extreme limb, because of the drastically reduced relative contribution of the wide-angle stray light to the observed intensity.

2.2. Zero point of the polarization scale

Fig. 1 shows the set of recorded [FORMULA] spectra at the 11 used center-to-limb positions as well as a representative Stokes I spectrum (panel to the upper left), obtained at [FORMULA]. Since the relative Stokes I spectrum varies only slowly with center-to-limb distance, we do not display all the I spectra here. As the spectral field of view was not sufficiently large to cover both the D2 and D1 lines at the same time, each spectral panel contains two partially overlapping recordings. Since the [FORMULA] zero-line offset due to varying instrumental polarization may be different in the two recordings, we have first shifted the zero point of the D1 recording such that it fits the overlapping portion of the D2 recording. The overlapping portions are not averaged but plotted separately on top of each other, but since the relative spectral structures reproduce in minute detail, they differ by less than the width of the solid curves in Fig. 1 and are therefore indistinguishable in the figure. The small, odd-looking spectral feature near 5893 Å is for instance no artifact but real, intrinsically solar, since it reproduces perfectly in the two overlapping recordings and is seen in all other ZIMPOL recordings that we have made of this spectral region. This spectral feature appears to be the combined effect of a depolarizing component, located at the position of an Ni I line, and a polarizing component, located at the position of an Fe I line.

[FIGURE] Fig. 1. Center-to-limb variation of the profile shapes. The Stokes I spectrum in the upper left-hand panel has been recorded at [FORMULA]. All the other panels display the polarized [FORMULA] spectra recorded at various µ, from 0.05 to 0.7. Note that the scale has been magnified as µ increases, to allow a comparison between the relative profile shapes. The horizontal dashed lines represent the level of the continuum polarization

As the next step in the reduction the joint zero point of both recordings has been shifted until the behavior of the [FORMULA] spectrum at the low and high wavelength portions as well as in the zero-crossing portion between the two lines is consistent with the predicted level of the continuum polarization and with previous observational (Stenflo & Keller 1997) and theoretical (Stenflo 1997) studies. In the April 1995 ZIMPOL I recordings by Stenflo & Keller (1997) for [FORMULA] the spectral coverage was much larger than here and included wavelengths sufficiently far from the D2 and D1 lines, where the line polarization has asymptotically reached the continuum polarization level. Since no useful empirical determinations of the continuum polarization are available, we use the theoretical continuum polarization values determined from the theory of Fluri & Stenflo (1999) and a model for the average quiet Sun by Fontenla et al. (1993). From our previous work we know how the continuum level is approached in the far line wings at the wavelengths covered by the present recordings. We also know rather well how the Ni I feature near 5893 Å is positioned with respect to the zero-crossing wavelength. This zero crossing is the result of quantum interference between the [FORMULA] and [FORMULA] upper states of the D2 and D1 transitions and has been studied theoretically rather extensively in the past (Stenflo 1980, 1994, 1997; Landi Degl'Innocenti 1998). Based on all this previous knowledge, it is possible to make a reliable estimate of the true position of the zero point of the polarization scale. This is the way in which the zero points of the various diagrams in Fig. 1 have been determined. The dashed horizontal lines in the diagrams represent the level of the theoretical continuum polarization.

2.3. Examples of [FORMULA] profiles

Inspection of the 2-D [FORMULA] spectra showed that infiltration of the longitudinal Zeeman effect due to [FORMULA] instrumental cross talk was significant only for the recordings with the four largest µ values. These recordings could still be used for our analysis after the "contaminated" portions of the slit were excluded or clipped in the data reduction process. For [FORMULA] we could use 63 % of the slit, for [FORMULA] we used 70 %, for [FORMULA] only 25 %, while for [FORMULA] we could use 83 %. Due to the small useful slit portion for [FORMULA], the low level of the measured polarization, and remnant magnetic-field effects that could not be cleanly excluded completely, the [FORMULA] spectrum for [FORMULA] is of lesser quality than the others but still good enough to be included here. Conservative wavelet smoothing has been applied to the spectra to suppress noise fluctuations while preserving all spectral structures.

The compressed scale of the plots in Fig. 1, needed to fit all the 11 [FORMULA] spectra into one figure, makes it difficult to appreciate the remarkable profile structure that is really there. To bring out some of this structure more clearly, we plot as an example in Fig. 2 the 2-D [FORMULA] image for the D1 recording at [FORMULA]. We can here see the main profile features of this line: the large-scale asymmetry between the wing polarizations (light on the left, dark on the right side), and the narrow polarization peak in the line core, surrounded by very narrow and well defined polarization minima.

[FIGURE] Fig. 2. 2-D spectra of Stokes I and [FORMULA] for Na I D1 at [FORMULA]. Darker areas mean stronger polarization (in the positive direction), lighter areas weaker or negative polarization. The slit is parallel to the limb. While there is some spatial variation of the polarized core peak maximum, the wing polarization remains spatially invariant. When averaging the [FORMULA] diagram along the spatial direction, one obtains the 1-D [FORMULA] profile in the second left panel of Fig. 1

Another phenomenon that is less pronounced but still visible in Fig. 2 is a spatial variation that is consistent with a Hanle effect interpretation. In a first approximation there are no spatial variations along the 50 arcsec long spectrograph slit. However, if we look carefully, we find that there are spatial variations of the core polarization peak, because the lower spatial portion of the core peak in Fig. 2 is darker than the upper portion, while the wing polarization and the minima surrounding the core do not show any such variation. This is consistent with the expectations from Hanle-effect theory (cf. Omont et al. 1973; Stenflo 1994, 1998), according to which the Hanle effect only operates in the Doppler core of spectral lines but not in their wings (beyond a few Doppler widths). This expectation has been empirically confirmed by Hanle observations in the Sr II 4078 and Ca I 4227 Å lines (Bianda et al. 1998b, 1999). The subtle variation along the slit of the core peak amplitude can therefore be understood in terms of spatially varying Hanle depolarization due to fluctuating magnetic fields. Similar and more pronounced spatially varying Hanle depolarization has previously been noted and displayed for the D2 line (Stenflo et al. 1998).

Such subtle spatial variations of the core peak polarization are sufficiently small here to be ignored when averaging along the slit to compress the 2-D spectra into the 1-D spectra of Fig. 1. To illustrate the spatially averaged profile shapes in better quantitative detail we show in the upper panel of Fig. 3 the 1-D spectrum that results from compressing the 2-D spectrum of Fig. 2. It is identical to the [FORMULA] spectrum in the second panel to the upper left of Fig. 1, but we are now able to better appreciate the details of the profile structure. For comparison we show in the lower panel the corresponding profile for [FORMULA].

[FIGURE] Fig. 3. Upper panel: 1-D spectrum of Stokes [FORMULA] for Na I D1 at [FORMULA], obtained from the corresponding 2-D spectrum in Fig. 2 by averaging along the slit. It is identical to the [FORMULA] spectrum in the second left panel of Fig. 1, but it brings out in greater detail the remarkably complex polarization structure of the D1 line. The horizontal dashed line represents the level of the continuum polarization. Lower panel: The corresponding profile for [FORMULA]. It is identical to the [FORMULA] spectrum in the fourth left panel of Fig. 1

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© European Southern Observatory (ESO) 2000

Online publication: March 9, 2000