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Astron. Astrophys. 364, 829-834 (2000)

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2. Observations and data reduction

The observations have been carried out at the German Vacuum Tower Telescope in the Observatorio del Teide, Tenerife, from 8 to 11 November 1998. We used the filter spectrometer TESOS (Kentischer et al. 1998) to take two-dimensional spectra of a sunspot and its immediate surroundings. The Doppler velocity has been derived from the Fe II line at 542.5 nm, which is formed in the deep photosphere, at an optical depth [FORMULA]. For the velocity measurements we used a Fourier phase method: The phase of the first Fourier component of the line profile provides the wavelength position (c.f., SS2000). At a spatial resolution of some 500 km, we have observed an isolated round sunspot (NOAA 8578). In the present analysis we concentrate on the data set, taken at a position angle of 11o. The observations and the properties of the spectral line used are described in more detail in SS2000.

2.1. Selection of bright and dark component

The intensity distribution in the penumbra is rather broad (e.g., Grossmann-Doerth & Schmidt 1981) and the mean intensity also depends on the spot radius. Therefore, a local definition of "bright" and "dark" has to be made. To this end, we converted the intensity data to polar coordinates relative to the spot center, and applied an azimuthal boxcar smoothing (box size [FORMULA] Mm) to the continuum intensity values. For each radius, values above the mean were defined as "bright", and vice versa. With this definition, about half of the pixels are bright, and the other half are dark (50.5%, 49.5%). An example of that procedure is shown in Fig. 1. The graph contains part of an azimuthal slice of the continuum intensity at a spot radius of 10.3 Mm (inner penumbra). A boxcar average (not shown in the figure) has been applied to distinguish locally between bright and dark features. The plusses denote the bright component and the triangles correspond to the dark one. At the smallest spatial scale there are some very weak local darkenings that are not recognized as "dark" filaments, but the vast majority of (locally) dark and bright structures is correctly identified. White light images and filtergrams (being recorded simultaneously by TESOS) were co-aligned to find the co-spatial information within the velocity maps. The velocities corresponding to bright and dark pixels are plotted in Fig. 2 for the same spot radius as in Fig. 1. Here, the entire azimuthal slice is displayed, to demonstrate the sinusoidal modulation of the LOS-velocity in azimuth. Note that in most cases, the velocities corresponding to the bright (dark) features are above (below) the local mean.

[FIGURE] Fig. 1. Penumbral intensity (normalized to quiet Sun) at a fixed spot radius (10.3 Mm) versus azimuthal distance. Bright features are marked with a plus sign and dark features with a triangle. For better visibility, only the first 40 Mm of the azimuthal slice is plotted.

[FIGURE] Fig. 2. Velocities at the locations of bright and dark pixels along the same azimuthal slice as in Fig. 1, except that entire slice is plotted.

2.2. Flow angle and flow velocity

We take advantage of the azimuthal variation of the line-of-sight velocity in order to determine the mean flow vector. In this paper we perform the analysis not only for the mean flow (as in SS2000), but also for the flow of the bright and dark component, separately. Neglecting an azimuthal component of the penumbral flow field, the line-of-sight measurement, [FORMULA], is related to the material flow on the Sun through

[EQUATION]

where [FORMULA] is the flow velocity, [FORMULA] is the heliocentric angle of the sunspot, [FORMULA] denotes the inclination angle of the flow with respect to the surface normal and [FORMULA] is the azimuth angle of the sunspot, with [FORMULA] pointing to disk center. Since the deviation of the penumbral flow field from axisymmetry is small, we describe the flow vector by azimuthal means.

From the measurements, [FORMULA], we extract the vertical and horizontal components of the line-of-sight velocity. The vertical contribution to the line-of-sight velocity is given by its azimuthal mean, [FORMULA],

[EQUATION]

and the horizontal contribution is described by

[EQUATION]

A(r) is the amplitude of the azimuthal sinusoidal variation. The mean, m, and the amplitude, A, are obtained by fitting the function [FORMULA] to the measured velocities along an azimuthal slice, taking into account the uncertainty [FORMULA] (see Sect. 2.3). The fit procedure is performed by minimizing

[EQUATION]

using the Marquardt algorithm (e.g., Bevington & Robinson 1992). The inclination angle, [FORMULA], is computed from Eqs. (2) and (3) as

[EQUATION]

The absolute flow velocity, [FORMULA], is given as

[EQUATION]

2.3. Error analysis

Several sources of error have to be considered for the determination of m and A (cf. Eq. (4)): Measuring the LOS velocity is spuriously affected by seeing variations while scanning the line profile (see SS2000) and by determination of the line position. The largest uncertainty, however, is introduced by the penumbral fine structure itself: The velocity vectors of the bright and dark components differ significantly. This introduces a deviation from axisymmetry, i.e. from a sinusoidal shape of azimuthal velocity slices. The root-mean-square values of these small-scale deviations can be used as a measure of the uncertainty [FORMULA] in Eq. (4).

Fig. 3 shows the radial dependence of [FORMULA] for all pixels (solid line), for the bright (dotted line) and for the dark component (dashed line). In comparison with Fig. 2, it is seen that the rms values within the penumbra are small compared to the amplitude of the azimuthal variation of the velocity. Note that for all radii, the rms of the velocity fluctuation is smaller in the individual components (bright and dark) than for all data points. This indicates that the fit parameters m and A adopt different values for the bright and dark component.

[FIGURE] Fig. 3. The standard deviation of the measured velocity fluctuations along azimuthal slices versus radial distance from spot center. The penumbra spans from the left end of the abscissa to the vertical line at some 16 Mm. The fluctuations mainly stem from the fine structure of penumbra and granulation.

The standard deviations of the fit parameters, [FORMULA] and [FORMULA], correspond to the diagonal of the error matrix, which is calculated by the Marquardt algorithm (cf. Eq. (4)). The uncertainties for the flow angle, [FORMULA], and for the flow velocity, [FORMULA], result from analyzing the error propagation. In addition to [FORMULA] and [FORMULA], one has to consider the error in the position angle, [FORMULA], which is taken from synoptic maps. We estimate this error to be [FORMULA] rad and ignore the slight variation of [FORMULA] across the spot. Using Eqs. (5) and (6), we obtain:

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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