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Astron. Astrophys. 346, 285-294 (1999)

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5. Data analysis and results

5.1. Methods and corrections

All data were decompressed to reverse the onboard compression scheme. Given the extreme accuracy we ultimately want to achieve, many correction and calibration procedures have to be applied. Basically, two approaches can be used to correct for possible instrument biases: We can either employ external information, such as a detector flat field to correct intensities or a detector grid image to correct geometric distortions, or exploit the data itself to estimate and correct the effective biases of the parameters. We will use here the second approach, but, whenever possible, a verification with additional information will be performed.

In the first approach, a flat-field removal would correct the measured counts with a known detector sensitivity bias factor before calculating the line and continuum parameters. In the second approach, the influence of the detector bias is treated as a "black box"; its effect is estimated afterwards by evaluating the whole raster. For instance, the flat-field fluctuations were removed by correcting for sinusoidal variations of different periodicities, which can be observed in averages over the whole extension of the east-west raster.

An exposure of the detector at a raster step [FORMULA] leads to a matrix of intensity counts [FORMULA], where [FORMULA] is the spatial pixel address and [FORMULA] is the spectral pixel address. The calculations of line and continuum intensities, line shift and widths are functions operating on subsets of this data cube, (i.e., on profiles): [FORMULA], where [FORMULA] mark the borders of a spectral window.

Let [FORMULA] be the average of f applied to spatial pixel i across the raster. Let [FORMULA] be an interval of length P symmetrically around i, and [FORMULA] a moving average of length P. Then [FORMULA] measures the effect of the detector bias in the spectral window [FORMULA] defined above at spatial address i on operator f during the raster. If the raster is unbiased, [FORMULA] will also estimate the real detector effect in an unbiased way.

[FORMULA] is a correction factor close to 1.0, and the correction vector along the slit is usually similar to a sinusoidal curve of spatial period P. The operator f is then corrected to be [FORMULA] for each pixel of the 2-dimensional field of view [FORMULA], [FORMULA], and [FORMULA]; these spatial periods were detected empirically. This procedure was, for instance, employed in producing Figs. 4 and 5 after the centroid position was deduced by integrating the line intensity across the spectral window and determining the 50% point with sub-pixel precision. This is adequate for an overview of two-dimensional velocity structures, if blends are of no major importance or have equal influence everywhere. The line width was estimated by using the relation between the full width at half maximum, FWHM, amplitude, A, and integral, I, of a Gaussian: FWHM = [FORMULA].

The determination of the detector curvature needed to correct the effect of the geometric distortion on the position of the Ne VIII line in the spectral regime was obtained with the help of September 22, 1996 data, because this raster covered a rather uniform area on the disk, i.e., without any possible bias from coronal hole or limb portions justifying the assumption that each spatial pixel along the slit should have seen similar regions of the Sun across the raster scan. The line position as a function of the spatial pixel is estimated by averaging the line centroids (obtained as outlined above) across the raster (i.e., the 83 exposures in the western portion) and is plotted in Fig. 6. Also shown is the spectral correction function resulting from the standard SUMER detector distortion correction (Moran 1997). The range of its application is recommended to be from pixel 19 to 339 in the spatial regime of Fig. 6, which shows detector pixel numbers in reversed order (0: south; 359: north). Within these limits, both detector corrections do not differ by more than [FORMULA] 0.2 pixel. This lends support to our detector curve, which was then subtracted in both data sets to deduce relative line shifts. Note, however, that a portion of the off-limb corona (spatial pixels [FORMULA] 339) is not covered by the standard procedure.

[FIGURE] Fig. 6. Curvature of the detector in the spectral regime for the line Ne VIII ([FORMULA]770) as a function of the spatial pixel position from September 22, 1996 data (solid line) and, for comparison, the corresponding curve (dotted line) obtained from the standard SUMER geometric distortion correction for the same detector position. (Note that the curvature is dependent on the detector position and therefore will vary for different spectral lines.) The limit of applicability of the standard procedure at position 339 (dashed-dotted line) is shown as well as the interval (from spatial pixel 225 to 286; horizontal bar) used for the absolute wavelength calibration in Sect. 5.2. It was placed at a pixel position where the spectral correction is close to zero (dashed line).

The temporal variations, due to thermal influences (heater cycle and temperature drifts), have then been subtracted. They have been estimated by averaging the residual line centroid positions (i.e., after the detector correction) across the north-south extension of the slit. (They have been shown in Fig. 1). The Doppler shift induced by the solar rotation speed of 2 km s-1 at the equatorial limb has been taken into account by calculating the LOS velocities for all pixels as a function of their nominal mapping onto the solar disk. Without this correction a clear solar rotation signal would be discernible demonstrating the velocity sensitivity of SUMER (see also Fig. 5 of Peter 1998). All these corrections are, however, not sufficient to deduce absolute line positions, because there may be systematic line shifts on the disk (solar effects), and because the wavelength positions on the detector may have moved between the observations (detector effects).

5.2. Absolute wavelength calibration

For an absolute wavelength calibration, an extended spectral window is required around Ne VIII (in second order) containing cool chromospheric UV lines (in first order). It can reasonably be assumed that these lines have no or very small solar Doppler shifts (Hassler et al. 1991; Brekke et al. 1997; Chae et al. 1998). The method for the detector curvature correction in the spectral regime along the slit as described above (cf., Fig. 6) is only suitable for single lines and has to be modified for a wide spectral window. To achieve this, the distortion correction will be performed in such a way that the average correction is close to zero. We selected an interval of 62 spatial pixels (approximately [FORMULA] 10% of the detector around the spatial pixel 256 in Fig. 6) to perform the calculation of an average profile for the extended spectral window. In this interval, the empirical detector curvature employed here is also very close to the one used by the standard SUMER destretching procedure. The last 83 exposures of both raster scans are taken because (a) the coronal hole scan contains an off-limb region there, (b) the thermal drift effects were stronger at the beginning of the scans than at the end. The spectra have simply been added in the spatial dimension over the 62 pixel interval. Even if this will induce a certain increase in line widths, it should not, with our parameter choice, lead to a net line shift.

For an identification of the numerous cold lines, a crude wavelength scale was first established for the mid-latitude scan on September 22 by fitting literature wavelengths of the peaks of the prominent lines Si II ([FORMULA]1533.431), Ne VIII ([FORMULA]770.409), C IV ([FORMULA]1548.202) (a linear fit has been used, i.e., it has been assumed that the spectral pixel size is constant). The resulting spectral pixel size in this first step is 42.4 mÅ. (The nominal pixel size, based on the optical design, was predicted to be 42.3 mÅ for detector A in this wavelength range.)

With the help of our preliminary wavelength scale, in a second step, 51 cold lines or line blends could be identified in the same data set. In the case of line blends, the expected wavelength was calculated as the average nominal position of the lines involved, weighted with their literature intensities. The pixel positions were fitted against the literature or expected wavelengths in a third-order polynominal fit, under the assumption that the pixel size is a quadratic function of the detector position. The result is a pixel size between 41.6 mÅ and 42.6 mÅ, with the maximum close to the middle of the detector. The differences between expected and observed wavelengths of the lines have always been smaller than 21 mÅ or 0.5 pixel and the standard deviation is 10 mÅ or 0.25 pixel. The uncertainty estimate given by the fitting routine for the wavelength range around 1540 Å is better than 3 mÅ. This defines a wavelength scale with variable pixel size against which the measured intensities can be plotted.

In a third step, we have to determine the wavelength offset which might be present between both days. The synthetic spectrum constructed in Sect. 2 and the spectra observed on September 21 and 22 in our selected spatial interval of 62 pixels (with the wavelength scale defined above for the second day) have therefore been compared by a cross-correlation analysis in the wavelength range from 1538.16 Å to 1543.28 Å. For this analysis, the interval containing the Ne VIII line in second order obviously has to be excluded (from 1540.38 Å to 1541.23 Å). The corresponding spectra are shown in Fig. 7. The highest correlation coefficient of the left portion of the spectrum of September 21 is 0.80 and of the right portion 0.98 (with a mean displacement of + 13 mÅ). The corresponding coefficients for the September 22 raster are 0.49 and 0.97, respectively, so only the shift corresponding to the right interval (+ 1 mÅ, thus essentially verifying the wavelength scale obtained in the second step) is taken into account. Consequently, both wavelength scales have been adjusted by a relative shift of 12 mÅ.

[FIGURE] Fig. 7a and b. Comparison of observed spectra in coronal hole (solid line) and quiet Sun (dotted line) regions a with a synthetic spectrum in the lower panel  b . It was built up, as described in the text, by combining all Si I and C I lines given by Kelly (1987) in this range. The interval marked by a horizontal bar covering the Ne VIII line was not considered when performing the cross-correlation analysis in Sect. 5.2. The synthetic spectrum is based on wavelength measurements in laboratories. By comparing the solar observations with this spectrum, the relativistic red shift of 0.62 km s-1 has been taken care of. The spectral radiance is given for the first order lines as in Fig. 2.

After applying these procedures, we have specified a wavelength for each spectral pixel of the detector transmitted on September 21 and 22, and thus the wavelengths for the pixels selected around the Ne VIII line ([FORMULA] as described in Sect. 5.1) are uniquely identified. As a result we obtain rectified wavelength scales, which can be used for determining the spectral position of the Ne VIII line under specific solar conditions in the next section.

5.3. Doppler shifts and rest wavelength of Ne VIII ([FORMULA]770)

The determination of the line centroids (measured in spectral pixel positions) and various corrections have been described in Sect. 5.1. A unique wavelength has been defined for each spectral pixel and both scans in Sect. 5.2. With this information, we are able to deduce the wavelength positions of the Ne VIII line for each spatial pixel on both days. In order to determine Doppler shifts, we have to define the rest wavelength first.

The western portion of the solar area surveyed on September 21, 1996 contains quiet Sun, coronal hole, and limb regions, as well as a small part of the off-limb corona. The line position observed off limb is assumed to have zero Doppler shift, because any bulk plasma motion must, for reasons of symmetry and for optically thin media, average out along the line of sight there. Optically thin conditions can be expected for Ne VIII in the corona at altitudes higher than 20" above the photosphere (Doschek et al. 1998). This provides us with a wavelength scale which is independent of a Ne VIII ([FORMULA]770) laboratory rest wavelength. The rest wavelength is thus defined as the average of the 400 positions off the limb between 980" and 1000" distance from the disk centre. At greater altitudes, the spectra become rather noisy and are dependent on a very small number of detector pixels near its edge, for which, moreover, the correction routines become unreliable. Hence they are disregarded for establishing the rest wavelength.

LOS Doppler velocities for each spatial pixel of both scans (83 raster positions each) can then be deduced with the help of the Doppler formula. The velocity structure across the Sun is obtained by averaging these velocities in concentric circles with a width of 1" around the solar disk centre. The result shown in Fig. 8 is similar to that of Fig. 3(b) of Hassler et al. (1999), who calibrated the wavelength of the Ne VIII line with an elaborate scheme using a Si II baseline. As for the off-limb corona, the mean wavelengths have also been calculated for two other typical regions on the disk, namely the "coronal hole" region between 880" and 900" (near the extreme of the blue shift), and the "quiet Sun" region between 280" and 560" from the disk centre (cf., Fig. 8). In Table 1, we list the wavelengths of the Ne VIII line for these regions.

[FIGURE] Fig. 8. Average line shifts of the Ne VIII ([FORMULA]770) line between 150" and 397" west on September 21, 1996 (right data set), and between 187" east and 66" west on September 22, 1996 (left data set). Dots indicate averages calculated in concentric circles around the solar disk centre of 1" width each. The left data set is characterized by quiet Sun conditions, whereas the right data set shows quiet Sun, coronal hole, and off-limb corona conditions. Note that the quiet Sun values on both days are in agreement, thus confirming the relative positions of the rectified wavelength scales. Notice also the blue shift immediately above the limb, probably caused by the opaqueness of Ne VIII there. Intervals used to determine typical average wavelengths and resulting line shifts (cf., Table 1) are indicated by horizontal bars. The limb is marked at 960" (dashed-dotted line) as seen in the continuum. Within the pointing uncertainty, this is consistent with the photospheric limb at 965".


Table 1. Wavelengths obtained for Ne VIII ([FORMULA]770) from mean line positions on concentric circles and the resulting LOS speeds (cf., Fig. 8)

With the assumptions of zero Doppler shift of Ne VIII in the off-limb corona and no average shift of the atomic lines, we find a rest wavelength for the line Ne VIII ([FORMULA] - [FORMULA]) of 770.428 Å. The line is then shifted predominantly towards blue in the other regions. By considering the uncertainties of the fit routines, which give 3 mÅ, and taking into account the Gaussian fit and cross-correlation results as well as the uncertainties of the atomic line data, we estimate an overall uncertainty level of 6 mÅ (1 [FORMULA]) for our wavelength determinations in first order, or 3 mÅ (1 [FORMULA]) in second order.

The average wavelength measured on the disk (770.426 Å) would indicate a redshift of 6.6 km s-1 if the old literature wavelength of 770.409 Å is assumed. This is, indeed, consistent with the results obtained by Brekke et al. (1997) and Chae et al. (1998).

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© European Southern Observatory (ESO) 1999

Online publication: May 6, 1999