Astron. Astrophys. 355, 769-780 (2000)

5. Analysis of the CDS data

5.1. Fitting the CDS spectral profiles

All results in this section were derived from integrated line intensities, taking their uncertainties into account. Because the way we obtained them was common in all following subsections, we will discuss it briefly here. We supposed that the spectra in individual spectral windows (covering the wavelength interval 1.33 Å for NIS I and 2.33 Å for NIS II) can be approximated by a sum of a constant representing the background (stray light) and L Gaussian components representing the individual spectral lines centered at wavelengths :

is the intensity observed in the i-th spectral pixel at the wavelength , is the amplitude, is the FWHM of the k-th Gaussian and .

To obtain the best fit of parameters , , and , we used the least square method and minimized the function:

where are the statistical errors in intensities collected in individual spectral pixels of the detector obtained from the photon statistics (Thompson 1997) and N is the number of spectral pixels. The searched integrated intensity of a given k-th line can be then easily calculated from fitted parameters and .

The statistical errors of the fitted parameters were calculated only for the parameters concerning the spectral line of interest which directly influence the searched integrated line intensity (i.e. , and ). These errors were obtained by a standard method described in detail in Press et al. (1989). Another source of errors is the unknown accuracy in the calibration curves of CDS which can introduce a systematic error into the line intensities. Because this error is unknown, it could not be included, but when one is interested in a ratio of two line intensities with similar wavelengths lying on the same detector, these errors tend to cancel each other. This is the case of the density sensitive pair Fe XIV. On the other hand, for a ratio of two distant lines or two lines lying on different detectors it can be quite significant. This is the case of the temperature sensitive pair Fe XVI/Si XII.

5.2. CDS temperature diagnostics

To get information on the temperature distribution of hot plasma in the system across the loop tops we used the intensity ratio of lines Fe XVI at 360.8 Å and Si XII at 520.7 Å which is temperature sensitive in the interval from 1 MK to 3.2 MK. The theoretical intensity ratio is also dependent on the relative chemical abundance of iron to silicon which is not very well known (Meyer 1985) and which can also vary from flare to flare (Fludra & Schmelz 1995). Moreover, as we already mentioned above, the observed intensity ratio can be quite significantly influenced by the systematic error resulting from the uncertainty in the CDS calibration. These reasons prevented us from calibrating the temperature using only theoretical data and we had to use another method based on our knowledge of the temperature obtained from SXT data. Therefore, we supposed that the maximum temperature of plasma in the PFL system, measured using the line ratio, corresponds to the plasma temperature measured by SXT at the same time.

The ratios of Fe XVI and Si XII lines were determined in 13 pixels along the x axis from pixel numbers 22 - 34 (see Fig. 6). To improve the S/N ratio the signal was integrated from 6 pixels along the y axis from pixels 52 - 57. The error bars correspond to 3 probability of the line fits and they do not contain any uncertainties in values of theoretically calculated emissivities. Their very variable length is given by the theoretical dependence of the temperature on the emissivities ratio. The emissivities were calculated using ADAS (Summers et al. 1996).

Fig. 6. The vertical thermal structure across the loop tops determined from the temperature sensitive ratio of Fe XVI 360.8 and Si XII 520.7 lines. The error bars correspond to probability of the fits. This graph is over-plotted by the normalized intensity profile of Fe XVI loop-like structure (dash-dot line) and by the errors in the intensities (dotted lines). In the upper left corner of the graph a part of the CDS raster was plotted. Its axis are in pixels which we refer all the measurements from CDS to. The dashed box is the region where the line intensities were measured.

The results of this analysis are presented in Fig. 6, where the intensity profile of the Fe XVI loop was also plotted. It is clearly visible that the temperature increases with height and reaches its maximum above the Fe XVI loop. Plasma with lower density is probably located here. This distribution of temperature in the PFL system is in full agreement with the classical formation theory of PFL systems. The course of temperature under the Fe XVI loop reflects the temperature of rare hot coronal plasma surrounding the PFL system rather then the temperature of plasma located in the loops visible in lines having lower formation temperatures.

5.3. CDS electron density diagnostics

The CDS data was used to determine the electron density of the hot part of the PFL system, using the density sensitive line pair of Fe XIV 334.2/353.8 (Mason et al. 1997, Mason 1998). This line pair is electron density sensitive in range approximately from  cm^-3 to  cm^-3, where also the expected values of electron densities of PFL lie.

The disadvantage of this line pair is that when plasma with temperature greater than  MK is present in the analysed region, the Fe XIV line at 353.8 Å is strongly blended with a very bright Ar XVI line. Fortunately the temperature analysis of the SXT data and also the comparison of observed spectra with the synthetic spectra calculated with CHIANTI (Dere et al. 1997, Mason 1998) showed that the plasma temperature in the analysed region is lower than  MK, so that the blending is not prohibitive. The theoretical dependence of the electron density on the intensity ratio was calculated using CHIANTI.

The electron densities were determined in 13 different positions along the x axis across the loop top, in pixel numbers 22 - 34 (see the raster in Fig. 6). To improve the S/N ratio the intensity was integrated from 6 pixels along the y axis from pixels 52 - 57. To determine the level of scattered light we connected the spectral windows containing the given diagnostic lines with the adjacent spectral windows, and the minimum signal from these two windows was taken as the background. The error bars correspond to 3 probability of the line fits. No uncertainties are included in the theoretical dependence of electron density on the intensity ratio. The course of the electron density across the loop top is shown in Fig. 7. These values were used to estimate the geometrical filling factor in the top of the loop system.

Fig. 7. The course of electron density determined from the ratio of intensities of the electron density sensitive line pair Fe XIV 334.2/353.8 across the top of the loop system. The error bars correspond to probability of the fits.

5.4. CDS emission measure diagnostics

The integrated intensities of allowed lines with different formation temperatures were used to obtain the plasma emission measure using the method originally designed by Pottasch (1963). We assumed an isothermal plasma at temperature T, which corresponds to the maximum of the contribution function (emissivity) of the given line. Then the integrated intensity of the line can be approximated by a simple formula:

where A is the elemental abundance relative to the hydrogen and I is the integrated intensity of the particular spectral line. The contribution functions were calculated using ADAS (Summers et al. 1996) and the elemental abundances for a solar flare were taken from Fludra & Schmelz (1995). These abundances are subject to a combined statistical and systematic uncertainty. The errors of integrated line intensities resulting from the photon statistics correspond to probability of the line fits.

The values of the emission measure were determined at the brightest regions at the tops of the loop-like structures visible in CDS raster in different spectral lines. These regions were one pixel broad in the x direction and the intensity was integrated from six pixels in the y direction. The results are summarized in Table 2 and the dependence of the emission measure on line formation temperature is shown in the upper plot in Fig. 8. Under the assumptions adopted in the Sect. 4 it is possible to determine the mean electron densities from these values of emission measure. We assumed that the size of the loop system along the line of sight corresponds to the apparent diameter of the loop system obtained from SXT and CDS images  cm. The dependence of the electron density, obtained in this way, on the formation temperature is shown in the lower plot in Fig. 8. The mean electron densities for lines with formation temperatures from 2.2 MK (Fe XVI) to 0.6 MK (Ca X) remain almost constant with the temperature ( cm^-3). In contrast, the mean electron density calculated from the emission measure derived from the intensity of O III is substantially smaller  cm^-3.

Fig. 8. The mean emission measures and electron densities obtained from allowed lines and SXT measurements versus the formation temperature of the lines. The emission measure is calculated for the brightest parts in the tops of corresponding loops (see Table 2).

Table 2. The results of the emission measure analysis.

© European Southern Observatory (ESO) 2000

Online publication: March 9, 2000
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