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Astron. Astrophys. 362, 737-745 (2000)

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3. Intensity statistics of the quiet Sun

Although the radiances of the two instruments agree well within their uncertainties (Pauluhn et al. 1999), CDS measures on average 30% higher radiance values than SUMER in the HeI line (for the data analyzed here). For the MgX 609 Å and 624 Å lines this difference is 17% and 9%, respectively. The important point is, however, that both instruments exhibit the same long-term temporal trends, suggesting that fluctuations from one scan to the next are solar, which allows us to combine the various observations to form histograms with extremely high S/N. It has to be noted that the 609 Å line is blended with an OIV line formed at a temperature of [FORMULA] K with a lower region of formation than the corona (Thompson & Brekke 1999), and thus its histograms do not represent a "pure" coronal line like those of the 624 Å line.

The intensities of each spectral line and of each instrument have been binned before forming histograms. The binsize has been chosen such as to provide a high enough resolution, but also to be larger than the instrumental noise. The resulting binsizes are 0.05 W/m2/sr for HeI , 0.005 for MgX 609 Å, 0.001 for MgX 624 Å, and 0.003 for NV and NeVIII .

We tested five statistical models of probability functions against the data: a Rayleigh distribution, a Maxwellian distribution, a superposition of two Gaussian distributions, a lognormal distribution, and a superposition of a lognormal and a Gaussian distribution.

The Rayleigh and Maxwell distributions are positively skewed functions defined on a positive data range. In the following µ and [FORMULA] denote the mean and the standard deviation, respectively, and x is the intensity.

Rayleigh:

[EQUATION]

Maxwell:

[EQUATION]

Double Gaussian:

[EQUATION]

Lognormal:

[EQUATION]

Lognormal+Gaussian:

[EQUATION]

The lognormal distribution is strongly peaked and positively skewed and well suited to model a frequency distribution with a sharp maximum and a pronounced tail. The Maxwellian and Rayleigh distributions have the advantage of the fewest free fit parameters (2 for unnormalized, 1 for normalized frequency distributions, where "normalized" means divided by the number of pixels). The lognormal has 3 and 2 free parameters, respectively. Finally, the double Gaussian fit and lognormal plus Gaussian combinations involve 6 free parameters for unnormalized distributions and 5 for normalized ones. Fits consisting of two distributions introduce a natural distinction between brighter and darker parts of the solar atmosphere, usually equated with the network and cell interiors, while a single distribution suggests that there is no principle distinction between various parts of the quiet Sun, at least not a simple one.

The lognormal, double Gaussian and lognormal+Gaussian fits to the observed distributions are plotted in Figs. 1-4. Fig. 1 shows the results for the HeI 584 Å line ([FORMULA] K). In the upper row the SUMER data and corresponding fits are plotted, while the lower row shows the same for the CDS data. Fig. 2 and Fig. 3 show the same but for the coronal lines MgX 609 Å and MgX 624 Å ([FORMULA] K). The fits based on the Maxwell and Rayleigh distribution functions are not plotted due to their relatively poor quality.

[FIGURE] Fig. 1. Upper row: SUMER frequencies of occurrence at HeI 584 Å, fitted with the 3 different models. Lower row: The same for the CDS frequencies of occurrence.

[FIGURE] Fig. 2. Upper row: SUMER frequencies of occurrence at MgX 609 Å, fitted with the 3 different models. Lower row: The same for the CDS frequencies of occurrence. (Note that this line contains a strong blend of a lower temperature line (OIV , [FORMULA] K).

[FIGURE] Fig. 3. Upper row: SUMER frequencies of occurrence at MgX 624 Å, fitted with the 3 different models. Lower row: The same for the CDS frequencies of occurrence.

[FIGURE] Fig. 4. Upper row: SUMER frequencies of occurrence at NV 1238 Å, fitted with the 3 different models. Lower row: The same for the NeVIII 770 Å line.

[FIGURE] Fig. 5a and b. Example of a graphical representation of the errors of the fits for the various model distributions. Shown are the absolute differences of the (normalized) fit-functions to the (normalized) radiance distribution measured by SUMER at 770 Å a , and the relative difference, or asymmetry, of fit and data, i.e. the differences divided by the sum b .

In Fig. 4 the results for the transition region lines NV 1238 Å ([FORMULA] K; upper panels) and NeVIII 770 Å ([FORMULA] K; lower panels) are shown (SUMER data only).

The [FORMULA]-differences are given in Table 1, where we computed [FORMULA] as the sum over the squared differences between the normalized observed and functional distributions.


[TABLE]

Table 1. Error of the fits ([FORMULA]-differences).


Table 2 gives the fractions of the intensities that are represented by the fits and, in the case of the 2-component fits, of the individual components, respectively (the latter given in brackets). Finally, Fig. 5 depicts the discrepancies between the fits and the data. As an example we present the results for the NeVIII 770 Å line, as observed by SUMER. In Fig. 5a the absolute difference is plotted, [FORMULA], while in Fig. 5b the asymmetry, [FORMULA], is shown, where F denotes the model distribution and f the data. The latter is a measure of the relative difference between fit and data. It is obvious that the Rayleigh and Maxwell distributions are not suitable to model the quiet Sun data. Also, the two Gaussians are somewhat better at representing the height of the maximum, whereas the tails can only be reproduced by a lognormal. It is clear from this figure that the Rayleigh and Maxwell distributions are far inferior to the rest in the quality of the fit produced. This is confirmed by Table 1.


[TABLE]

Table 2. Fractions of the radiance contained in the components of the fits.


A careful study of Figs. 1-5 and Table 1 and Table 2 reveals the following:

1) The double Gaussian fit (6 free parameters) captures more than 80% of the intensity and seems to be a reasonable approximation over the main body of the distribution. It is, however, unable to account for the prominent tail of the distributions, in spite of the large number of free parameters.

2) The lognormal distribution (3 free parameters) alone covers more than or nearly 90% of the intensity in all temperature ranges. The errors in reproducing the original data intensities vary from 2-12%. The lognormal distribution is able to represent the tails of the distributions quite accurately. It provides an almost perfect fit for the transition region and chromospheric lines, with a steady decrease in quality towards higher temperatures.

3) The combined model of a lognormal and a Gaussian (6 free parameters) provides the best fit to the quiet Sun intensity data distributions for most lines, in particular those formed at coronal temperatures. It reproduces the original distributions to within 0.5-6%. In the colder lines the Gaussian gives less than a 15% contribution to the total, while for the coronal lines - especially for the CDS measurements - the Gaussian represents a large fraction of the total intensity. However, the gain in precision over the single lognormal fit is only achieved at the price of doubling the number of free parameters. Also, the lognormal+Gaussian combination suffers from the fact that the contribution of the Gaussian component differs by up to a factor of 2 between SUMER and CDS data (Table 2), although this may be due to the different spatial resolutions of the two instruments. More serious is that, unlike in the case of the double Gaussian, the relative strengths and locations of the two components vary strongly from fit to fit. This suggests that the distinction between the two components is arbitrary and not particularly stable, so that the improvement of some of the fits is mainly a consequence of the increased number of free parameters.

It is interesting that for the coronal lines, i.e. those with the poorest lognormal fits, the lognormal+Gaussian fits are significantly better for the SUMER than for the CDS data. We checked whether this is due to the difference in spatial resolution between the two instruments by degrading the SUMER data obtained with detector A, where raster scans over an area of 60"[FORMULA]300" had been made, to the CDS resolution. They were smoothed by a suitable point-spread-function and rebinned to the CDS spatial pixel size as described by Pauluhn et al. (1999). The histograms of all data of these 7 rasters in HeI and 6 rasters in MgX were then fitted using the Gaussian and lognormal functions. In general the errors of the fits increased relative to those found for the undegraded data. The fact that both MgX lines measured by CDS are better represented by the sum of two Gaussians than by a single lognormal also turns out to be a result of the poorer spatial resolution of CDS, since the degraded SUMER data exhibit the same effect. Finally, the lower resolution of the degraded SUMER data lead to a change in the partition of intensity contents in the 2-component models for HeI 584 Å and MgX 609 Å: the lower intensity ("cells" in the case of the two Gaussians) Gaussian component then contained approximately half of the total intensity. This also agrees with the findings from fits to CDS and SUMER (see Table 2). This shows that the main differences between the histograms obtained from SUMER and CDS scans are due to the difference in spatial resolution between the two instruments. However, only for the lines formed at higher temperature do the distributions change with resolution, for the chromospheric HeI 584 Å line there was no improvement at higher resolution. It therefore seems that the spatial elements contributing to the EUV emissions of lower temperature regions can be resolved, whereas for coronal lines the available SUMER resolution is not sufficient, and the size of these heating elements lies at or below SUMER's resolution of 1".

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© European Southern Observatory (ESO) 2000

Online publication: October 24, 2000
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