3. Correlations and discussion
3.1. Soft X-rays
Fig. 2 is a soft X-ray image of extremely long exposure resulting from a combination of all data of the Al.1 filter taken during the first time interval. The weak structures of the quiet corona are emphasized by the gray scale. The contours indicating the average longitudinal strength of the magnetic field per resolution element outline the magnetic network in the photosphere and chromosphere having a scale of 20'000 to 50'000 km. The network elements increase in peak strength in the lower left corner, the direction to active region 7842. The pronounced network structure and the highest peak being only 80 gauss indicate that the field of view comprises a section of the photosphere that is completely quiet. In soft X-rays, however, some high reaching coronal loops of the distant active region are projected into the eastern half of the image. They produce a foreground through which the enhanced X-ray structures of the low corona are still visible.
In Fig. 2 the rich X-ray structure of the quiet corona is evident. Although some detail is lost in the image because of the long time-averaging, X-ray loops connecting bipolar magnetic elements are discernible; e.g., at the bottom of the image, just to the left of the middle. Other X-ray features, e.g. at positions 170"/25" and 200"/40", are located between bipolar network elements, suggestive of connecting loops.
The power spectrum of the right half of the X-ray image is presented in Fig. 3a. Each line (column) was transformed and the result averaged over all columns (lines) for an East-West (North-South) spectrum. The spectra decrease with wavenumber k up to rad/Mm, where the spectrum becomes constant. This value corresponds to scale sizes of 5500 km (.6) and is close to the Nyquist number , where is the effective resolution. At higher wavenumbers the spatial resolution of the SXT instrument is increasingly oversampled, and the spectrum is flattened by noise. Thus the soft X-ray observations contain significant structure down to the resolution limit.
The spectrum in Fig. 3a is not a simple power-law. The slope below rad/Mm has a power-law index of -1.4 0.2. For rad/Mm the slope decreases with . It is similar to reported by Gómez et al. (1993) for active regions. The break point corresponds to a scale of 25'000 km, suggestive of the supergranular scale.
3.2. Radio waves
The power spectrum of the radio image at 2 cm (Fig. 3b) has a power-law part with an index of -1.8. The low-wavenumber break at 0.143 rad/Mm corresponds to a scale size of 44'000 km. Note the sharp cut-off at high wavenumbers due to the beam size. Contrary to soft X-rays (Fig. 3a) where the spectrum at large wavenumbers is dominated by noise, the radio image does not contain much noise at scales below the beam size. As the beam is narrower in the North-South direction (.8), the power-law extends to higher wavenumbers than in East-West direction. The difference between North-South and East-West in Fig. 3b clearly demonstrates that structures as small as the resolution limit have been detected in the 2cm image. The existence of fine structure in the 2cm emission of the quiet corona at this small scale is surprising, as radio emissions at 20 cm wavelength, where a considerable fraction is coronal, appear to be resolved at resolution (Gary & Zirin 1988). Radio wave scattering in the solar atmosphere may be the main reason. The observations agree with the -law of radio wave scattering and a plausible model of the inhomogeneity in the solar atmosphere (Bastian 1994).
Fig. 4 confirms that enhanced radio emission is often, but not always, associated with regions of enhanced magnetic field in the network and vice versa (as was shown by Erskine & Kundu, 1982, at 6 cm). The cross-correlation of the radio intensity with the absolute magnetic flux is significant at all observed wavelengths. Table 1 gives the peak values near zero lag. Peak correlation is often shifted by several arcseconds similar to the X-rays. The cross-correlation is best and peaks closest to zero lag for the short wavelength, originating from the smallest height. The correlation of the radio image with the magnetic field is clearly less than the one found in an old, decayed active region by Gary & Zirin (1988) at 6 cm.
Table 1. Properties of images. The X-ray values are given in Data Numbers (3 photons) per second and pixel and refer to the right part of Fig. 2. The radio values are in degrees kelvin and are calculated from the primary beam image produced only from the data taken in the three-frequency mode. Thus the values for the 3 radio frequencies are based on the same integration time. The errors are discussed in the text.
Table 1 also indicates that the 1.3cm and 2.0cm images correlate very well. The 2.0cm and 3.6cm maps correlate well, but are displaced from each other. The correlation between the 1.3cm with 3.6cm maps is less pronounced.
A comparison of X-ray emission and photospheric magnetic field in Fig. 2 indicates that most, but not all, enhanced magnetic field regions are sites of elevated X-ray emission. In the left half of Fig. 2, dominated by the loop-shaped X-ray emission of bipolar regions, the bipolar regions are often at the footpoints of the elongated X-ray brightenings. In the right half of the image the peak of the X-ray emission is also slightly displaced or between bipolar regions, suggesting that the X-ray emission generally originates in magnetic loops rooted in the photospheric region of enhanced average magnetic field.
The two-dimensional cross-correlation of the X-ray emission and the magnetic field is shown in Fig. 5a. The statistical significance of the correlation is determined by a Student's t-test. The degrees of freedom of the statistical problem are given by the number N of independent picture elements of the image with the lower spatial resolution. In all cross-correlations the field was chosen to be centered in the upper middle of Fig. 2. The field does not include the projected strong sources in the lower left corner of Fig. 2. For Fig. 5a and an effective resolution of the SXT amounting to .3, . The test on the observed peak correlation of 0.15 then indicates a significant correlation. It confirms the association of the coronal X-ray emission with enhanced photospheric magnetic field and the magnetic network in general.
The statistical error of the cross-correlation is of the order of , where is the signal-to-noise ratio of the noiser data set. The statistical error in Fig. 5a then is 0.01. However, the cross-correlation coefficient is dominated by relatively few intense structures. Therefore, the values given in Table 1 refer to a particular quiet region and may not be general. For this reason, the errors given in Table 1 are not representative.
The error in the position of the peak correlation is determined by the spatial resolution and the statistical noise in the two images. This is better than the .1 accuracy with which the shifts have been read out. Note that in Fig. 5a the total displacement of the peak correlation from zero lag is . Fig. 2 suggests that this is mainly caused by the X-ray emission originating between the two elements of bipolar regions. Since these individual shifts appear to be random, the strongest elements then dominate the direction of displacement in the cross-correlation. Thus the net shift of peak correlation is the result of a few elements, and the variation from region to region may be much larger than the statistical error.
The cross-correlation of the X-ray image and the simultaneous radio intensity at 2 cm is shown in Fig. 5b. It shows the weakest of all cross-correlations. The correlation peaks at 0.07 is still significant. The peak is displaced by only from zero lag. The small displacement of the peak indicates that the soft X-ray emission and 2cm radio emission are indeed correlated. The correlation seems to have at least two peaks. The second peak, displaced by , is even higher. Thus there seem to be two preferred directions of displacement and the correlation is smeared out over many lags, reducing the central peak, which is smaller than the cross-correlation with the magnetogram. An example of the cross-correlation of radio images with the magnetogram is shown in Fig. 6. The peak values and peak displacements are given in Table 1.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998