4.1. X-Ray Structures of the quiet corona
Coronal soft X-rays are a combination of free-free radiation and (more importantly in the present case) line emissions. The flux density may be written as an integral over the emitting volume V,
where is the filter function of the telescope depending on wavelength , is the emissivity per unit emission measure, and is the electron density. The factors of Eq.(1) have been determined and integrated over the wavelengths observed by the SXT filters (Tsuneta et al. 1991, Fig. 9). The result can be written in the form
where the integral is along the depth of a pixel. The response function for the Al.1 filter has a broad peak at K, is rather flat above, and drops off rapidly below about K with a slope of approximately It is reduced by a factor of one hundred at K.
A formal temperature of K results from the ratio of the X-ray counts observed in the two filters. This temperature is a weighted average over the X-ray emitting material. It indicates that most of the emitting plasma is at a temperature in the regime of the response function. This makes clear that not much plasma exists in the quiet corona at temperatures above K.
If the kinetic pressure, , were constant in the field of view, an enhanced temperature would increase the X-ray intensity observed by SXT as . One may thus expect that regions of enhanced emission were hotter. On the contrary, the observed temperature of bright regions is generally similar or smaller than in the network cells. Therefore, the kinetic pressure must be larger in the X-ray bright regions of the quiet corona. In active regions, a correlation between SXR flux and gas pressure has been found by Yoshida & Tsuneta (1996). To reproduce the observed quiet Sun X-ray structures, the density must be enhanced in the sources above the network.
4.2. Structures in the radio emission
The radio emission of the quiet Sun is generally agreed to be mostly thermal free-free radiation (e.g. Chiuderi Drago et al., 1983). Nevertheless, the radiation is not known well enough to exclude occasional contributions of non-thermal emissions. In the absence of such evidence thermal emission is assumed here for interpretation. Intensities, I, are usually expressed in brightness temperature, , in radio astronomy using the Rayleigh-Jeans approximation for conversion
where is the observing frequency and the Boltzmann constant. In isothermal black bodies equals the kinetic temperature. The brightness temperature is the solution of the transfer equation
Neglecting the magnetic field, the optical depth of free-free emission is given by (e.g. Benz 1993)
The Gaunt factor, , is a slowly varying function of electron temperature, , and density amounting to about 8 in the upper chromosphere.
Eqs. (4) and (5) indicate that the effect of a temperature increase on the radio intensity is model dependent. Models of the upper chromosphere and transition region are still vividly disputed (e.g. Zirin et al. 1991). Chiuderi Drago et al. (1983) find increasing radio brightness from the network center (Vernazza et al. 1981, VAL model B), average network (VAL model D) to network elements (VAL model F) similar to the observed relative values. We note, however, that the VAL models yield an absolute brightness temperature that grossly exceeds the observed values (Zirin et al. 1991, Bastian et al. 1996).
The following investigation is based on the models of Fontenla et al. (FAL, 1993) which agree reasonably well with radio observations (Bastian et al. 1996). The differences in total brightness due to the slightly non-vertical ray path are less than 3 percent. Nevertheless, models should only be taken as a qualitative guide. Fig. 7 indicates the altitude of origin of the three radio wavelengths for the model quiet Sun. The contribution of the corona to the 3.6 cm emission is 5-10%, and an order of magnitude less at 1.3 cm.
The average model is now disturbed to find the necessary deviations for the observed radio structures. In Fig. 8 the temperature and density of the whole atmosphere (FAL model C) were multiplied by constant factors. Although based on unconfirmed models, the Figs. 8a and b give an estimate on the changes in temperature and density necessary to interpret the observations. The observed variations in brightness temperature (Table 1) can roughly be related to these changes in temperature and/or density. For all the models we have tried, the calculations predict an increase in brightness temperature for higher plasma temperature and/or density. The largest effect for a given change is at the longest wavelength. This is consistent with the observed increase of the contrast with wavelength as indicated in Table 1.
In addition, the comparison of observations with models requires that the factorial change increases with wavelength. If for example the temperature is constant in FAL model C, an rms fluctuation in density by 10.2%, 17.6%, 16.4%, and 31.3% is necessary to produce the standard deviations in brightness of 1.3cm, 2.0cm, 3.6cm, and X-rays, as reported respectively in Table 1. These values correspond to half the average density enhancement between cell interior and network at the source heights of the various radiations. It is noted, however, that the quantitative results depend strongly on the chromospheric model. The VAL model, for example, requires a factor of two smaller fluctuations to account for the radio variations.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998