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Astron. Astrophys. 358, 741-748 (2000)

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3. Results and discussion

3.1. Emission measure distributions

A widely used method for the analysis and interpretation of solar observations is based on differential emission measure distributions (DEM ), e.g. Raymond & Doyle (1981). In the present study differential emission measures were produced using the CHIANTI software package (Del Zanna et al., 1997, Dere et al., 1997), the ionization equilibrium values of Arnaud & Rothenflug (1985) and the coronal abundances of Feldman et al. (1992). The DEM is given by:


where [FORMULA] is the electron density, and [FORMULA] is the reciprocal of the line of sight temperature gradient. We found it necessary to increase the abundance of oxygen by 0.2 dex, in order to obtain consistency between the oxygen and neon spectral lines.

Despite the large number of lines observed, several of these were unsuitable for DEM analysis due to blending problems or low intensities. For example, NeVI 558.59 Å is blended with NeVII 558.609 Å while FeXV 327.03 Å was found to have a low intensity in all the regions. The two helium lines are not used as they are both optically thick.

The emission measures produced using the remaining lines are shown in Fig. 2 for the internetwork, network and bright network areas. The differential emission measures are plotted versus the temperature where the product of the G(T) function and the DEM is at a maximum. Note that the NeVII 561.73 Å line is absent from both the internetwork and network regions.

[FIGURE] Fig. 2. Differential emission measures of (from left to right) the bright network, the network and the internetwork region. Temperatures shown are the product of G(T) [FORMULA] DEM(T) .

Raymond and Doyle (1981) found that, assuming a power law of the form EM[FORMULA], the emission measures for the network and internetwork had slopes ([FORMULA]) of 1.4 and 2.1 respectively. Recently, Griffiths et al. (1999) found that this slope varied from between [FORMULA]1.8 in the highest intensity regions up to [FORMULA]2.4 in the lowest intensity regions, for temperatures in the range of 5.2 [FORMULA] log Te [FORMULA] 6.2. Griffiths et al. argue that the trend of these results suggests a decrease in the emission measure slope with increasing intensity. Applying the same technique to our results we find slopes of 1.3 for the bright network, 1.6 for the network and 1.8 for the internetwork, between temperatures of 5.4 [FORMULA] log Te [FORMULA] 6.0. We also see a decrease in slope with increasing intensity, with the bright network region showing the lowest value of [FORMULA], although there may be some abundance related uncertainties in these values. Following the argument of Griffiths et al. (1999) these results suggest that different density structures are contributing to each internetwork and network region.

3.2. Filling factors of the emission

One of the outcomes of the image segmentation technique, is that it allows us to estimate the percentage area (spatial density) occupied by the internetwork, network and bright network emission in each line. In Fig. 3 we plot these area percentages as a function of the line formation temperature. Each of the distributions has been fitted with a second order polynomial.

[FIGURE] Fig. 3a - d. Percentage of a normal network emission, b internetwork emission, c bright network emission, and d combined network (normal network and bright network) emission.

Fig. 3a indicates that the area of network emission does not change significantly with temperature, with only a marginal decrease towards higher temperatures. However an examination of the rasters of eight representative low and high temperature lines, shown in Fig. 4, indicates that the network shape changes considerably at or about the temperature of the NeVII line. This is despite the fact that the overall percentage area of network emission remains more or less constant. Feldman (1987) used the fact that the network in low and high temperature lines is very different in appearance, e.g. OIII and SiX in our figure, to argue that unresolved fine structures extend in temperature up to at least that of the NeVII line, but not to coronal temperatures.

[FIGURE] Fig. 4a and b. a Rasters showing internetwork, network and bright network emission for the four low temperature lines, OIII , OIV ,NeIV , OV and b internetwork, network and bright network emission for the four high temperature lines, NeVII , SiVIII , MgIX , SiX . The bright network areas are delineated with white lines. In the rasters of OV and SiX we also delineate the internetwork areas with black lines.

In Fig. 3 the internetwork emission area shows a maximum increase at the temperature of the NeVI and NeVII lines. In comparison the combined network area (i.e. the area of emission from the normal and bright network) shows a minimum at the temperatures of NeVI and NeVII , before increasing again to reach a maximum at the temperature of FeXVI . Habbal & Grace (1991) interpreted a similar minimum in their estimates of spatial densities to indicate that within the structure of bright points there exist two favoured temperature populations of magnetic structures: one below 3 [FORMULA] 105K and one at coronal temperatures. We argue that our results also show the presence of two populations within the (combined) network, a low transition region group and a coronal group. However, we must be careful in making these assumptions as our technique for seperating the areas can assign a non-negligible number of pixels to the wrong distribution. While this technique still gives a valid estimate of the proportion of area in each distribution it can produce some uncertainties. An examination of the rasters plotted in Fig. 4, confirms that there is a minimum in the amount of network area at the temperatue of NeVII . It is also possible to see from these rasters that the amount of network area decreases from larger values at low temperatures to increase again at higher temperatures, albeit with a different appearance. This increase in network area at higher temperatures has been taken to signify the opening out of flux tubes or funnels (Gallagher et al., 1998, Patsourakos et al., 1999).

We note that the CaX and MgX lines show slightly anomalous values for the amount of combined network and internetwork area. For example, MgX shows a value that is significantly smaller than the values of other coronal lines in the combined network plot, while CaX shows a value significantly higher than similar temperature lines, such as SiVIII . These variations are possibly explained by the fact that both lines are blended to a small degree, MgX with a line of OIV 624 Å and CaX with the NeVI 558 Å line. These values should therefore be treated with some caution.

A plot of bright network emission, Fig. 3c, shows a rapid increase in the area above a temperature of log Te = 5.7 K. Again this can be qualitatively confirmed by examining the rasters in Fig. 4. Some bright points (e.g. at X=160, Y=70 arcsec) in this figure show no appreciable expansion with temperature, but overall the rapid expansion of the bright network to higher temperatures indicates the opening out of flux tubes, either in the form of funnels or loop structures into the corona. In order to examine the magnetic structure of the network we use an MDI magnetogram which was taken on the same day as the CDS observations (see Sect. 2.1). This is plotted in Fig. 5. We have overlain the magnetogram with bright network contours of OV and SiVIII in black and normal network contours of the same lines in white. In the magnetogram the intensity has been scaled to [FORMULA] 50 G to bring up the weaker field structure. Magnetic fields with values greater than this will thus be saturated in these images, appearing as either purely black or white, depending on their polarity. In Table 3 we show the distribution of magnetic field within the bright network, normal network and internetwork areas defined using the OV contours. The maximum magnetic field per pixel is 330.7 Gauss. It can be seen that almost 70% of the total magnetic field lies within the bright and normal network.

[FIGURE] Fig. 5. High-resolution MDI magnetograms showing the field concentrations in the quiet Sun. The magnetogram has been scaled to highlight weaker magnetic features. In the upper panel the magnetogram has been contoured with the OV (629 Å) bright network (black lines) and normal network (while lines) while the bottom panel shows the corresponding SiVIII (319 Å) contours.


Table 3. The distribution of magnetic field and radiative losses within the bright network, normal network and internetwork.

It is striking that the bright network emission in both the OV and SiVIII lines is concentrated precisely in those areas where there is a strong magnetic field, as indicated by the black and white patches. Note that the (coronal) bright point at approximately X=160, X=70 arcsec (160,70) corresponds to a strong unipolar magnetic region. No strong opposite magnetic polarity is visible. Similar bright network points are apparent at (60, 150), (40,85) and (95, 175). The area of emission over these bright points increases in the SiVIII line, signifying the opening out of a coronal funnel. As this magnetogram was obtained less than two minutes after the raster images, the magnetic structure did not have time to change significantly. These brightenings may be due to local density enhancements, caused by localized heating, along individual field lines emanating from patches of purely unipolar magnetic flux. In these regions there appear to be no magnetic bipoles present. This is in contradiction to the arguments of Dowdy (1993) and Falconer et al. (1998) that network brightenings over unipolar regions only appear to be present due to a failure to resolve the opposite polarity flux in the magnetograms. That argument does not seem to hold in this situation.

However other regions at (25,35) and (60,40) do show brightenings over bipolar magnetic field regions. Following the definition of Dowdy (1993) these are considered to be locations of network loops. The expansion of the bright network area in these locations indicates the expansion of network loops as they arch up into the corona. Furthermore, both loops and funnels must reach coronal temperatures as the expansion of these features in the bright network can be seen all the way from OIII up to SiX at log Te = 6.1 K (see Fig. 4). Note, however, that certain bright points in the low temperature lines do not show equivalent brightenings in the higher temperature lines, e.g. at (140, 175), (185, 185), (200, 160). These brightenings are present in OV (Fig. 5), but not in the higher temperature SiVIII line. They can similarly be seen in all the low temperature lines in Fig. 4 but not in the high temperature lines. These brightenings are interpreted as internally heated network loops without any strong thermal connection with the large-scale corona. In fact, the brightening at (140, 175) may be occurring over a unipolar region, suggesting a possible low temperature funnel structure.

3.3. Radiative losses

In Table 3 we present the total radiative losses calculated using our emission measures and the radiative loss function of Cook et al. (1989). The radiative loss function was scaled to the coronal abundances of Feldman et al. (1992). The radiative losses have been integrated over the temperature range log Te = 5.0-6.3 K (see Fig. 6a). Scaling these radiative losses to the area fractions shown in Fig. 3 for the three regions results in the curves shown in Fig. 6b. From this figure it is clear that the bright network contributes little to radiative losses below log Te = 5.7 K, due to its small spatial extent at these temperatures. However, between log Te = 5.7 and log Te = 6.3 as much as 30% of the total radiative losses is accounted for by bright network emission, and thus by continuous layers in a stratified atmosphere as suggested in the previous section. The integrated value of the scaled radiative losses over log Te = 5.0-6.3 K is shown in the final column of Table 3. The sum of these integrated values gives a value of 5.46[FORMULA]105 ergs cm-2 s-1 for the whole quiet Sun, agreeing well with the value of 6[FORMULA]105 ergs cm-2 s-1 calculated by Dere & Mason (1993).

[FIGURE] Fig. 6a and b. a The radiative losses shown for each of the three regions; bright network (solid line), normal network (dot-dash line) and cell (dashed line). b The scaled radiative losses for the same regions.

3.4. Geometry of the emission

Chae et al. (1998) showed that the variation with temperature of the relative cross-sectional area of a flux tube can be estimated from measurements of redshift velocity, if the pressure is presumed constant in the transition region and if the mass flux carried by the downward motion is conserved along the flux tube.

Using this technique they were able to produce estimates of the relative area of a flux tube from their redshift measurements which they then fitted with the functional form (see Dowdy et al., 1987 and Rabin, 1991);


where A(T) is the area of the flux tube as a function of temperature, A(Th) is the area at the hot end of the flux tube (in their case Th = 106 K), [FORMULA] is the value of the shape factor and [FORMULA] is the amount of constriction of the flux tube, [FORMULA]=A(Th)/A(Tc), with A(Tc) being the area at the cool (usually chromospheric) end. [FORMULA] is also sometimes referred to as the loop expansion factor. The shape factor [FORMULA], as its name suggests, determines the shape of the flux tubes (see Dowdy et al., 1985, 1987).

From their redshift measurements Chae et al. (1998) calculated values of [FORMULA] = 31 and [FORMULA] = 3.6 for the constriction and shape factor respectively. Inserting these values into Eq. 4 gives the curve with the short dashes shown in Fig. 7a. Similarly, using other redshift values from Brekke et al. (1997) we were able to calculate values of [FORMULA]=114 and [FORMULA]=2.2 and to produce the curve with the dots shown in Fig. 7a. Rabin (1991) showed that the hotter transition region plasma (5.0 [FORMULA] log Te [FORMULA] 6.0) can be explained as a conductive transition between the corona and the chromosphere if he modelled a bowl(horn) shaped funnel with [FORMULA]=4, [FORMULA]=2.5 and Th=[FORMULA] K. Using these estimates of [FORMULA], [FORMULA] and Th in Eq. 4 gives the dot-dash line shown in Fig. 7a.

[FIGURE] Fig. 7a and b. a Estimates of normalised cross-sectional areas calculated from the redshift values of Brekke et al. (1997), dotted line, Chae et al. (1998), dashed line and from the Rabin (1991) model parameters (dot-dash line). b The normalised area of the bright network emission overplotted with a thin black line fit, using the function in Eq. 4. Also plotted are the equivalent estimated areas from the Rabin coefficients (dot-dash line), the Brekke coefficients (dotted line) and the Chae coefficients (dashed line) together with a fit to the observed bright network area above log Te=5.6 K (long dashes).

In Fig. 7b we plot as triangle symbols the normalised area of bright network emission as a function of temperature. Fitting these values with the functional form shown in Eq. 4, where in this case we chose Th = [FORMULA] K, the temperature of FeXVI , gives values of [FORMULA] = 6.7 and [FORMULA] = 2.3. Inputing these values back into Eq. 4 results in the curve shown in Fig. 7b as a continuous thin black line. Using the estimated values of [FORMULA] and [FORMULA] from Chae et al. (1998), Brekke et al. (1997) and Rabin (1991), but with Th=[FORMULA] K to compare with our results, gives the different dashed and dotted curves shown in Fig. 7b.

It is clear from Fig. 7b that the estimated cross-sectional areas are similar in shape to the fit of our measured bright network emission areas. This is particularly true in the case of the areas estimated from the Brekke et al. (1997) redshifts, especially at temperatures above [FORMULA]log Te=5.6 K. If we only fit the measured values above log Te=5.6 K with the functional form of Eq. 4 (i.e. if we remove the effects of the low temperature lines) we obtain the curve shown in Fig. 7b with long dashes. We note that this curve has a very similar shape to the estimated areas from the Brekke et al. (1997) redshifts as can be seen in the figure. This new curve has [FORMULA]=2.36 and a large [FORMULA]=2625. The fact that the estimated cross-sectional areas (from Chae et al.,1998 and Brekke et al., 1997) have a similar shapes to the observed area expansion of the bright network strongly suggests that what we are seeing in these bright network regions (at least above [FORMULA]log Te=5.6 K) is an expansion of flux tubes, either in loops or funnels, with a corresponding downflow of redshifted material.

Some recent work of Teriaca et al. (1999), Peter (1999) and Peter & Judge (1999) has found evidence for a net blueshift above temperatures of log Te=5.7 K, in contradiction to Chae et al. (1998) and Brekke et al. (1997). Peter (1999) discusses these blueshifts and points out that they may be evidence for downward propagating waves induced by nanoflares in loops, similar to the model by Hansteeen (1993). This is also the conclusion reached by Teriaca et al. (1999). However Peter (1999) mentions another possibility, in that the observed blueshifts may be the solar wind in the transition region. He further suggests that the redshifted material may originate in the cool loops below temperatures of 5 [FORMULA] 105 K, with the higher temperature blueshifted material (the slow solar wind) originating in the regions in between (i.e. the coronal funnels). In producing the cross-sectional estimates of the Brekke et al. (1997) redshifts we assumed that there was a constant downward flux present. However if above 5 [FORMULA] 105 K there are blueshifts present (as shown by Peter & Judge,1999 etc.) we can assume that above this temperature there is a constant upward flux of material. If so, then the fact that the estimated cross-sectional areas from Brekke et al. (1997) above [FORMULA]5 [FORMULA] 105 K (i.e. log Te=5.7) are so similar to our fits of the observed bright network emission area would lead us to conclude that if there were blueshift velocities present, of the same magnitude as the Brekke et al. (1997) redshifts above [FORMULA]5 [FORMULA] 105 K, then they would provide the same close correspondence with our observed results.

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Online publication: June 8, 2000