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Astron. Astrophys. 342, 563-574 (1999)

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3. The analysis of NIXT data

3.1. The method

The approach to deriving the physical conditions inside the loops seen by NIXT, in a single narrow spectral band, is to compare the brightness distribution along the observed loop structures to that resulting from hydrostatic loop models. The SXT observations confirm that the emitting structures do not change significantly during the observation over a time scale of [FORMULA] minutes, according to the general evidence that such structures live over time scales longer than the radiative and conductive cooling times (hours).

Therefore, in agreement with the hypothesis of Paper I, we consider a hydrostatic model of a coronal loop, semicircular, of constant cross section, symmetric with respect to its apex; the equations are those of plasma hydrostatic equilibrium and of energy balance among radiative losses, thermal conduction and a phenomenological term of local heat input in the plasma. The lower boundary of the loop is set at [FORMULA] K. We refer to Paper I for further details.

The models are uniquely identified by [FORMULA], the plasma pressure at the base of the arch and by L the loop semi-length, which determine [FORMULA], the maximum plasma temperature, located at the loop apex; these parameters are linked by the scaling laws of Serio et al. (1981).

Paper I shows that the brightness distribution along the loops, as observed by NIXT, depends on the loop base pressure. For high base pressure ([FORMULA], for loops with [FORMULA] cm) the loop brightness is high and has a sharp maximum at the loop base, whereas the brightness is much more uniform (and lower) at lower pressure.

Paper I concludes that bright footpoints can high pressure coronal regions in NIXT pictures. On the other hand, faint loop structures, if detected at all, are expected to be observed as entire loops in NIXT images, given the uniform brightness distribution predicted by the modeling.

Given the property that the profile of the brightness along the loop depends mostly on the plasma pressure, the loop plasma pressure can be determined by matching the NIXT brightness distribution, synthesized from the loop models, to the observed one. However obtaining a brightness distribution directly comparable with models is, in practice, very difficult, because the loop orientation is unknown. Furthermore the brightness profiles derived along the loops typically show fluctuations which make an accurate comparison even more difficult.

Therefore we have chosen to take the brightness contrast ([FORMULA]) of the footpoints to the apex of the loop as indicator of the brightness distribution shape. Operatively the loops that we analyze are those for which we are able to mark an arc between two footpoints, and this is possible only as long as the footpoints are brighter than the rest of the loops. This implies that our selected loops will all have [FORMULA], i.e. they are relatively high pressure loops. Of course, this implies a selection effect, but it is unavoidable also because low pressure loops are faint.

Even the contrast [FORMULA] depends moderately on the loop orientation with respect to the observer, as shown in Fig. 4, which reports the contrast [FORMULA] between the luminosity in the NIXT band at the base and at the apex of model loops of various lengths vs. the base pressure. The luminosity is obtained by integrating the emission per unit volume on the labeled bin size (in pixels) along the loop. Each column shows the resulting contrast for a different extreme orientation: side view and top view, respectively. The contrasts are higher for loops seen from the top, as may be expected since the footpoint luminosity includes a significant vertical section of the loop.

[FIGURE] Fig. 4a and b. Luminosity contrast of loop footpoint with respect to the loop apex in the NIXT band vs pressure, as predicted from loop model. Each solid line pertains to model loops with the same length. The lowest line is for the shortest loop ([FORMULA]), higher lines are for longer and longer loops with the labeled step of increasing length. The three rows of figures are for different sizes (in pixels, in the figure labels) of the length step along the loop over which the luminosity is integrated. In the first column the contrast is computed assuming the loop as seen from above, the second as seen with a front view.  

For a given semilength and for low contrasts, [FORMULA] generally increases with the pressure as expected. However, above a contrast threshold, which depends on the spatial integration step, on the loop orientation and on the loop length, the contrast has a maximum and then decreases. For such high contrasts therefore we do not have a univocal pressure diagnostics. This limit does not affect our results: we have been able to find univocal pressure values for all our selected loops. Since all selected loops are well inside the solar disk we have neglected the effect of different orientations with respect to the loop axis.

The loop models yield results per unit cross sectional area. The expected total loop luminosity is then obtained by multiplying the model brightness integrated along the loop by the loop cross-sectional area. This luminosity can be compared to the luminosity measured on the calibrated images. The ratio of the two provides a quantitative measurement of the volume fraction effectively occupied by the emitting plasma, i.e. the loop plasma filling factor in the instrument band. This ratio is expected to be always smaller than (or at most equal to) unity, i.e. the model luminosity should always be larger than (or equal to) the measured luminosity. Errors in the aspect determination and on the loop cross section however allow for ratios larger than 1.

3.2. The results

3.2.1. The pressure

In order to apply the pressure diagnostics we first evaluate the half-length L of the selected loops. Assuming semicircular loops (such an assumption is not critical), the half-length can be obtained from the distance between the footpoints d, [FORMULA]. For d we have taken the linear distance between the brightest pixels at the two opposite sides of a loop. This was not possible for region D , for which, instead, we have taken the distance between the magnetic poles from the magnetograms. As error on d we take the root sum square of the uncertainties on the positions of the two footpoints. To be conservative, each uncertainty is estimated as half of the full width half maximum (FWHM/2) of the one of two brightness profiles (the more well-behaved) perpendicular to the magnetic field lines, taken close to the footpoints. The values obtained for L are shown in Table 1, and are all between [FORMULA] and [FORMULA] cm.

Then we extract the brightness distributions along each loop. We first mark a sequence of pixels which follow the loop structure as close as possible and connect them by a line. In order to increase the signal to noise ratio, we consider a strip of pixels centered on this path, and divide this strip into sectors with approximately the same area (i.e. including the same number of pixels). The FWHM of the transveral brightness distribution described above is taken as the strip width x. As sector size along the loop we take the characteristic length in which the brightness decreases from the maximum to 1/3 of the maximum, and in such a way that: i) the number of sectors is odd, so that the central sector, usually the one containing the brightness minimum, is univocally and well identified; ii) the number of sectors is [FORMULA], enough to obtain a proper brightness distribution.

The pixel brightness has then been averaged on each sector and the resulting brightness distributions along the paths are shown in Fig. 5.

[FIGURE] Fig. 5. Loop brightness profiles along the paths drawn in Fig. 3. The brightness is averaged in the sectors marked in Fig. 3.

As described in Sect. 3.1, the loop pressure is evaluated from the brightness contrast between the footpoints and the apex. Before computing this contrast, the background brightness is subtracted from the brightness along the loops. The background brightness for each loop has been evaluated in a small region outside of the loop, close to the loop apex, the point of minimum brightness. Its value changes from region to region, and ranges from 10% to 40% of the brightness at the loop apex.

For each loop we obtain two values of the contrast, one for each footpoint ([FORMULA] and [FORMULA]), which in general differ from each other. We have then taken the average [FORMULA] as the best contrast estimate and [FORMULA] as its uncertainty. We have estimated this uncertainty to be dominant with respect to others, such as fluctuations, statistical noise, and it includes that due to the chance alignment of other structures. We have verified on strips wider than 2 pixels, that halving the strip width changes the contrast values by less than 10%.

In order to compare homogeneous quantities, the plasma emission synthesized from the models is binned along the loop, with bin sizes of the same length as that of the loop sectors selected on the NIXT image, i.e the number of pixels y in Table 1. The resulting contrasts are shown in Table 1.

Data and models can now be compared on homogeneous ground. For a detailed comparison, we have generated a grid of loop models as fine as to minimize the uncertainty due to the discreteness of the grid. For each estimated loop length, the base pressure of the model loops range between 0.1 and 10 [FORMULA], increasing with logarithmic step [FORMULA]. The jump of pressure from one model to the next is [FORMULA] whereas the average uncertainty from the data is [FORMULA]%. The brightness contrast between the base and the top sector is computed considering the two possible extreme loop orientations with respect to the observer, as seen from the front and from above (see Sect. 3.1). The loop morphology is taken as good evidence that loops in regions A  and D  are seen from the front, and therefore their contrast can be compared to the contrasts in the first column of Fig. 4, and that the others are seen from above (second column in Fig. 4).

The best value of the pressure of the observed loop with contrast [FORMULA] is the pressure of the model whose contrast is the closest to [FORMULA]. The uncertainty on the pressure is determined from [FORMULA] in the same way. The pressure values p are shown in Table 1, together with the loop maximum temperature [FORMULA] estimated from the scaling laws of Serio et al. (1981).

We have verified that the other loop orientation reduces the pressure by a factor [FORMULA] for loops A  and D , and increases it by a factor [FORMULA] for the others. The values of the NIXT filling factor, discussed in the following, are instead much less affected by this assumption, except for that of loop A , which becomes larger than unity if the loop is assumed to be seen from above, clearly an unreasonable result.

The results can be summarized as follows:

Loop A: The resulting relatively low pressure value is consistent with the loop being a large scale structure (the diameter is [FORMULA]% of the solar radius).

Loops B, C, E: Although their dimensions are typical of active region loops, the pressure values are somewhat lower.

Loop D: This loop of intermediate size has an intermediate pressure value.

For all the structures the temperature estimated from the scaling laws is close to that to which NIXT is most sensitive.

In general we notice that the smallest selected loops tend to yield higher pressures than the largest ones. We cannot exclude a selection effect of the instrument passband, which tends to detect more easily structures with plasma at temperatures close to the temperature of maximum sensitivity. From the point of view of the scaling laws this is equivalent to fixing the product [FORMULA], so that smaller L implies larger p.

3.2.2. The volume filling factor

The length L and the pressure p uniquely identify a loop model among those of Serio et al. (1981). We now compare the total observed loop luminosity to that expected from the corresponding model. For a homogeneous comparison with the image calibrated brightness, we integrate the emission per unit volume derived from the model on the sector length y, multiplied by the width x, and divided by the same sector section z as the one used for the data.

We find that the model brightness is sistematically larger than the measured one, consistent with the expectation of plasma volume filling factors generally smaller than 100%. The estimated loop volume filling factor (in the NIXT band), computed as the ratio of the measured and expected brightnesses, is shown in the last row of Table 1. The uncertainties take into account also an average data calibration error of 20%. The results are:

Loop A: the filling factor is relatively high, comparable to unity.

Loops B, C, D, E: the filling factor is very small, between 0.1% and 1%.

Such a wide range of filling factor values, spanning three orders of magnitude, excludes the presence of dominant calibration systematic effects on the results.

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© European Southern Observatory (ESO) 1999

Online publication: February 22, 1999
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