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Astron. Astrophys. 348, 261-270 (1999)

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3. Data analysis

3.1. DEM analysis with CDS

The line intensities reported in Table 1 have been used to determine the coronal hole DEM , defined as

[EQUATION]

Element abundances have been taken from Feldman et al. 1992 (coronal values). Ion fractions come from Arnaud & Raymond 1992 (Fe ions) and Arnaud & Rothenflug 1985 (other ions). To determine the DEM of the emitting plasma we used the iterative method described by Landi & Landini 1997 to which we refer for details.

The DEM curve is usually defined in a temperature range including plasma from chromospheric to coronal conditions; above the coronal temperature, where no more lines are detected by the instruments, the DEM function is very often extrapolated to a very low value at a very high temperature ([FORMULA]). This causes the DEM to have a sharp maximum around the coronal temperature. Landi & Landini 1998 have criticized this assumption, since a decreasing DEM at high temperatures implies an increasing temperature gradient [FORMULA], which would produce a physically unacceptable high temperature. It would also pose problems to the energy balance equations, because the temperature gradient is linked to the heat conductive flux. Consequently, Landi & Landini 1998 proposed to truncate the DEM at a maximum temperature [FORMULA] beyond which the DEM is not defined; in this way [FORMULA] does not increase when the temperature approaches [FORMULA].

In this paper we will use both methods to determine the DEM from the observed EUV line intensities in the coronal hole. These DEM curves will be then tested by computing the radio brightness temperature and comparing it with the observations.

The DEM fits are displayed in Fig. 3 together with the data points. It must be pointed out that, in the data points plotted in the figure, the Fe, Mg and Si abundances from Feldman 1992 have been lowered by a factors ranging from 3.4 to 4.0, getting a good agreement with the Grevesse & Anders 1991 photospheric values. This indicates that the First Ionization Potential effect (FIP effect - see e.g. Haisch et al. 1996) cannot be applied to this dataset, in agreement with the findings of Del Zanna & Bromage 1999.

[FIGURE] Fig. 3. DEM for the coronal hole region. Left : usual definition for the DEM . Right : DEM truncated at [FORMULA] according to Landi & Landini 1998. Experimental DEM constrains provided by each observed line are overplotted on the DEM .

The strong peak in the DEM curve displayed in Fig. 3 (left) occurs at temperatures slightly smaller than [FORMULA] K, while the DEM curve displayed in Fig. 3 (right) is defined up to [FORMULA] K.

3.2. Radio brightness temperature

The radio brightness temperature [FORMULA] is related to the plasma parameters through the transfer equation:

[EQUATION]

where

[EQUATION]

is the differential radio optical depth. In this equation l is the coordinate along the ray path and n is the refractive index:

[EQUATION]

where [FORMULA] is the critical density. Since the coronal hole is located near the disk center, the ray path is a straight line, [FORMULA] and Eq. 4 may be re-written as

[EQUATION]

[FORMULA] is the optical depth of the level where [FORMULA] and [FORMULA]. At very short wavelengths, where the refractive index [FORMULA] in the TR and corona, [FORMULA] can be directly derived from the EUV lines intensity through the DEM without any further assumption and the integral in Eq. 3 is performed up to [FORMULA]

At the frequencies considered in this paper this approximation does not hold, as the critical density is located in the high TR or in the corona.

In order to take into account the refractive index n in Eq. 4, we must know the electron density profile in the upper part of the solar atmosphere. This has been done in the two following ways:

  • (a) from the simple assumption of a constant electron pressure in the TR (hereafter defined as [FORMULA]), getting [FORMULA]

  • (b) by combining the DEM profile with the assumption of the hydrostatic equilibrium:

    [EQUATION]

    getting

    [EQUATION]

    where [FORMULA] K cm -1 and C is an integration constant related to the electron pressure at the temperature [FORMULA]: [FORMULA].

The lower temperature in the above integral has been arbitrarily set [FORMULA] K: this value does not affect the calculations since all considered radio frequencies have their critical level at temperature [FORMULA] and therefore the integral of the transfer equation stops at higher temperature.

For every value of [FORMULA], used as an upper limit of the integral in Eq. 8, there exists a corresponding minimum value of the electron pressure [FORMULA], given by

[EQUATION]

Pressure values lower than [FORMULA] would give [FORMULA] at [FORMULA].

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© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999
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