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

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4. Comparison between EUV and radio observations

The radio brightness temperatures have been computed from Eqs. 3 and 4, by substituting [FORMULA] with [FORMULA] and using [FORMULA], derived from the assumptions [FORMULA] and [FORMULA] mentioned above, in the refraction undex. Both the DEM functions displayed in Fig. 3 have been used in the calculation.

No substantial difference is found between the results obtained using the two density profiles (a ) and (b ) mentioned above. This is because the electron density only enters the refractive index. In the following we will therefore show only the results obtained using the hydrostatic equilibrium assumption, mentioning the other ones when necessary. The assumption of hydrostatic equilibrium in a coronal hole will be discussed in the next section.

Fig. 4 displays the radio brightness temperatures calculated using both the DEM curves reported in Fig. 3 and three values of the electron pressure [FORMULA]. The results obtained using the standard DEM definition show a rather good agreement with the observations at high frequencies, where the curves are almost independent of [FORMULA]. At low frequencies, on the contrary, the curves exceed the observed [FORMULA] and, moreover, they present a cut-off at a frequency decreasing with decreasing electron pressure.

[FIGURE] Fig. 4. Comparison between computed and observed radio brightness temperature. Values of [FORMULA] are [FORMULA] (dashed line), [FORMULA] (dotted line) and [FORMULA] (full line) cm-3 K-1. Left : standard DEM definition (up to [FORMULA] K); Right : Landi & Landini 1998 DEM definition, truncated at [FORMULA].

It could therefore appear that a good agreement with the observations could be found in the whole observed range of wavelengths, by decreasing furthermore the electron pressure. However an electron pressure lower than [FORMULA] cm-3 K-1 leads to [FORMULA] at [FORMULA] K and is therefore unacceptable.

The results obtained using the Landi & Landini 1998 DEM curve show the same problem of the cut-off at low frequencies. Moreover the computed [FORMULA] are lower than the observed ones in the whole frequency range and it is clear, from the trend of the curves, that there are no values of the electron pressure [FORMULA] able to provide radio brightness temperatures in agreement with the observations.

The reason of this disagreement is due, in our opinion, to have neglected the presence of an isothermal corona at the top of the TR. In the derivation of the DEM from EUV line intensities, the presence of an isothermal corona is in fact not properly considered as the coronal contribution to the intensity of EUV lines formed at temperatures close to [FORMULA] is included in the DEM .

The density profiles derived from the DEM , irrespective to the adopted assumption on the electron pressure trend, are therefore abruptly truncated at [FORMULA]. If the value of the electron density at [FORMULA], which depends on the assumed value of electron pressure, turns out to be larger than the critical density at a given frequency [FORMULA], no contribution to the [FORMULA] will be given by the atmosphere at all frequencies [FORMULA], thus producing the sharp cut-off noticed in Fig. 4.

If, on the contrary, the presence of a nearly isothermal corona, where [FORMULA] and hence the DEM cannot be defined, is assumed above the level where [FORMULA], we have still an emitting plasma, whose density slowly decreases with height up to extremely low values, that can provide a non zero [FORMULA] at any frequency.

Having this in mind, we have repeated the calculations using the DEM up to [FORMULA] and adding above this level an isothermal corona at [FORMULA].

If hydrostatic equilibrium is assumed, the coronal radio optical depth [FORMULA] can be analytically calculated obtaining:


when the critical density is located in the corona ([FORMULA]) and


when the radiation crosses the whole corona ([FORMULA]). In the above equations [FORMULA] is the scale height, [FORMULA] and [FORMULA] is the the electron density, derived from the assumed electron pressure trend, at [FORMULA].

The electron pressure [FORMULA] and the coronal temperature [FORMULA] (the upper limit of the integral in Eq. 8) are left as free parameters to be determined from the fit of radio data.

The fit of the radio spectrum has been done for three sample values of the coronal temperature, [FORMULA] and [FORMULA] K and several values of the electron pressure in the TR spanning from [FORMULA] to [FORMULA] cm-3 K. The results, obtained assuming the hydrostatic equilibrium in the whole considered portion of the atmosphere (assumption (b )) are compared with the observations in Fig. 5.

[FIGURE] Fig. 5. Comparison between computed and observed radio brightness temperature,in the assumption of hydrostatic equilibrium. [FORMULA] is the electron electron pressure at [FORMULA] K in units of [FORMULA] cm-3 K.

Unfortunately the lack of resolution at 164 MHz does not allow a precise determination of the coronal temperature, but only an upper limit, [FORMULA] K.

Very good fits are obtained for coronal temperatures [FORMULA] K assuming a value of the electron pressure [FORMULA] cm-3 K at [FORMULA] K. Identical results are obtained if a constant electron pressure (assumption (a )) is assumed in the TR with [FORMULA] cm-3 K. It must be pointed out that an electron pressure [FORMULA] cm-3 K at [FORMULA] K, in the hydrostatic equilibrium assumption, leads to electron pressure values of [FORMULA] cm-3 K at [FORMULA] K, thus indicating that the parameter affecting the radio data is the electron pressure at bottom of the corona and not its trend in the TR.

From these values of the coronal electron pressure the following electron densities are derived at the base of the corona: [FORMULA] cm-3 ([FORMULA] K) and [FORMULA] cm-3 ([FORMULA] K).

These values of the density can be compared with those obtained from density sensitive line ratios available in the dataset.

4.1. Density diagnostic

Electron densities have been measured at TR and coronal temperatures using the [FORMULA] 625.8/554.5 and [FORMULA] 341.9/(349.8+349.9) line ratios: the theoretical ratios have been calculated using the CHIANTI database (Dere et al. 1997, Landi et al. 1999), taking into account the effects of photoexcitation from photospheric radiation, as described by Young et al. 1999.

The resulting density values are reported in Table 3, together with the electron pressure [FORMULA] where [FORMULA] is the temperature of ion formation given by Eq. 1.


Table 3. Electron density and pressure measurement for the coronal hole.

It is interesting to note that neither the constant electron pressure nor hydrostatic equilibrium assumptions are satisfied by the data shown in the table, as the electron pressure varies by more than one order of magnitude between [FORMULA] K and [FORMULA] K, in conflict with our previous calculations.

It must be pointed out however that the density determination from the [FORMULA] is quite uncertain because the line is weak and blended with the [FORMULA] 624.94 Å , moreover its wavelength falls in a region of the detector where the calibration is rather uncertain (Landi et al. 1997).

The density values at the basis of the corona, determined from the fit of radio data are larger by a factor of about 2.5 than those derived from the [FORMULA] measurement reported in Table 3. They also exceed the average values ([FORMULA]) in polar coronal holes, reported by Fludra et al. 1999a, 1999b) but they agree very well with those determined from [FORMULA] line ratio by Del Zanna & Bromage 1997 in this same coronal hole (the elephant trunk hole), observed two rotations before. It must be noticed however that Del Zanna & Bromage 1997 and Fludra et al. 1999a, 1999b did not include photoexcitation from photospheric background radiation as a populating mechanism for the [FORMULA] ground levels. Neglecting this process leads to slightly overestimate the electron density.

The discrepancy between the electron density inferred from [FORMULA] and that derived from the fit of radio data can be, at least in part, ascribed to one of the following reasons: (i) The regions of the coronal hole where EUV line intensities and radio brightness temperature have been averaged are not the same. and, (ii) the scarce angular resolution at low frequencies overestimates the radio brightness temperature in the hole, thus requiring a higher electron density in the corona.

Another serious physical reason of the overestimation of the electron density derived from radio data will be discussed in Sect. 5.

4.2. Synthetic EUV spectra

The three models used to fit the radio data shown in Fig. 5 have been then used to compute the intensity of the EUV lines listed in Table 1: the resulting [FORMULA] ratios are plotted in Fig. 6 as a function of wavelength.

[FIGURE] Fig. 6. Comparison between theoretical and observed EUV line intensities.

Fig. 6 shows that synthetic EUV intensities of transition region lines are not affected by the choice of the [FORMULA] value, while coronal line intensities are very sensitive even to small changes of [FORMULA]. For this reason they are able to provide further constraints to the temperature in the coronal hole, which could not be achieved from our radio data, since only an upper limit of [FORMULA] at the lowest frequency is available. Fig. 6 shows that the best agreement between theoretical and observed intensities is reached at [FORMULA] K, which is consistent with the results obtained from radio data. Moreover, the top panel in Fig. 6 is able to supply a lower limit to the measured coronal temperature: EUV line intensities computed assuming [FORMULA] K are in fact too low.

It must be pointed out that the computed line intensities using both the DEM curves shown in Fig. 3, without considering the contribution from an isothermal plasma, are of course in very good agreement with the observations, being the curves derived from the best fit of the observed line intensities. This indicates that, contrary to what we found for radio data, EUV line intensities can be equally well reproduced by an atmosphere with or without an isothermal corona, provided that the upper limit of the DEM definition is properly set.

4.3. Reliability of the results

In the present work the new standard CDS intensity calibration (revised in late December 1998) has been used. To evaluate the effect of possible calibration uncertainties in our results, the whole study has been repeated using the old, pre-flight intensity calibration (Bromage et al. 1997), and theoretical radio brightness temperatures have been re-calculated. The differences between the resulting [FORMULA] values are negligible, being smaller than 1%.

Similar comparisons have been carried out using different sets of ion fractions (Shull & Steenberg 1982, Mazzotta et al. 1998). Again, differences on the resulting [FORMULA] values are negligible.

This means that the results of the present work are not affected by uncertainties neither in EUV intensity calibration nor in the adopted ion fractions.

The accuracy on the radio brightness temperature has been estimated around [FORMULA] K ([FORMULA] 10% of the measured [FORMULA] vlaues); moreover, another source of uncertainty in the present study comes from having compared EUV and radio data averaged over two different portions of the hole. In Sect. 2.2 this uncertainty has been estimated to be smaller than 15%: this is the main limitation to the reliability of the results obtained in the present work.

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© European Southern Observatory (ESO) 1999

Online publication: July 16, 1999