We now consider the problem of discrimination between the chromospheric and coronal contributions to the polarized emission at different wavelengths bands and discuss the appropriate methods for magnetic field estimates. We rewrite Eqs. (11)-(12) as
Results of model calculations (Eqs. (13)-(14)) for a constant magnetic field (with ) are presented in Fig. 1 versus frequency for a plage region with parameters given in Eq. (10). As one can see from Fig. 1, clear dominance of either chromosphere or corona, both in intensity and polarization, exists only for high or low frequencies respectively. When one of these contributions becomes dominant, we can use Eqs. (13) and (14) (or Eq. 4) for magnetic field estimates in the corona and chromosphere:
|Figure 1: The modelled relative contributions of chromospheric and coronae emission to intensity (Stokes parameter ) and polarization (Stokes parameter ) for the plage region atmosphere. The spectral index is calculated for the total intensity (Stokes parameter )|
At long cm-wavelengths the coronal contribution becomes significant (Fig. 1), so that we could use Eq. (15a) to estimate the coronal magnetic field. However, the magnetic field in the photosphere and the corona varies continuously. Therefore we establish, for a simplified model atmosphere, the relation between the value of an inhomogeneous field (or its parameter ) at some height h and its estimated value , as derived from Eq. (4). We show that for magnetic fields varying between the chromosphere and corona the estimate from Eq. (4) gives the value of the magnetic field at the height of the transition region (TR), i.e. near the base of the corona, independent from wavelength and the magnetic field profile.
We can approximate such an inhomogeneous atmosphere with a simplified model for and as
(a) in any optically thick isothermal medium ( ), the radiation temperature ( ) equals the kinetic temperature: (black body radiation), with the absence of polarization ( );
(b) in the optically thin emission from the corona ( the polarization spectrum is proportional to the coronal brightness , and grows with , which is an observational signature of the coronal origin of polarization;
(c) in the optically thick regime ( ) polarization quickly decreases, as for any black-body radiation;
(d) in the intermediate emission regime where the polarization spectrum has a sharp maximum near wavelength determined by 1.
The bottom of the corona corresponds in our model to a normalised radiodepth of (Eq. 17), so that the model magnetic field strength at this level becomes:
As was shown above, the main contribution to the polarized emission originates
near the thin transition region. For this reason we can assume, that a
magnetic field is uniform in the region of formation of polarized emission
and use the equations above (Eq. (4) and Eqs. (13)) for the uniform case (
), as a first approximation.
Thus Eq. (4) can be used to estimate the magnetic field:
From a single-frequency observation we have an another estimate
for the magnetic field from Eqs. (13):
From the single frequency observations of plage regions we are unable to determine either n or Q. They can be provided by our model simulation for observations (see Table 1 and Fig. 1). In addition, we can use simplified equations for the limiting cases at high and low emission frequencies:
(a) at long wavelengths with coronal domination in intensity for q<<1, we have and (optically thin coronal radiation), that Eq. (15) gives the magnetic field near the transition region:
In order to illustrate both limiting cases, we calculated typical values for the expected Q-factor for Westerbork cm) and Nobeyama cm) plage observations. We find for a typical plage area(s) , and . Therefore, with Nobeyama observations at 17 GHz the chromospheric contribution is important for more precise estimates.
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