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3 Origin of polarization in the solar atmosphere

Brightness contrasts above the magnetic network from recent VLA - Yohkoh (SXT) observations of quiet Sun regions with arc sec resolution (at $\lambda $ = 1.3, 2.0, and 3.6 cm) were obtained by Benz et al. ([1997]). Model simulations with current optical reference atmospheres (Fontenla et al. [1993]) were inconclusive. In order to solve the problems of model simulations we use here the results of a multi frequency study (Bogod & Grebinskij [1997]) in a wide range of mm - cm band microwave brightness observations with moderate (arcmin) spatial resolution.

3.1 Model simulations of the quiet Sun and plage emission

The solar atmosphere at microwave frequencies can be considered (see Grebinskij [1987]; Zirin et al. [1991]) as a two-component medium with an optically thin corona and an optically thick chromosphere with small temperature gradients:

\begin{displaymath}T_{\rm b}(\lambda )=T_{\rm e,cor}\tau _{\rm cor}(\lambda )+T_{\rm b,chr}(\lambda).
\end{displaymath} (5)

Zirin et al. ([1991]) found the best fit to the quiet Sun brightness spectra with $T_{\rm b}=140077\cdot \nu ^{-2.1}+10880$ K in the frequency range 1.4 - 18 GHz, which corresponds to an optically thick and isothermal chromosphere with $T_{\rm e,chr}$ = 10880 K. Bogod & Grebinskij (1997) used a wider frequency range (1 - 90 GHz) and reported a slight power-law chromosphere electron temperature gradient $T_{\rm e,chr}(t)\sim t^{-q}$ from the inversion of the chromosphere emission spectrum at the short microwave lengths ( $\lambda< 3$ cm) for the quiet Sun and plage regions. We simulate these spectra by using a two component model (slightly anisothermic chromosphere and optically thin corona) of the solar atmosphere with several free parameters. Here, we would totally neglect all possible contributions to observed microwave emission from a solar plasma at transition temperature range. The matter of plasma is confined into unresolved small scale loops (see Grebinskij [1987] and Appendix B). For coronal layers the coronal brightness temperature follows:

$\displaystyle T_{\rm b,cor}(\lambda )$ = $\displaystyle \lambda ^{2}<tT_{\rm e}>_{\rm cor},$  
  < $\displaystyle tT_{\rm e}>_{\rm cor}\equiv \int\limits_{0}^{t_{\rm cor}}T_{\rm e}(t){\rm d}t,$ (6)

where $<tT_{\rm e}>_{\rm cor}$ is related to the column emission measure $%
EM(T_{\rm c})=\int N_{\rm e}^{2}{\rm d}l$ according to $<tT_{\rm e}>_{\rm cor}\simeq 2\
10^{-22}EM_{\rm cor}T_{\rm c}^{-0.5}$ and can be directly determined by the inversion of observed brightness spectra (see Bogod & Grebinskij [1997]).

For the chromosphere, we use a power-law electron temperature distribution, as

\begin{displaymath}T_{\rm e,chr}(t)=\left\vert
T_{\rm o}, & t\...{t}{t_{\rm o}})^{-q}, & t\geq t_{\rm o}
\end{array}\right. .
\end{displaymath} (7)

The modelled brightness spectrum of the chromosphere is obtained from Eqs. (1) and (7)

\begin{displaymath}T_{\rm b,chr}(\lambda )=T_{\rm o}(1-{\rm e}^{-\tau _{\rm o}})...
...-q,\tau _{\rm o}),~~\tau _{\rm o}\equiv \lambda ^{2}t_{\rm o},
\end{displaymath} (8)

where $\Gamma $ is the partial Euler Gamma function. Our model chromosphere is justified by the following reasons. For a power-law kinetic temperature distribution (Eq. 7) in the chromosphere the emission brightness temperature $%
T_{\rm b,chr}({\lambda })$ at fixed $\lambda _{*}$ corresponds to the electron temperature at the level with $\tau (\lambda _{*})\simeq
1$, or $t_{*}\simeq \lambda _{*}^{-2}$. From Eq. (7) we find $T_{\rm b,chr}(\lambda _{*})\simeq T_{\rm e,chr}(t_{*})=T_{\rm o}(t_{\rm o}\lambda
_{*}^{2})^{q}$ with the power-law brightness spectrum with spectral index n=2q. Thus, in order to match the observations, one should use: (i) the temperature index q, which is mostly defined by the shape of the observed brightness spectrum at the short microwave lengths; (ii) the chromospheric temperature parameter$T_{\rm o}$, which is the expected value of the electron temperature at the top of the chromosphere ( $T_{\rm e}\simeq
10-20\ 10^{3}$ K); (iii)the final adjustment of the optical depth scale $t_{\rm o}$ of the electron temperature $T_{\rm e}(t)$ with the chromosphere depth t.

The model has 4 free parameters ( $T_{\rm o},t_{\rm o},q$ and $<tT_{\rm e}>_{\rm cor}$). The same model is used both for the quiet Sun and plage emission, but with different parameters. These are found by modelling the observed brightness spectra.

As observational spectra for model matching, we use here a combination of OVRO (Zirin et al. [1991]) and compiled (Bogod & Grebinskij [1997]) mm-band observations for different quiet Sun regions (as reference) in the frequency range of 87 - 1.4 GHz. For plage area(s) we use observations from Metsähovi at 87 - 22GHz (Urpo et al. [1987]), Nobeyama maps at 17 GHz, RATAN at 1.4 - 16 GHz (Bogod & Gelfreikh [1980]; Akhmedov et al. [1982]), and Westerbork at 5 GHz (Kundu et al. [1977]). With plage data we use the reported results of reduction of observed brightness excesses for moderate beam resolution.

The results of the brightness spectra simulations are well fitted by simple model with the following parameters:

$\displaystyle {\rm Quiet~Sun:}~~~T_{\rm o}$ = $\displaystyle 15000~{\rm K},~~~t_{\rm o}=10^{-2.5},$  
q = $\displaystyle 0.095, ~~~<tT>_{\rm cor}\ =\ 95.$ (9)
$\displaystyle {\rm Plage:}~~~T_{\rm o}$ = $\displaystyle 15000
~{\rm K},~~~t_{\rm o}=10^{-1.9},$  
q = $\displaystyle 0.095, ~~~<tT>_{\rm cor}\ =\ 800.$ (10)

The results of the fits are presented in Tables 1 and 2. They are well inside reported error bars of the observed brightness temperatures at all frequencies and much better fit the observations than optical reference models (see Bastian et al. [1996] for a review).

Taking into account the parameters above we can simplify the Eq. (8). With q<<1 and $\tau _{\rm o}\equiv t_{\rm o}\lambda
^{2}<<1$ at $\lambda < t_{\rm o}^{-0.5}\simeq
10-18$ cm we can use the approximation $\Gamma (1-q,\tau _{\rm o})\simeq 1$ and $%
\tau _{\rm o}<<\tau _{\rm o}^{q}$ and obtain the total brightness spectrum

\begin{displaymath}T_{\rm b,tot}(\lambda )=t_{\rm o}^{q}T_{\rm o}\lambda ^{2q}+<tT_{\rm e}>_{\rm cor}\lambda ^{2}.
\end{displaymath} (11)

The condition of $\tau _{\rm o}<<1$ is valid through the entire frequency range ($\nu > 10$ GHz) of chromosphere plage emission with dominating coronal contribution. At longer wavelengths one should use Eq. (8) with $ T_{\rm b,chr}\simeq T_{\rm o} $. In Table  2 we give the observed and modelled brightness excess temperatures ($\Delta T$) of a plage region relative to the quiet Sun brightness at different wavelengths, together with the observed and modelled quiet Sun brightness temperature ( $T_{\rm QS}$) and spectral index the $n(\lambda )={\rm d}\log T_{\rm}/{\rm d}\log\lambda $ for the modelled total brightness.


Table 2: Quiet Sun brightness $T_{\rm b}$, observed and modelled
  Observed Modelled
$\nu$ $T_{\rm obs}$ $\sigma_{\rm obs}$ $T_{\rm mod}$ $T_{\rm obs} - T_{\rm mod}$
(GHz) (103 K) (103 K) (103 K) (103 K)
1.4 70.5 3.0 70.6 -0.1
1.6 63.8 2.8 58.6 5.2
1.8 52.2 2.5 50.0 2.2
2.0 42.9 1.9 43.5 -0.6
2.4 32.8 1.4 34.7 -1.9
2.8 27.1 1.1 29.1 -2.0
3.2 24.2 1.1 25.3 -0.9
3.6 21.7 1.1 22.5 -0.8
4.2 19.4 0.8 19.7 -0.3
5.0 17.6 0.8 17.2 0.4
5.8 15.9 0.7 15.6 0.3
7.0 14.1 0.6 14.0 0.1
8.2 12.9 0.6 13.0 -0.1
9.4 12.2 0.6 12.3 -0.1
10.6 11.3 0.5 11.7 -0.4
11.8 11.0 0.5 11.3 -0.3
13.2 10.8 0.5 10.9 -0.1
14.8 10.8 0.6 10.5 0.3
16.4 10.7 0.7 10.2 0.5
18.0 10.3 0.5 10.0 0.3
22.2 9.0 0.5 9.46 -0.5
36.8 7.8 0.5 8.40 -0.6
77.1 7.25 0.5 7.28 -0.03
87.0 7.2 0.5 7.11 0.09

The polarized emission component $V(\lambda )$ can be now obtained from Eqs. (2) and (A12). For a simple model with constant magnetic field one can obtain for the chromosphere and corona:

\begin{displaymath}V_{\rm tot}(\lambda )=\beta _{\rm chr}qt_{\rm o}^{q}T_{\rm o}\lambda ^{2q}+\beta
_{\rm cor}<tT_{\rm e}>_{\rm cor}\lambda ^{2}.
\end{displaymath} (12)

Here the magnetic fields are characterized by $\beta _{\rm cor}$ and $\beta _{\rm chr}$.

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