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4 Methods of magnetic field measurement

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

$\displaystyle I_{\rm obs}$ =$\displaystyle I_{\rm chr} + I_{\rm cor},$
$\displaystyle V_{\rm obs}$$\textstyle = V_{\rm chr} +
V_{\rm cor}$$\displaystyle =\beta _{\rm chr}q I_{\rm chr} + \beta _{\rm cor}
I_{\rm cor},$ (13)


\begin{displaymath}I_{\rm chr}=\tau _{\rm o}^{q}T_{\rm o},~~\tau _{\rm o}=t_{\rm...
^{2},~~I_{\rm cor}\ =\ <tT_{\rm e}>_{\rm cor}\lambda ^{2}.
\end{displaymath} (14)

We note from Eqs. (13) that the contributions of the chromosphere and corona to intensity and polarization are different: if, for example, $\beta _{\rm chr}=\beta _{\rm cor}$, then the contribution of the chromosphere to $V_{\rm obs}$ is reduced by a factor q relative to contribution from corona. For $q\ll 1$, as found by our model, the reduction is significant. The chromospheric contribution both to intensity and polarization will always dominate the coronal contribution at shortest wavelengths, because the coronal brightness decreases faster (with $\lambda ^{2}$) than that of the chromosphere (with $\lambda ^{2q}$) - see Eq. (14). Thus, there are 3 typical frequency ranges (a, b and c): (a) the corona dominates both in intensity and polarization, (b) the chromosphere dominates both in intensity and polarization and (c) mixed regime with a dominating chromosphere in intensity and dominating corona in polarization.

Results of model calculations (Eqs. (13)-(14)) for a constant magnetic field (with $\beta _{\rm chr}=\beta _{\rm cor}$) 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:

\begin{displaymath}\lbrack B_{\rm cor}]=\frac{5350}{\lambda }\frac{V_{\rm cor}}{...
\frac{5350}{\lambda }\frac{V_{\rm chr}}{qI_{\rm chr}}\cdot
\end{displaymath} (15)

\includegraphics[width=8.8cm,clip]{ds8262f1.eps}\end{figure} Figure 1: The modelled relative contributions of chromospheric and coronae emission to intensity (Stokes parameter $%
I_{\rm chr}/I_{\rm cor}$) and polarization (Stokes parameter $V_{\rm chr}/V_{\rm cor}$) for the plage region atmosphere. The spectral index $n\equiv {\rm d}\log I_{\rm tot}/{\rm d}\log
\lambda $ is calculated for the total intensity (Stokes parameter $%
I_{\rm tot}\equiv I_{\rm chr}+I_{\rm cor}$)

4.1 Coronal emission

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 $\beta (h)$) at some height h and its estimated value $[\beta ]$ , 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 $T_{\rm e}(t)$ and $\beta (t)$ as

\begin{displaymath}T_{\rm e}(t)=T_{\rm chr}+T_{\rm cor}{\rm e}^{-t/t_{\rm cor}},~~~\beta (t)=\beta
_{\rm o}(1-{\rm e}^{-t/t_{\beta }}).
\end{displaymath} (16)

This model may be treated as an atmosphere with an optically thick chromosphere ( $T_{\rm chr}(t)={\rm const}$) and a hot corona ( $T_{\rm cor}>>T_{\rm chr})$ with the optical depth $\tau _{\rm cor}=\lambda ^{2}t_{\rm cor}$ depending on the changing magnetic field. Its strength is constant ( $\beta (t)=\beta _{\rm o}$) in deep layers ( $t>t_{\beta }$), with a rapid drop in the top layers ( $%
t<t_{\beta }$), as it is expected for solar magnetic fields. The radiation transfer Eqs. (1)-(2) give then

\begin{displaymath}I(\lambda )=T_{\rm chr}+\frac{\tau _{\rm cor}}{1+\tau _{\rm cor}}T_{\rm cor}\equiv
I_{\rm chr}+I_{\rm cor},
\end{displaymath} (17)

\begin{displaymath}V(\lambda )=\beta _{\rm o}\frac{\tau _{\rm cor}}{(1+\tau _{\r...
1+(1+\tau _{\rm cor})t_{\beta }/t_{\rm cor}}T_{\rm cor},
\end{displaymath} (18)

which leads to the estimate of the magnetic field through Eq. (4):

\begin{displaymath}\lbrack \beta (\lambda )]\equiv \frac{V(\lambda )}{\lambda
...1}{1+(1+\tau _{\rm cor})t_{\beta
}/t_{\rm cor}}\beta _{\rm o}.
\end{displaymath} (19)

The following results (see also Grebinskij [1985]) are typical for the polarization spectrum $V(\lambda )$ forming in an inhomogeneous medium:

(a) in any optically thick isothermal medium ( $T_{\rm e}=T_{\rm chr}$), the radiation temperature ( $T_{\rm rad}$) equals the kinetic temperature: $%
T_{\rm rad}=T_{\rm chr}$ (black body radiation), with the absence of polarization ( $%
V_{\rm chr}=0$);

(b) in the optically thin emission from the corona ( $\tau _{\rm cor}<<1)$ the polarization spectrum $V(\lambda )$ is proportional to the coronal brightness $I_{\rm cor}=\tau _{\rm cor}T_{\rm e,cor}$, and grows with $\lambda ^{3}$, which is an observational signature of the coronal origin of polarization;

(c) in the optically thick regime ( $\tau _{\rm cor}>>1$) polarization quickly decreases, as for any black-body radiation;

(d) in the intermediate emission regime where $\tau \approx 1$ the polarization spectrum has a sharp maximum near wavelength $\lambda _{\rm cr}$determined by $\tau _{\rm cor}(\lambda _{\rm cr})\simeq $ 1.

The bottom of the corona corresponds in our model to a normalised radiodepth of $%
t=t_{\rm cor} $ (Eq. 17), so that the model magnetic field strength $%
\beta (t_{\rm cor}) $ at this level becomes:

\begin{displaymath}\beta _{\rm TR}\equiv \beta _{\rm o}(1-\exp (-t_{\rm cor}/t_{\beta })),
\end{displaymath} (20)

so that the magnetic field (Eq. 16b) in the transition region is governed by $%
\gamma =t_{\rm cor}/t_{\beta }$:

\begin{displaymath}\beta_{\rm TR}=\beta_{\rm o},~\gamma>>1;~~~\beta_{\rm TR}=\gamma
\beta_{\rm o},~\gamma<<1.
\end{displaymath} (21)

The estimated magnetic field (Eq. 19), as expected from the emission spectra (Eqs. 17, 18) through Eq. (4), corresponds to the model field at TR well. For an optically thin model corona (with $\tau _{\rm cor}<<1$), we find from Eq. (19):

\begin{displaymath}\lbrack \beta (\lambda )]=\beta _{\rm o},~\gamma >>1;~~~[\beta
(\lambda )]=\gamma \beta _{\rm o},~\gamma <<1,
\end{displaymath} (22)

which are the same as for the simple model (Eqs. (16) and (21)). Thus, we conclude, that the method (Eq. 4) gives a field value corresponding to the coronal base near the transition region:

\begin{displaymath}\lbrack \beta (\lambda )]\equiv \frac{V(\lambda )}{\lambda
^{2}{\rm d}I/{\rm d}(\lambda ^{2})}\simeq \beta _{\rm TR}.
\end{displaymath} (23)

4.2 Practical estimate methods

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 ( $%
\beta _{\rm cor}=\beta _{\rm chr}=\beta _{\rm o}$), as a first approximation. Thus Eq. (4) can be used to estimate the magnetic field:

$\displaystyle \lbrack B]_{\rm G}$ = $\displaystyle 10700\frac{\rho }{n\lambda _{\rm cm}},$  
n $\textstyle \equiv$ $\displaystyle \frac{{\rm d}\log I_{\rm obs}}{{\rm d}\log \lambda },~~~\rho =\frac{V_{\rm obs}}{%
I_{\rm obs}},$ (24)

where n is the spectral index of the brightness temperature spectrum $I_{\rm obs}$, and $V_{\rm obs}$ is the observed radiative temperature of polarized emission. We present the typical values of the spectral index in Table  1 and Fig.  1.

From a single-frequency observation we have an another estimate for the magnetic field from Eqs. (13):

\begin{displaymath}\lbrack B]_{\rm G}=5350\frac{1}{\lambda _{\rm cm}}Q(\lambda
)\frac{V_{\rm obs}}{I_{\rm obs}},
\end{displaymath} (25)

where we have introduced the scaling factor $Q(\lambda )$ as

\begin{displaymath}Q=\frac{1+\frac{I_{\rm chr}}{I_{\rm cor}}}{1+q\frac{I_{\rm chr}}{I_{\rm cor}}},1<Q<\frac{1}{q%
\end{displaymath} (26)

Here $I_{\rm chr},\ I_{\rm cor}$ are partial contributions to the total observed brightness $I_{\rm obs}=I_{\rm chr}+I_{\rm cor}$. For the case of a homogeneous magnetic field, the two estimates should give the same result, so that n=2/Q.

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 $I_{\rm chr}/I_{\rm cor}<<1$for q<<1, we have $Q\simeq 1$ and $n\simeq 2$ (optically thin coronal radiation), that Eq. (15) gives the magnetic field near the transition region:

\begin{displaymath}\lbrack B]_{\rm G}=\frac{5350}{\lambda}\frac{V_{\rm obs}}{I_{\rm PL}},\ %
I_{\rm PL}\equiv \delta I_{\rm PL}+I_{\rm QS},
\end{displaymath} (27)

where $I_{\rm PL}$ is the observed total intensity of the plage region. The observed brightness excess $\delta I_{\rm PL}$ above the quiet Sun level $I_{\rm QS}$is related to the coronal parameters by

\begin{displaymath}<tT_{\rm e}>_{\rm PL}\simeq \lambda ^{-2}\delta I_{\rm PL}+<tT_{\rm e}>_{\rm QS}.
\end{displaymath} (28)

The value of $<tT_{\rm e}>_{\rm PL}$ is useful for checking if model and observations are consistent, because from a previous study (Bogod & Grebinskij [1997]) we have a priori values for these parameters close to $%
<tT_{\rm e}>_{\rm PL}\ \simeq\ 900$ and $ <tT_{\rm e}>_{\rm QS}\ \simeq\ 100$. (b) chromospheric domination in intensity $I_{\rm chr}/I_{\rm cor}>>1$ at short wavelengths. We have Q>>1, because q<<1. In the limiting case Q=1/q and n=2q<<1. This case may be important for measurements of umbral magnetic fields, where the observed polarization at the mm wave band may be extremely small for the relatively strong magnetic fields, due to the reduction of the coronal emission measure relative to plage areas as known from EUV observations.

In order to illustrate both limiting cases, we calculated typical values for the expected Q-factor for Westerbork $(\lambda =6.0$ cm) and Nobeyama $%
(\lambda =1.76$ cm) plage observations. We find for a typical plage area(s) $%
Q(\lambda =6.0~{\rm cm})=1.2$, and $Q(\lambda =1.76~{\rm cm})=3.9$. Therefore, with Nobeyama observations at 17 GHz the chromospheric contribution is important for more precise estimates.

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