2 Radiation transfer in inhomogeneous atmospheres

In order to retrieve magnetic field from intensity and polarization spectra emitted by an inhomogeneous solar atmosphere we use the radiation transfer equations for the Stoke's parameters of total intensity $% I=(T_{\rm r}^{+}+T_{\rm r}^{-})/2$ and $V=(T_{\rm r}^{+}-T_{\rm r}^{-})/2$ for circular polarization. These equations for the bremsstrahlung emission mechanism in an inhomogeneous plane layered atmosphere (see Appendix A) are:

$$I(\lambda )=\lambda ^{2}\int\limits_{0}^{\infty }T_{\rm e}(t){\rm e}^{-\lambda ^{2}t}{\rm d}t,$$ (1)

$$V(\lambda ) \lambda ^{2}\int\limits_{0}^{\infty }\beta (t)\triangle T(t){\rm e}^{-\lambda ^{2}t}{\rm d}t,$$ =

$$\triangle T(t) \int\limits_{0}^{\infty }\frac{{\rm d}T_{\rm e}(t+t^{\prime })}{{\rm d}t}% {\rm e}^{-\lambda ^{2}t^{\prime }}{\rm d}t^{\prime },$$ = (2)

where $\beta =2\nu _{B}\cos \alpha /\nu \ll 1$, $\nu _{B}=2.8\ 10^{6}B$is the electron gyrofrequency, $\nu =c/\lambda$, $\alpha$ is the angle between the magnetic field vector $\vec{B}$ and the line of sight and $T_{\rm e}$ is the electron temperature. The integration is carried out with respect to the argument t, the frequency-independent radio depth measure as a function of the current geometrical depth l from the top, based on the isotropical optical depth $% \tau (l)$

$$t\equiv \frac{\tau (l)}{\lambda ^{2}}=\frac{1}{\lambda ^{2}}%... ...0}^{l}\frac{N_{\rm e}^{2}}{T_{\rm e}^{3/2}}{\rm d}l^{\prime },$$ (3)

where $\ \mu ^{\rm o}\simeq 0.2\cdot \nu ^{-2}N_{\rm e}^{2}T_{\rm e}^{-3/2}$ is the bremsstrahlung opacity in an isotropic plasma with electron density $N_{\rm e}$.

The solution of the inverse problem given by Eqs. (1), (2) gives the magnetic field $B\equiv (5350/\lambda )\beta (t)$ and the electron temperature $T_{\rm e}(t)$ in the solar atmosphere as functions of t. We call such solutions "tomographic" models (see Bogod & Grebinskij [1997]) because they represent the only direct information, which may be extracted from brightness spectra without additional assumptions. The "physical" models, i.e. spatial distributions of the electron number density $N_{\rm e}(h)$, electron temperature $T_{\rm e}(h)$ and field B(h), may be retrieved from the tomographic models by solving the inversion problem Eq. (3), which requires the assumptions on the relation between $% T_{\rm e}(h)$ and $N_{\rm e}(h)$. This approach is quite different from the standard model method which starts with the information obtained from optical observations and then does forward calculations to fit microwave observations in the following steps: $T_{\rm e}(h)$, $N_{\rm e}(h)\Rightarrow t(T_{\rm e})\Rightarrow T_{\rm e}(t)\Rightarrow T_{b}(\nu )\$ (see Avrett [1997] for review).

It was shown by Bogod & Grebinskij ([1997]), that Eq. (1) may be effectively inverted to find $T_{\rm e}(t)$ by the differential deconvolution method (DDM). However, that technique is not applicable to Eq. (2), which cannot be reduced to a Laplace transform. Thus, to retrieve magnetic fields from spectral observations, we should use some estimates, well known from previous studies by Bogod & Gelfreikh ([1980]) in the form of (see Appendix A):

$$\lbrack \beta (\lambda )]=\frac{V_{\rm obs}(\lambda )}{\lambd... ...\rm G}\equiv \frac{5350}{\lambda _{\rm cm}}[\beta (\lambda )].$$ (4)

Here and below we use the square brackets to denote physical qualities, which are obtained by inversion from radio brightness spectra and to distinguish its presumed physical values (denoted without square brackets) in the atmosphere. This expression gives an exact solution of Eqs. (1), (2) for any inhomogeneous atmosphere with uniform magnetic field (see Appendix A).

In order to obtain such estimates from observations of solar atmosphere emission in the case of inhomogeneous magnetic fields, one should answer on two questions: (i) Which layer of the atmosphere (corona or chromosphere) gives the main contribution to the polarization at a given wavelength $\lambda$? (ii) What is the relation between the estimated magnetic field [ $B(\lambda )$ ] (Eq. 4) and its real value in the case of an inhomogeneous field B(h)?

In order to answer these questions, we use a priori information about the distribution of temperature in the solar atmosphere, which may be extracted from observed brightness spectra by Eq. (1) and modelling.

 

Table 1: Radio brightness excesses of plage areas, observed and modelled