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1 Introduction

The structure of coronal magnetic fields plays a crucial role in problems of heating and flare energy release, but our knowledge of it is rather limited as compared to photospheric level. The magnetic field in the corona cannot be measured with confidence from Zeeman splitting of coronal lines. Three main techniques are widely discussed in this context (Lee et al. [1998]): optical observations of vector magnetic fields in the photosphere and their extrapolation into the corona (McClymont et al. [1997]); EUV/X-ray observations which can reveal the projected paths of magnetic field lines via the density contrast between neighboring bundles of field lines (Yan & Sakurai [1997]); and spectral-polarization radio observations in the frequency domain of gyroresonance emission mechanism, which are sensitive to both the strength and the direction of the coronal magnetic field (Lang et al. [1993]; Abramov-Maximov et al. [1996]; Vourlidas et al. [1997]; see White & Kundu [1997] for review). These different techniques are complementary, each have its own advantages and disadvantages and should therefore be combined in order to make further progress.

The above techniques are effective when we deal with magnetic fields either at photospheric or the coronal levels, but the crucial intermediate region of the upper chromosphere is not accessible to those studies. Here we try to fill this gap by a detailed analysis of polarization-intensity spectra of free-free (i.e. bremsstrahlung) microwave emission from active regions (Bogod & Gelfreikh [1980]). Such techniques, combined with 2D microwave imaging, provide direct microwave magnetograms at the interface between chromosphere and corona (Grebinskij et al. [1998]) and may be important for an independent check of chromospheric and coronal fields interpolated from photospheric fields.

The magnetoionic theory (Ratcliffe [1959]) predicts two modes of propagating electromagnetic radiation, ordinary and extraordinary, with different opacities. The intensity of polarized emission is proportional to the longitudinal component of the magnetic field, which may be retrieved from spectral-polarization observations by inversion techniques (Gelfreikh [1972]). There are several important deficiencies which limit the practical usage of the methods during the past years. The thermal microwave emission from solar active regions is generated by two basic mechanisms, i.e. gyroresonance line (at the second and third harmonic of the electron gyrofrequency $\nu
_{B}=2.8$ $10^{6}\ B$ Hz), and free-free continuum emission. Polarization effects for free-free emission at high frequencies are very weak and the circular polarization degree $\rho \equiv
V/I$ is ranging between 0.1 and 10$\%$, both for weak (prominences, plage areas, coronal holes) and strong (sunspot) fields. At longer wavelengths the polarization increases as $\rho
\sim \lambda $ ( $V\sim \lambda ^{3},\ I\sim \lambda ^{2}$) for optically thin emission (Grebinskij [1985]). At the frequencies lower then ( $\nu \leq 3\nu _{B}$) the gyroresonance emission becomes dominant above sunspot regions. Thus, the possibilities of measurements (with free-free emission) of magnetic fields at longer cm waves are limited above active regions. The relative contributions of both emission mechanisms were widely discussed in the seventies with the first RATAN, Westerbork and VLA observations (see Alissandrakis et al. [1980] for review). The gyroresonance emission dominates above sunspots at cm waves, where the strong sunspot fields are strongly localized so that the corona is mainly radiating free-free emission above plage regions and between spots (see Gary & Hurford [1994]).

Thus, the main domain of free-free emission lies at short cm wavelengths. Observations at $\lambda =1.76$ cm (Nobeyama Radioheliograph) meet this condition for all active regions with magnetic fields below B=2000 G, which correspond to the third gyroresonance harmonic. In that frequency range the main observational problem concerns the instrumental limitations of measurements of the weak polarization effects. The most existing instruments (OVRO, VLA) acquire and process data in a left-right hand polarization modes and have an insufficient accuracy of polarization measurements above $10\%$. One-dimensional imaging with RATAN provides a high dynamic range of about 1000:1, but its fan-beam pattern diminishes the observed contrast. Single dish observations (RT-22 of CrAO) have an insufficient angular resolution (minutes of arc). For these instruments only a few reliable results of free-free polarization measurements have been reported: around $\lambda \approx 1$ cm with the RT-22 (CrAO) radio telescope (Apushkinskiy & Topchilo [1976]) for polar prominences; at $%
\lambda =6$ cm with the WSRT interferometer (Kundu et al. [1977]) for plages; at 2.3 - 4.0 cm with RATAN (Bogod & Gelfreikh [1980]) for plage regions; at decimetric wavelength with RATAN (Borovik et al. [1999]) for a coronal hole.

This situation has changed with the construction of the Nobeyama Radioheliograph. Its high operation frequency of 17 GHz gives opportunities for studying rather strong fields $B\leq 2000$ G, and its high sensitivity of polarized imaging (with $\rho
_{\rm rms}\leq 1\%$) opens possibilities for measuring weak fields also. The design principles of the Nobeyama Radioheliograph (Shibasaki et al. [1991]) make it especially appropriate for these goals: (a) its data processing (Hanaoka et al. [1994]) is based on the transformation of recorded data (in left-right polarizations) to the Stoke's parameters I,V in the primary processing stage (rough imaging before CLEAN), that gives a high dynamical range (26 dB) both for $I\ $and V images, and (b) its high acquisition rate (0.1 s) providing many possibilities for improvement of the images (dynamic range through averaging, resolution enhancements up to 10 arcsec etc.).

The main goal of this paper is to develop the practical methods of solar magnetic field measurements, which are based on an analysis of free-free emission recorded with Nobeyama radio maps ( $microwave\ magnetography$) and RATAN - Metsahovi radiospectroscopy. The method uses an improved treatment of radiation transfer equations and realistic model atmospheres, based on mm waves Metsahovi observations (Urpo et al. [1987]). This includes a number of problems, discussed below: the derivation of the decoupled radiation transfer equations for the Stoke's parameters I and V (see Appendix A); the problem of inversion of those equations for the extraction of magnetic fields (Sect. 2); a development of simplified model atmospheres to distinguish contributions of the chromosphere and the corona in the observed spectra (Sect. 3); the development of practically estimates of magnetic fields (Sect. 4).

We illustrate and discuss these techniques with results of published and new (Nobeyama) observations (Sects. 5 and 6). Some further results, related to the study of strong sunspot fields with Nobeyama imaging, will be discussed separately (see Grebinskij et al. [1998]).

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