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

= | |

(13) |

where

(14) |

We note from Eqs. (13) that the contributions of the chromosphere and corona to intensity and polarization are different: if, for example, , then the contribution of the chromosphere to is reduced by a factor

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:

(15) |

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

(16) |

This model may be treated as an atmosphere with an optically thick chromosphere ( ) and a hot corona ( with the optical depth depending on the changing magnetic field. Its strength is constant ( ) in deep layers ( ), with a rapid drop in the top layers ( ), as it is expected for solar magnetic fields. The radiation transfer Eqs. (1)-(2) give then

(17) |

(18) |

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

(19) |

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

(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:

(20) |

so that the magnetic field (Eq. 16b) in the transition region is governed by :

(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 ), we find from Eq. (19):

(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:

(23) |

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:

= | |||

n |
(24) |

where

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

(25) |

where we have introduced the scaling factor as

(26) |

Here are partial contributions to the total observed brightness . For the case of a homogeneous magnetic field, the two estimates should give the same result, so that

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:

(27) |

where is the observed total intensity of the plage region. The observed brightness excess above the quiet Sun level is related to the coronal parameters by

(28) |

The value of 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 and . (b) chromospheric domination in intensity at short wavelengths. We have

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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