## 3. Molecular Stokes profilesHere we present the forward spectral synthesis of the Stokes parameters of the MgH and TiO transitions. The set of radiative transfer equations is solved using the formulation given by Rees et al. (1989) as implemented in the code STOPRO (in an extended and improved version relative to that described by Solanki et al. 1992, cf. Frutiger et al. 2000). The code has been further updated to allow for molecular line computations. Calculations of the wavelength shifts of the Zeeman components and their theoretical strengths were included in accordance with the theory by Schadee (1978). Also, molecular number densities are calculated under the assumption of the chemical equilibrium of 270 compounds consisting of the 33 most abundant atoms (Tsuji 1973; L. Hänni, private communication). The code has also been extended to include the influence of blends (important near band heads), so that we can carry out the first spectral synthesis calculations of molecular and atomic Zeeman-split lines. As in the case of atomic lines, the magnetic splitting of molecular lines depends on the strength of the interaction of the internal magnetic moment of the molecule with the external magnetic field. It is sufficient to consider the molecular Zeeman effect if the coupling between internal angular momenta is strong, i.e. the magnetic splitting is less than rotational and multiplet splitting. If the relevant angular momentum interacts stronger with the external field than with the molecular fields, i.e. decoupling of the momenta takes place, the molecular Paschen-Back effect must be taken into account. In the following subsections we present calculations of the TiO and MgH band systems, whose splittings at the strengths reached by sunspot magnetic fields are good examples of the Zeeman and Paschen-Back effects, respectively. Here we concentrate on the results of the spectral synthesis and on their comparison with the data. Details about the code and the molecular physics of the considered bands are given in a separate paper (Berdyugina et al. 2000). ## 3.1. The TiO -systemThe electronic states of the TiO
-system
are under strong spin-orbit coupling,
and, thus, the system is a good example of the ordinary molecular
Zeeman effect up to very strong fields. The splitting of the levels is
symmetrical and proportional to the field strength for a given total
angular momentum number J. It is larger for low J and decreases with
increasing J. The splitting is also proportional to the sum of the
projections of the orbital and spin quantum numbers. Larger splitting
is therefore expected for the state,
especially for the third multiplet level. Our calculations show also
that lines of the Q- and R- branches show Stokes Since the largest Zeeman effect is expected for lines with small
values of J, the best spectral region for studying molecular magnetic
splitting is at the beginning of rotational branches. The strongest
and least blended band head of the TiO
-system observed in the spectrum of
sunspot umbra is (0,0)R
The Stokes ## 3.2. The MgH green systemThe MgH - system is an example of the partial Paschen-Back effect at moderate field strengths. The orbital angular momentum of the electrons in the lower electronic state () is zero, and this results in decoupling of the electron spin from the internuclear axis. Also, the electron spin interacts very weakly with the angular momentum of nuclear rotation in the lower state. Therefore, at low rotation the decoupling of the spin starts at rather moderate field strengths, less than 1000 G. Because of shifts of magnetic sublevels in the lower electronic state, the line splitting is no longer symmetrical with respect to the zero field wavelength, and the strengths of the Zeeman components are not balanced (Schadee 1978; Illing 1981). This results in wavelength shifts of lines. The imbalance of the transition probabilities of single rotational lines leads to increased strengths of lines of the satellite branches. Therefore, it is essential to consider a rotational transition and its satellite transition as a single line. The main rotational line and its satellite line have a common upper level and similar energies of the lower levels. At field strengths above 1000 G, the Zeeman components of the two lower levels start to merge and the satellite line gains strength. The two lines thus start to imitate a normal Zeeman splitting. This effect must be taken into account when interpreting molecular line splittings in terms of magnetic field strengths. Since the splitting is not linearly proportional to the field strength, it is not surprising that Wöhl (1969), who assumed proportionality, obtained significantly lower magnetic field strengths from MgH lines than from iron lines. The strongest lines of the MgH green system are observed in the
(0,0) band. We find that early rotational lines of the P-branch are
less blended in sunspot spectra and, thus, more suitable for study.
The calculations of Stokes Four MgH features are found to be useful for future investigations:
5197.3, 5199.6, 5200.9, 5201.7 Å. Although they are all blended
by atomic and TiO lines, the MgH transitions dominate the Stokes
© European Southern Observatory (ESO) 2000 Online publication: January 29, 2001 |