First polarimetric observations and modeling of the FeH F<SUP>4</SUP>Delta-X<SUP>4</SUP>Delta system

International audienceContext: Lines of diatomic molecules are typically much more temperature and pressure sensitive than atomic lines, which makes them ideal, complementary tools for studying cool stellar atmospheres as well as the internal structure of sunspots and starspots. The FeH F^4Delta-X^4Delta system represents such an example that exhibits in addition a large magnetic field sensitivity. However, the current theoretical descriptions of these transitions including the molecular constants involved are only based on intensity measurements because polarimetric observations have not been available so far, which limits their diagnostic value. Furthermore, the theory was optimized to reproduce energy levels and line strengths without taking the magnetic sensitivities into account. Aims: We present for the first time spectropolarimetric observations of the FeH F^4Delta-X^4Delta system measured in sunspots to investigate their diagnostic capabilities for probing solar and stellar magnetic fields. In particular, we investigate whether the current theoretical model of FeH can reproduce the observed Stokes profiles including their magnetic properties. Methods: The polarimetric observations of the FeH F^4Delta-X^4Delta system in Stokes I and V are compared with synthetic Stokes profiles modeled with radiative transfer calculations. This allows us to infer the temperature and the magnetic field strength of the observed sunspots. Results: We find that the current theory successfully reproduces the magnetic properties of a large number of lines in the FeH F^4Delta-X^4Delta system. In a few cases the observations indicate a larger Zeeman splitting than predicted by the theory. There, our observations have provided additional constraints, which allowed us to determine empirical molecular constants. Conclusions: The FeH F^4Delta-X^4Delta system is found to be a very sensitive magnetic diagnostic tool. Polarimetric data of these lines, in contrast to intensity measurements, provide us with more direct and detailed information to study the coolest parts of sunspot and starspot umbrae, and cool active dwarfs