Abstract
Aims. We derive information about the dynamics of the stellar photosphere, including pulsation, from a coherent interpretation of the linear polarisation detected in the spectral lines of the Mira star χ Cyg. Methods. From spectropolarimetric observations of χ Cyg, we performed a careful analysis of the polarisation signals observed in atomic and molecular lines, both in absorption and emission, using radiative transfer in the context of polarisation produced through two mechanisms: intrinsic polarisation and continuum depolarisation. We also explain the observed line doubling phenomenon in terms of an expanding shell in spherical geometry, which allows us to pinpoint the coordinates over the stellar disc with enhanced polarisation. Results. We find that the polarised spectrum of χ Cyg is dominated by intrinsic polarisation and has a negligible continuum depolarisation. The observed polarised signals can only be explained by assuming that this polarisation is locally enhanced by velocity fields. During the pulsation, radial velocities are not homogeneous over the disc. We map these regions of enhanced velocities. Conclusions. We set an algorithm to distinguish the origin of this polarisation in any stellar spectra of linear polarisation and to find a way to increase the signal by coherently adding many lines with an appropriated weight. Applied to the Mira star χ Cyg, we reached the unexpected result that during the pulsation, velocities are radial but not homogeneous over the disc. The reason for these local velocity enhancements are probably related to the interplay between the atmospheric pulsation dynamics and the underlying stellar convection.