Transverse kink oscillations of inhomogeneous prominence threads: numerical analysis and H$\alpha$ forward modelling

Prominence threads are very long and thin flux tubes which are partially filled with cold plasma. Observations have shown that transverse oscillations are frequent in these solar structures. The observations are usually interpreted as the fundamental kink mode, while the detection of the first harmonic remains elusive. Here, we aim to study how the density inhomogeneity in the longitudinal and radial directions modify the periods and damping times of kink oscillations, and how this effect would be reflected in observations. We solve the ideal magnetohydrodynamics equations through two different methods: a) performing 3D numerical simulations, and b) solving a 2D generalised eigenvalue problem. We study the dependence of the periods, damping times and amplitudes of transverse kink oscillations on the ratio between the densities at the centre and at the ends of the tube, and on the average density. We apply forward modelling on our 3D simulations to compute synthetic H$\alpha$ profiles. We confirm that the ratio of the period of the fundamental oscillation mode to the period of the first harmonic increases as the ratio of the central density to the footpoint density is increased or as the averaged density of the tube is decreased. We find that the damping times due to resonant absorption decrease as the central to footpoint density ratio increases. Contrary to the case of longitudinally homogeneous tubes, we find that the damping time to period ratio also increases as the density ratio is increased or the average density is reduced. We present snapshots and time-distance diagrams of the emission in the H$\alpha$ line. The results presented here have implications for the field of prominence seismology. While the H$\alpha$ emission can be used to detect the fundamental mode, the first harmonic is barely detectable in H$\alpha$. This may explain the lack of detections of the first harmonic.