The effect of the environment on the P1/P2 period ratio for kink oscillations of coronal loops

Aims. The P1/P2 period ratio of transversal loop oscillations is currently used for the diagnostics of longitudinal structuring of coronal loops as its deviation from 2 is intrinsically connected to the density scale-height along coronal loops and/or the sub-resolution structure of the magnetic field. The same technique can be applied not only to coronal structures, but also to other oscillating magnetic structures. Methods. The oscillations in magnetic structures are described by differential equations whose coefficients depend on the longitudinal structure of the plasma. Using a variational principle written for the transversal component of the velocity vector, developed earlier by McEwan et al. (2008, A&A, 481, 819), we investigate how the different temperature of the environment compared to the temperature of the magnetic structure will influence the P1/P2 ratio for typical coronal and prominence conditions. The possible changes are translated into quantities that are used in the process of remote plasma diagnostics in the solar atmosphere. Results. Using a straightforward, yet comprehensive, procedure we show that under coronal conditions the effect of the temperature difference between the plasma inside and outside the magnetic structure can change considerably the period ratio; in the case of coronal loops the change in the period ratio can reach even 40%. We also show that once dispersive effects are taken into account, with oscillation periods shorter than the cut-off period (determined by the density and temperature difference) the domain where the model can be applied is reduced. In the case of prominences embedded in the hot corona, the effect of the environment is negligeable given the high density and temperature difference between the chromospheric prominences and corona. Using a numerical approach, we show that our analytical discussion produces a robust result. We also discuss what implications our model has on seismological (or diagnostics) techniques in the solar corona. Conclusions. Our analysis shows that the period ratio P1/P2 is sensitive to the temperature difference between the loop and its environment and this effect should always be taken into account when estimating the degree of density structuring with period ratio method of coronal loops.