Transverse oscillations of two parallel coronal loops

Context. Collective oscillations of two or more coronal magnetic loops are observed very often. Aims. We study the eigenmodes of oscillations of a system consisting of two parallel magnetic loops. Methods. The linearised MHD equations for a cold plasma are solved analytically in bicylindrical coordinates using the long-wavelength approximation. A dispersion equation determining the frequencies of eigenmodes is derived and solved analytically. Results. Two solutions of the dispersion relation were found. The higher frequency corresponds to the antisymmetric mode polarised in the direction parallel to the line connecting the loop centres, and the symmetric mode polarised in the perpendicular direction. Depending on the polarisation of modes corresponding to the lower frequency, the systems of two parallel loops are classified as standard and anomalous. In standard systems the lower frequency corresponds to the symmetric mode polarised in the direction parallel to the line connecting the loop centres, and the antisymmetric mode polarised in the perpendicular direction. In anomalous systems the lower frequency corresponds to the antisymmetric mode polarised in the direction parallel to the line connecting the loop centres, and the symmetric mode polarised in the perpendicular direction. The limiting case of two identical loops is studied. The results for this case are compared with recent numerical results.