Aims: We determine the effect of an elliptical shape on the period ratio for the standing transversal oscillations of a longitudinally stratified coronal loop throughout its emergence from the low solar atmosphere into the ubiquitously magnetised corona.
Methods: Under the assumption that elliptical curvature has a negligible effect on eigenfrequencies, the equation that describes the projection of a density profile onto a magnetic flux tube with elliptical shape is obtained in a gravitationally stratified atmosphere. The effect of the elliptical shape on the period ratio of the fundamental mode to the first harmonic (P1/P2) at various stages of emergence is determined, assuming that the oscillation periods are much shorter than the characteristic time scale of loop emergence.
Results: We find that there are two separate cases of elliptical shape that occur, the minor ellipse and the major ellipse. It is then shown how the period ratio P1/P2 is dependent upon the ellipticity (∊), the parameter characterising the stage of emergence (λ) and the density scale height (H). Ellipticity is found to make an important contribution to P1/P2 for the minor ellipse when compared to its counterpart of standing oscillations of stratified loops with semi-circle or circle-arc shape. The major ellipse was found to have a lesser effect on the period ratio of standing oscillations. We also find the value of P1/P2 is dependent upon the stage of emergence of the loop, where the greatest contribution from emergence to the ratio of P1/P2 is when the loop is almost fully emerged. The important implication for magneto-seismological interpretations of the observations of oscillating coronal loops is that measurements of ellipticity and stage of emergence should supplement observations of oscillation periods and should be considered when applying observed frequencies of the fundamental mode and first harmonic to determine the diagnostic properties of these oscillating loops, e.g. the density scale height or strength of magnetic field. Neglecting the determination of ellipticity and stage of emergence may result in a 35% error in estimating density scale height.