Analytical determination of coronal parameters using the period ratio P/2P

School of Mathematics and Statistics, University of St Andrews, St Andrews, Fife KY16 9SS, Scotland e-mail: bernie@mcs.st-and.ac.uk; toni.diaz@uib.es

Received: 5 June 2007
Accepted: 30 November 2007

Abstract

Context. In transverse coronal loop oscillations, two periodicities have been measured simultaneously and are interpreted as the fundamental kink mode (with period P₁) and the first harmonic (with period P₂).
Deviations of the period ratio from unity provide information about the extent of longitudinal structuring within the loop.

Aims. Here we develop an analytical approximation that describes the shift in in terms of the ratio of the length  of a coronal loop and the density scale height .

Methods. We study the MHD wave equations in a low β plasma using the thin tube approximation. Disturbances are described by a differential equation which may be solved for various equilibrium density profiles, obtaining dispersion relations in terms of Bessel functions. These dispersion relations may be used to obtain analytical approximations to the periods P₁ and P₂. We also present a variational approach to determining the period ratio and show how the WKB method may be used.

Results. Analytical approximations to the period ratio are used to shed light on the magnitude of longitudinal structuring in a loop, leading to a determination of the density scale height. We apply our formula to the observations in Verwichte et al. (2004) and Van Doorsselaere et al. (2007), obtaining the coronal density scale height.

Conclusions. Our simple formula and approximate approaches highlight a useful analytical tool for coronal seismology. We demonstrate that is linked to the density scale height, with no need for estimates of other external parameters. Given the accuracy of current observations, our formula provides a convenient means of determining density scale heights.