Solar Physics & upper Atmosphere Research Group University of Sheffield A possible method for measuring the plasma density profile along coronal loops G.Verth & R. Erdélyi Space & Atmosphere Research Group, University of Sheffield, Department of Applied Mathematics, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, England (UK) & Abstract A technique is proposed to measure the variation in plasma density along coronal loops using the observations of transverse oscillations. The calculations are based on a theoretical model of an isolated thin magnetic flux tube for which the density is allowed to vary along its length both inside and outside the tube boundary. By observation, coronal tube radii are much smaller than their length. This allows us to show that the square of the eigenfrequencies of the standing kink modes are given by the eigenvalues of a Sturm-Liouville problem for a second order ordinary differential equation. By measuring how the amplitudes of these oscillations vary along a coronal loop, the inverse problem can then be solved to give the density profile along that loop. We discuss the feasibility of detecting density stratification of coronal loops with TRACE observations. Model We have adapted a previous model for prominence fibrils [1] to describe the fundamental standing kink mode in a coronal loop. A loop in its equilibrium state is modeled by a straight cylindrical magnetic flux tube of total length 2L and radius R. Plasma densities both inside and outside the loop are assumed to depend only on z coordinate. The magnetic field is uniform everywhere and directed along the z-axis. The linearised MHD equations for this model are shown below. Basic assumptions We use the cold plasma approximation and consider the fundamental standing kink mode with loop footpoints fixed in the photosphere. We also neglect viscous effects, gravity and loop curvature, but take density stratification into account. We assume the perturbations to be proportional to Applying an asymptotic expansion method and boundary conditions on the side surface of the tube (continuity of pressure and radial velocity component across the boundary r=R) we reduce the problem to a single equation for function A (z), which is the maximum amplitude of the oscillation at the tube’s boundary, i.e., r = R. Example We shall now solve this equation numerically for an exponentially stratified atmosphere. Taking loop curvature into account, the expression for density from z=0 (loop top) to z=L (loop footpoint) is where H is the scale height.. Varying the value of H and keeping the loop length and amplitude at z=0 and z=L fixed gives the following plot. Discussion It was calculated that for TRACE to identify the expected density stratification in a typical coronal loop with 2L=H (where H=60-70Mm) then there would have to be a relative error in amplitude measurement of less than 5%. The best available estimate of the relative error in such a measurement is approximately 30% [2]. This is due to background noise and the difficulty of ascertaining a loop's true length and geometry from a 2D projection. Therefore, TRACE observations of transverse loop oscillations would not be reliable in identifying the expected density stratification in the corona using this model. References [1] Dymova, M. and Ruderman, M.S. : 2004 (private communication) [2] Aschwanden, M.J., De Pontieu, B., Schrijver, C. and Title, A. : 2002, Solar Phys. 206, 99 - magnetic permeability - velocity components - perturbed magnetic field - perturbation in magnetic pressure