Analytical determination of coronal parameters using the period ratio P₁/2P₂

<p>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 P1/2P2 from unity provide information about the extent of longitudinal structuring within the loop.</p> <p>Aims. Here we develop an analytical approximation that describes the shift in P1/2P2 in terms of the ratio L/Λc of the length 2L of a coronal loop and the density scale height Λc.</p> <p>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.</p> <p>Results. Analytical approximations to the period ratio P1/2P2 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.</p> <p>Conclusions. Our simple formula and approximate approaches highlight a useful analytical tool for coronal seismology. We demonstrate that P1/2P2 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.</p&gt

Magneto-seismology of solar atmospheric loops by means of longitudinal oscillations

There is increasingly strong observational evidence that slow magnetoacoustic modes arise in the solar atmosphere. Solar magneto-seismology is a novel tool to derive otherwise directly un-measurable properties of the solar atmosphere when magnetohydrodynamic (MHD) wave theory is compared to wave observations. Here, MHD wave theory is further developed illustrating how information about the magnetic and density structure along coronal loops can be determined by measuring the frequencies of the slow MHD oscillations. The application to observations of slow magnetoacoustic waves in coronal loops is discused.Comment: 4 pages, 2 figures, to appear in Proceedings of IAU Symp 286, Comparative Magnetic Minima, C. H. Mandrini, ed

Non-axisymmetric oscillations of stratified coronal magnetic loops with elliptical cross-sections

We study non-axisymmetric oscillations of a straight magnetic tube with an elliptic cross-section and density varying along the tube. The governing equations for kink and fluting modes in the thin tube approximation are derived. We found that there are two kink modes, polarised along the large and small axes of the elliptic cross-section. We have shown that the ratio of frequencies of the first overtone and fundamental harmonic is the same for both kink modes and independent of the ratio of the ellipse axes. On the basis of this result we concluded that the estimates of the atmospheric scale height obtained using simultaneous observations of the fundamental harmonic and first overtone of the coronal loop kink oscillations are independent of the ellipticity of the loop cross-section

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 longwavelength 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

On the vertical and horizontal transverse oscillations of curved coronal loops

Kink oscillations of curved coronal loops with the density varying along the loop are studied in the thin tube approximation. The equilibrium magnetic field is assumed to be potential, and the field potential and flux function are used as curvilinear coordinates. It is also assumed that the loop expansion is weak, and the solution to the problem is looked for in the form of power series with respect to the tube expansion parameter lambda << 1. The main result of the study is that the eigenfrequencies of the vertical and horizontal tube oscillations are, in general, different, their difference being proportional to lambda. As an example a simple equilibrium with the magnetic field magnitude exponentially decaying with the height is considered. The implication of the obtained results for the interpretation of observational data is discussed

Effect of longitudinal magnetic and density inhomogeneity on transversal coronal loop oscillations

Context. Observations of post-flare transversal coronal loop oscillations by TRACE have given us an excellent opportunity to implement magneto-seismological techniques for probing the plasma fine structure of the Sun's upper atmosphere. Aims. We investigate the combined effect of magnetic and density stratification on transversal coronal loop oscillations. Methods. A coronal loop will be modelled as an expanding magnetic flux tube with arbitrary longitudinal plasma density. The governing equation of the fast kink body mode is derived and solved by analytical approximation and numerical methods. Results. It is found that even a relatively small coronal loop expansion can have a significant and pronounced effect on the accuracy of the plasma density scale height measurements derived from observation of loop oscillations. Conclusions. To conduct more accurate and realistic magneto-seismology of coronal loops, the magnetic field divergence should be taken into account

Resonantly damped oscillations of longitudinally stratified coronal loops

Soon after coronal loop oscillations were observed by TRACE spacecraft for the first time in 1999, various theoretical models have been put forward to explain the rapid damping of the oscillations of these intriguing objects. Coronal loop oscillations are often interpreted as fast kink modes of a straight cylindrical magnetic flux tube with immovable edges modelling dense photospheric plasma at the ends of the loop. Taking this model as a basis we use cold plasma approximation and consider the tube to be thin to simplify the problem and be able to deal with it analytically. In its equilibrium state the tube is permeated by a homogeneous magnetic field directed along the tube axis. We include the effect of stratification in our model supposing that plasma density varies along the tube. There is also density inhomogeneity in the radial direction confined in a layer with thickness much smaller than the radius of the tube. Considering the system of linearized MHD equations we study the dependence of the spectrum of tube oscillations and its damping due to resonant absorption on the parameters of the unperturbed state. The implication of the obtained results on coronal seismology is discussed

Torsional Alfvén waves: magneto-seismology in static and dynamic coronal plasmas

Aims: We study the properties of torsional Alfvén waves in coronal loops so that they may be exploited for coronal seismological applications. Methods: The governing equation is obtained for standing torsional Alfvén waves of a dynamic, gravitationally stratified plasma. The footpoints are assumed to obey line-tying conditions necessary for standing oscillations. Solutions are found in a number of different but typical scenarios to demonstrate the possibilities for both temporal and spatial magneto-seismology exploitation of waveguides with the standing torsional Alfvén oscillations. Results: It is found that the frequency of the standing Alfvén oscillation increases as the stratification of the plasma increases. The ratio of the periods of the fundamental modeand the first overtone is also found to change as the stratification of the plasma increases. Further, the eigenfunctions of the higher overtones of the standing oscillations are found to experience a shift of their anti-nodes. The influence of a dynamic plasma on the amplitudes of the mode is also investigated. The amplitude of the torsional Alfvén mode is found to increase as the plasma within the coronal loop experiences cooling

Electronic structure and optical properties of lightweight metal hydrides

We study the electronic structures and dielectric functions of the simple hydrides LiH, NaH, MgH2 and AlH3, and the complex hydrides Li3AlH6, Na3AlH6, LiAlH4, NaAlH4 and Mg(AlH4)2, using first principles density functional theory and GW calculations. All these compounds are large gap insulators with GW single particle band gaps varying from 3.5 eV in AlH3 to 6.5 eV in the MAlH4 compounds. The valence bands are dominated by the hydrogen atoms, whereas the conduction bands have mixed contributions from the hydrogens and the metal cations. The electronic structure of the aluminium compounds is determined mainly by aluminium hydride complexes and their mutual interactions. Despite considerable differences between the band structures and the band gaps of the various compounds, their optical responses are qualitatively similar. In most of the spectra the optical absorption rises sharply above 6 eV and has a strong peak around 8 eV. The quantitative differences in the optical spectra are interpreted in terms of the structure and the electronic structure of the compounds.Comment: 13 pages, 10 figure