Helioseismology - Analysis of Oscillation Data

Analysis of Oscillation Data

The data from time-series of solar spectra show all the oscillations overlapping. Thousands of modes have been detected (with the true number perhaps being in the millions). The mathematical technique of Fourier analysis is used to recover information about individual modes from this mass of data. The idea is that any periodic function f can be written as a sum of multiples of the simplest periodic functions, which are sines and cosines (of different frequencies). To find out how much (the amplitude) of each simple function goes into f, one applies the Fourier transform: at each point the value of this transform is obtained by computing a particular integral involving a modified version of f.

The simplest modes to analyse are the radial ones; however most solar modes are non-radial. A nonradial mode is characterized by three wavenumbers: the spherical-harmonic degree l and azimuthal order m which determine the behaviour of the mode over the surface of the star and the radial order n which reflects the properties in the radial direction (see the diagram on the top right for an example). Note that if the Sun were spherically symmetric, the azimuthal order would exhibit degeneracy; however the rotation of the Sun (along with other perturbations), which leads to an equatorial bulge, lifts this degeneracy. By convention, n corresponds to the number of nodes of the radial eigenfunction, l indicates the total number of nodal lines on spheres, and m tells how many of these nodal lines cross the equator.

In general the frequencies of stellar oscillations depend on all three wave numbers. It is convenient, however, to separate the frequency into the multiplet frequency, obtained as a suitable average over azimuthal order m and corresponding to the spherically symmetric structure of the star, and the frequency splitting .

Analyses of oscillation data must attempt to separate these different frequency components. In the case of the Sun the oscillations can be observed directly as functions of position on the solar disk as well as time. Thus here it is possible to analyze their spatial properties. This is done by means of a generalized 2-dimensional Fourier transform in position on the solar surface, to isolate particular values of l and m. This is followed by a Fourier transform in time which isolates the frequencies of the modes of that type. In fact, the average over the stellar surface implicit in observations of stellar oscillations can be thought of as one example of such a spatial Fourier transform.

Note that the oscillation data, rather than a continuous function, amount to values constrained by experimental error evaluated at a grid of positions and times. When computing transforms, values of this "function" outside this grid have to be interpolated and the integrals approximated by finite sums, a process inevitably introducing further errors. Details of the numerical methods used are included with the transformed data for purposes of comparison and constraining errors.

This discussion is adapted from the Jørgen Christensen-Dalsgaard lecture notes on stellar oscillations.