Extensive observations of solar oscillations over the past two decades have resulted in a large body of precisely, and possibly accurately, determined oscillation frequencies of the Sun, as well as information about the amplitudes and damping times of the modes. The goal of helioseismology is to use these data to gain an understanding of the physical processes that determine the present structure and dynamics of the solar interior.
In principle, the helioseismic inverse problem consists of inferring, from the observed solar signal (e.g. a time series of Doppler maps of the solar surface) the desired physical information, or at least obtaining constraints on the physical properties. Throughout the slow variation of solar properties with time, e.g. associated with the solar magnetic 22-year cycle, must be kept in mind. In practice, the analysis is carried out through a large number of steps, each of which giving rise to potential problems that may affect the outcome. These involve spatial analysis to separate different modes, temporal analysis to determine the frequencies and analysis of the frequencies to infer the properties of the solar interior.
Here I concentrate on the latter step, often characterized as helioseismic inversion; it is important, however, to keep data properties resulting from the preceding steps in mind when carrying out the inverse analysis. The analysis is normally separated in a determination of the spherically symmetric component of solar structure, e.g. characterized by the internal sound speed and density, and departures from spherical symmetry which are regarded as small perturbations. The frequency dependence on solar structure is nonlinear; although asymptotic techniques exist which allow approximate determination of the sound speed directly from the frequencies, in most cases the analysis is carried out by linearising the frequency dependence on structure around a reference solar model and determining the required corrections to that model. In this way the sound speed and density in most of the solar interior have been inferred with good precision and reasonable resolution. The more interesting, but less well-defined, questions concern the physics responsible for this structure, particularly the reasons for the remaining very significant differences between the inferences and numerically computed solar models. Answering these questions necessarily involves additional assumptions, or prejudices.
Departures from spherical symmetry include effects of rotation, as well as an aspherical component of structure. These can with sufficient precision be regarded as small perturbations, linearly related to the perturbations in the frequencies, resulting in inverse problems from which for example rotation can be inferred as a function of position in the Sun. The results again point to interesting physical issues concerning the origin of the solar internal dynamics, including the interaction between rotation and the convective motions in the outer 28 % of the Sun, the spin- down from a normally assumed state of faster rotation when the Sun was formed, and the interaction between magnetic fields and rotation.
I plan to give an overview of the steps involved in these various inverse analyses, and to discuss the interpretation of the results in terms of our physical understanding of the structure and evolution of stars.
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