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This chapter shows how the oscillations of a star are related to its structure. It also describes some of the important properties of stellar oscillations that allow us to use seismic data to determine stellar properties. Stellar oscillations are the response of a star to a mechanical disturbance, and thus the equations of stellar oscillations are derived by perturbing the equations of stellar structure. While a star may be spherically symmetric under equilibrium conditions, it need not retain spherical symmetry when it is perturbed, and as a result one can no longer avoid taking into account the three-dimensional nature of a star.

This chapter determines stellar properties from seismic data. It discusses how the average seismic properties discussed in the previous chapters are used in determining the mass and radius of a star. The chapter provides two ways of determining stellar properties: the “direct method” and grid-based modeling. The first method produces a set of model-independent results derived from a set of equations. On the other hand, the grid-based model, which can make up for the shortcomings of the direct method, derives its results from a fairly dense grid of stellar models of different metallicities that cover a wide range of masses and evolutionary stages.

This chapter discusses different ways in which one can use the frequencies to determine stellar properties. It considers straightforward comparisons of observed frequencies with those of stellar models, as well as the limitations and challenges of such comparisons. Different approaches are used to model stars whose frequencies have been observed. Some methods are similar to what is done if only the nonseismic properties of a star are known. Other methods make use of the fact that the average asteroseismic data give a good indication of the mass and radius of the star being modeled, and so these methods construct models around that mass and radius. Yet other methods involve optimization schemes. However, even the most complex optimization schemes used thus far do not explore the effects of all inputs and parameters that can be varied while constructing stellar models.

This chapter discusses some of the other ways mode frequencies can be used, and what additional information they can provide about a star. It first considers how specific combinations of mode frequencies can be used to determine stellar properties. In addition, the properties of a star, such as the acoustic depth of the convection-zone base, are determined by fitting models of the signatures left by acoustic glitches to the frequencies or the second differences. This chapter discusses how one can use signatures of acoustic glitches to infer properties of the star in question. Period spacings of red giants are also potentially very useful, and this chapter discusses how to make use of them.