Structure of the stellar interior

The oscillation frequencies of a star are determined solely by the variation with radius of the pressure p, density , and adiabatic exponent . The condition of hydrostatic equilibrium gives one relation between these variables, leaving two unknown functions of radius to be determined from the data. A stellar model can therefore be defined as the set {} of values of (say) the density and adiabatic exponent , at radii r_i. There are several techniques for determining constraints on that model from given a set of eigenfrequencies. One approach is to construct families of models with different masses, compositions, ages and parameterizations of internal physical processes, and to compare their predicted frequencies with those observed. Another is to try to construct a `seismological model' without reference to stellar evolution theory. Both approaches are valuable.

  

Figure 2.2: Inversions to determine the structure of a with a convective core. (a) Localizing kernels indicating the region of the star that contributes to the particular combination of frequencies used in the inversion. (b) Comparison of the corrections to determined from the inversion with the actual values. The horizontal error bars are the widths of the localizing kernels and give the uncertainty in locating the radial position of the changes; the vertical error bars correspond to the error in the data (after Gough and Kosovichev 1993).

A technique that has been deployed in geophysics, solar physics and other (theoretical models of) stars is the so-called optimally localized averaging method (Backus and Gilbert 1968; Gough 1985). The principle is as follows: Any one eigenfrequency (where i stands for the combination ) is determined by an integral over the star's interior, , where K_i(r) is the kernel for the mode i. Kernels for different modes have different dependencies on the internal structure. A combination of the frequencies is then given by , where the averaging kernel is . By a suitable choice of the c_i one can construct localized kernels K₀(r,r₁) designed to be large in the vicinity of a particular radius r_k and small throughout the rest of the star, thereby gaining information about the structure of the star near r=r_k. The method is iterative: a guess is made at a reference stellar model whose eigenfrequencies and corresponding eigenfunctions are calculated. These eigenfunctions are then used to produce localized kernels, and the same combination of the actual data is used to determine the corrections to the structure variables. This technique has been applied successfully to solar data and to simulated data from an evolved star of with a small convective core (Gough and Kosovichev 1993). In the case, the input frequencies were restricted to modes with and n=5 to 28, and random errors of the order of 0.3Hz were incorporated, which represent the errors one might expect to be present in the data from STARS. The initial guess was a solar model, chosen deliberately as a poor approximation to the star to provide a severe test of the procedure. The result of this inversion is given in Fig. 2.2 which shows both the localizing kernels and the relative changes in the function . As can be seen from the diagram, the actual differences are quite accurately recovered by this technique; in particular, the discontinuity at the edge of the convective core is clearly detected. These differences can be used to produce an improved model, and the process repeated until one obtains a `seismological model' (or models) that fits the frequencies to within the observational errors.