The ratio of small to large separations of acoustic oscillations as a diagnostic of the interior of solar-like stars

By considering stellar models with the same interior structure but different outer layers we demonstrate that the ratio of the small to large separations of acoustic oscillations in solar-like stars is essentially independent of the structure of the outer layers, and is determined solely by the interior structure. Defining the scaled Eulerian pressure perturbation psi(l)(omega, t) = rp'/(rhoc)(1/2) we define the internal phase shift delta(l)(omega, t) through the relation omegapsi/(dpsi/dt) = tan(omegat - pil/2 + delta(l)). The delta(l) are almost independent of acoustic radius t = integral dr/c outside the stellar core and can be determined as a continuous functions of omega from partial wave solutions for the interior - that is solutions of the oscillation equations for any omega that satisfy the Laplace boundary condition at a sufficiently large acoustic radius t(f) outside the stellar core. If the omega are eigenfrequencies then they satisfy the Eigenfrequency Equation omegaT = (n + l/2)pi + alpha(omega) - delta(l)(omega) where alpha(omega) is the l independent surface phase shift (Roxburgh & Vorontsov 2000). Using this result we show that the ratio of small to large separations is determined to high accuracy solely by the internal phase shifts delta(l)(omega) and hence by the interior structure alone. The error in this result is estimated and shown to be smaller than that associated with the errors in the determination of the frequencies (approximate to0.1-0.3 muHz) from the upcoming space missions MOST, COROT and Eddington.