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

Abstract

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 we define the internal phase shift through the relation . The are almost independent of acoustic radius outside the stellar core and can be determined as a continuous functions of ω from partial wave solutions for the interior – that is solutions of the oscillation equations for any ω that satisfy the Laplace boundary condition at a sufficiently large acoustic radius tf outside the stellar core. If the ω are eigenfrequencies then they satisfy the Eigenfrequency Equation where is the 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 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 (≈0.1–0.3 μHz) from the upcoming space missions MOST, COROT and Eddington.