Frequency dependence of the large frequency separation of solar-like oscillators: influence of the helium second-ionization zone

The large frequency separation (Δν) between modes of the same degree and consecutive orders in a star is approximately proportional to the square root of its mean density. To determine Δν as accurately as possible, a mean large frequency separation (<Δν>) computed over several orders is often used. It is, however, known that Δν varies with frequency in a second-order effect. From observations, it has been shown that this frequency dependence is more important for main-sequence stars than it is for red giant stars. Here we use YREC models to verify and explain this observational result. We find that for stars with R ≳ 8 R_⊙, the effect of the helium second-ionization zone (He II zone) is relatively small. For these stars, the deep location of the He II zone induces a frequency modulation covering only a few Δν, while the amplitude of the modulation is low due to the relatively weak and extended He II layer, causing a shallow wide depression in the first adiabatic exponent (Γ₁). For less evolved stars, the He II zone is located closer to the surface, and it is more confined, i.e. a deep narrow depression in Γ₁. This causes frequency modulations with relatively high amplitudes covering up to about 20Δν, inducing a relatively large frequency modulation. Additionally, we find that for less evolved stars, the He II zone is stronger and more localized for more massive stars and for stars with low metallicities further increasing the amplitude of the frequency modulation.