Asymptotic and measured large frequency separations^⋆

¹ LESIA, CNRS, Université Pierre et Marie Curie, Université Denis Diderot, Observatoire de Paris, 92195 Meudon Cedex, France
e-mail: benoit.mosser@obspm.fr
² Université de Nice-Sophia Antipolis, CNRS UMR 7293, Observatoire de la Côte d’Azur, Laboratoire J.L. Lagrange, BP 4229, 06304 Nice Cedex 04, France

Received: 25 September 2012
Accepted: 9 December 2012

Abstract

Context. With the space-borne missions CoRoT and Kepler, a large amount of asteroseismic data is now available and has led to a variety of work. So-called global oscillation parameters are inferred to characterize the large sets of stars, perform ensemble asteroseismology, and derive scaling relations. The mean large separation is such a key parameter, easily deduced from the radial-frequency differences in the observed oscillation spectrum and closely related to the mean stellar density. It is therefore crucial to measure it with the highest accuracy in order to obtain the most precise asteroseismic indices.

Aims. As the conditions of measurement of the large separation do not coincide with its theoretical definition, we revisit the asymptotic expressions used for analyzing the observed oscillation spectra. Then, we examine the consequence of the difference between the observed and asymptotic values of the mean large separation.

Methods. The analysis is focused on radial modes. We use series of radial-mode frequencies in published analyses of stars with solar-like oscillations to compare the asymptotic and observational values of the large separation. This comparison relies on the proper use of the second-order asymptotic expansion.

Results. We propose a simple formulation to correct the observed value of the large separation and then derive its asymptotic counterpart. The measurement of the curvature of the radial ridges in the échelle diagram provides the correcting factor. We prove that, apart from glitches due to stellar structure discontinuities, the asymptotic expansion is valid from main-sequence stars to red giants. Our model shows that the asymptotic offset is close to 1/4, as in the theoretical development, for low-mass, main-sequence stars, subgiants and red giants.

Conclusions. High-quality solar-like oscillation spectra derived from precise photometric measurements are definitely better described with the second-order asymptotic expansion. The second-order term is responsible for the curvature observed in the échelle diagrams used for analyzing the oscillation spectra, and this curvature is responsible for the difference between the observed and asymptotic values of the large separation. Taking it into account yields a revision of the scaling relations, which provides more accurate asteroseismic estimates of the stellar mass and radius. After correction of the bias (6% for the stellar radius and 3% for the mass), the performance of the calibrated relation is about 4% and 8% for estimating, respectively, the stellar radius and the stellar mass for masses less than 1.3 M_⊙; the accuracy is twice as bad for higher mass stars and red giants.