Free Access
Volume 572, December 2014
Article Number A11
Number of page(s) 13
Section Stellar structure and evolution
Published online 19 November 2014

Online material

Appendix A: Inertia ratios

For a better understanding of the inertia ratios between a p-type and a g-type mode, we follow the development of Goupil et al. (2013) based on the asymptotic method developed by Shibahashi (1979).

In the asymptotic regime and neglecting the size of the evanescent zone, Goupil et al. (2013) show that the inertia in the envelope of the stars varies as Ienv ≃ (c2/ 2πν)τp and in the core as Icore ≃ (a2/ 2πν)τg, so we can express the total inertia as (A.1)with with kr the radial wavenumber and ν the frequency of the mode in μHz. Here, c is a normalisation constant, and a is related to c by (Eqs. (16.49) and (16.50) from Unno et al. 1989) : (A.2)where (c/a)2 is a function of ν of period 1 /τp = Δν which varies between 4 (p-modes) and 1/4 (g-modes). Finally when

comparing the inertia of a mode trapped in the envelope (Ip) and of a mode trapped in the core (Ig), we have (A.3)This inertia ratio is then a function of the ratio τg/τp = ng/np, the number of mixed-modes by large separation.

Appendix B: Qualitative comparison to Kepler spectra

We present in Fig. B.1 some power spectra obtained with Kepler along with our 1.5 M RGB theoretical power spectra to show the main tendencies discussed in this paper. Concerning the height ratios and the limit for the detectability of mixed modes in our theoretical power spectra, we found the same tendencies in the observed ones. At the beginning of the red-giant branch, dipole mixed modes have heights that are comparable to p-type modes. Higher on the RGB, dipole mixed modes are partially resolved, and their heights present a clear modulation compared to the heights of p-type modes. At the level of model C, only the p-type modes have significant heights. There are more visible mixed modes in the observed spectra, owing to the presence of rotational multiplets, but without any consequence for their height and width.

thumbnail Fig. B.1

Theoretical and observed power spectra of Kepler stars with similar masses (from top to bottom: 1.44, 1.48, 1.47 M), Δν, and νmax. The heights in theoretical power spectra are in (m/s)2/μHz. The heights for observed spectra are given in ppm2/μHz divided by a factor 6000 to have scales similar to the theoretical spectra.

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© ESO, 2014

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