Issue 
A&A
Volume 549, January 2013



Article Number  A75  
Number of page(s)  13  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201220266  
Published online  21 December 2012 
Online material
Appendix A: Properties of eigenfunctions and mode inertia
A.1. Displacement eigenfunctions
Fig. A.1
Displacement eigenfunctions as a function of the normalized radius r/R. In blue the horizontal component to the displacement eigenfunction z_{2} (Eq. (7)) and in red the radial component to the displacement eigenfunction z_{1} (Eq. (6)). From top to bottom ℓ = 1 modes 1 and 3 of Table 2. 

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The displacement eigenfunctions computed for two modes of the selected pattern (see Table 2) for model M1 are shown as a function of the normalized radius r/R in Fig. A.1. They correspond to the modes with the largest and smallest maximum amplitudes in the pattern. The inner part is dominated by the horizontal displacement z_{2} and oscillates with a large number of nodes, as is typical of a highorder gravity mode. The largest maximum amplitude corresponds to the most gdominated mode whereas the smallest maximum amplitudes arise for the pm modes ν_{1} and ν_{6}.
The maximum amplitude of z_{2} occurs deep in the gcavity, at the same radius for all modes of the pattern. The region of nonnegligible amplitude defines the radius of a seismic rotating core which is here found to be independent of the mode (r/R ~ 0.02) and far smaller than the upper turning radius of the inner gravity resonant cavity (x_{2} ~ 0.08, Table 3)
A.2. Behavior of β and β_{core} with ζ
For ℓ = 1 modes of red giants, the term z_{2}z_{1} in β (Eq. (13)) plays almost no role because in the core and in the envelope (see Fig. A.1). As a result, we have where ζ is defined in Eq. (10). The linear dependence of β with ζ is verified in Fig. A.2. Furthermore, for all modes, , 2z_{1}z_{2} in the gcavity (see Fig. A.1), then β_{core,nl} ≈ β_{core} where for l = 1 modes, we derive hence (A.6)Numerical values for model M1 confirm that β_{core} increases linearly with ζ with a slope 1/2 (Fig. A.2). For gm modes (ζ ~ 1), β_{core} dominates with a nearly constant value of 0.5. Pm modes correspond to the teeth of the sawtype variation in β_{env} and the ratio β_{env}/β_{core} ~ 0.25 (Fig. A.2).
Fig. A.2
Top: β_{env} and β_{core} as a function of ν/Δν for model M1. Bottom: same as top for the ratio β_{core} as a function of ζ (black open dots). The approximation β = 1 − (1/2) ζ using the numerical values of ζ is represented with red crosses. 

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A.3. An approximate expression for ζ
This section determines an approximate expression of ζ = I_{core}/I as a function of ν/Δν. The derivation is based on results of an asymptotic method developed by Shibahashi (1979) to which we refer for details (see also Unno et al. 1989).

The envelope (~ppropagative cavity) is characterized by . Using Eq. (16.47) from Unno et al., it is straightforward to derive the following approximate expression (A.7)The constant c can be determined by the condition ξ_{r} = 1 at the surface. (A.8)with (A.9)where we have assumed σ^{2} ≫ N^{2} in Eq. (27). In the process of deriving the amplitude of arising in front of the sinusoidal term in Eq. (A.7), one can neglect in front of σ^{2} in the expression for k_{r} (i.e. k_{r} ~ σ/c_{s}). However this is not the case when k_{r} is in the phase of the sinusoidal term where we keep the expression Eq. (A.9). The inertia in the envelope can then be approximated as where we have defined (A.12)and the mean large separation is (A.13)The factor f is of order unity and represents the difference between the integration from x_{3} and from the center. We take f = 1 unless specified otherwise. The last equality in Eq. (A.10) is obtained assuming στ_{p} ≫ 1.

The core (~gpropagative cavity) is characterized by . Again, the asymptotic results lead to the following expression (A.14)where a is a constant that is determined by the resonant frequency condition between the p and g cavities, and (A.15)and we have used (A.16)Recalling that στ_{g} ≫ 1, therefore the inertia in the core can be approximated as (A.17)where we have defined (A.18)

The ratio q ≡ I_{env}/I_{core} is then approximated by (A.19)

We obtain the ratio c/a (from Eqs. (16.49) and (16.50) of Unno et al.) as (A.20)where we have used στ_{p} ~ 2n_{p}π. A exponential term is present in the Unno et al expression with the argument being an integral over the evanescent region between the p and gcavities. As this region is quite narrow in our models for the considered modes, the exponential is taken to be 1. Nevertheless the width of the evanescent region depends on the considered mode, and in some cases, for accurate quantitative results, it might be necessary to include effects of the evanescent zone with a finite width.

We now use the approximate expressions Eqs. (A.12) and (A.18) in order to derive for the ratio τ_{p}/τ_{g} in terms of observable quantities (A.23)where for convenience we have defined y = ν/Δν, and (A.24)with the period spacing for g modes (A.25)We also write (A.26)so that we obtain
© ESO, 2012
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