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Volume 549, January 2013
Article Number A75
Number of page(s) 13
Section Stellar structure and evolution
Published online 21 December 2012

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Appendix A: Properties of eigenfunctions and mode inertia

A.1. Displacement eigenfunctions

thumbnail Fig. A.1

Displacement eigenfunctions as a function of the normalized radius r/R. In blue the horizontal component to the displacement eigenfunction z2 (Eq. (7)) and in red the radial component to the displacement eigenfunction z1 (Eq. (6)). From top to bottom  = 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 z2 and oscillates with a large number of nodes, as is typical of a high-order gravity mode. The largest maximum amplitude corresponds to the most g-dominated mode whereas the smallest maximum amplitudes arise for the p-m modes ν1 and ν6.

The maximum amplitude of z2 occurs deep in the g-cavity, at the same radius for all modes of the pattern. The region of non-negligible 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 (x2 ~ 0.08, Table 3)

A.2. Behavior of β and βcore with ζ

For  = 1 modes of red giants, the term z2z1 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, , 2z1z2 in the g-cavity (see Fig. A.1), then βcore,nl ≈ βcore where for l = 1 modes, we derive hence (A.6)Numerical values for model M1 confirm that βcore increases linearly with ζ with a slope 1/2 (Fig. A.2). For g-m modes (ζ ~ 1), βcore dominates with a nearly constant value of 0.5. P-m modes correspond to the teeth of the saw-type variation in βenv and the ratio βenv/βcore ~ 0.25 (Fig. A.2).

thumbnail Fig. A.2

Top: βenv and βcore as a function of νν for model M1. Bottom: same as top for the ratio βcore 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 = Icore/I as a function of νν. 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 (~p-propagative 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 ξr = 1 at the surface. (A.8)with (A.9)where we have assumed σ2 ≫ N2 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 σ2 in the expression for kr (i.e. kr ~ σ/cs). However this is not the case when kr 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 x3 and from the center. We take f = 1 unless specified otherwise. The last equality in Eq. (A.10) is obtained assuming στp ≫ 1.

  • The core (~g-propagative 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 στg ≫ 1, therefore the inertia in the core can be approximated as (A.17)where we have defined (A.18)

  • The ratio q ≡ Ienv/Icore 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 στp ~ 2npπ. 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 g-cavities. 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.

  • The ratio q is then eventually approximated by (A.21)

  • For the relative core inertia ζ = Icore/I, (A.22)

  • We now use the approximate expressions Eqs. (A.12) and (A.18) in order to derive for the ratio τp/τg in terms of observable quantities (A.23)where for convenience we have defined y = νν, 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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