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Appendix A: How rotational splittings are fitted

A.1. Large separation and gravity mode spacing

The first step for identifying the red giant oscillation spectrum is, as for all stars showing solar-like oscillations, the correct identification of the radial mode pattern, in order to locate precisely the location of the theoretical pure dipole pressure modes. The fit of the radial modes depends mainly on the accurate determination of the large separation. According to the universal red giant oscillation pattern (Mosser et al. 2011b), the surface offset and the curvature of the ridge are functions of the large separation. In practice, small residuals due to glitches (Miglio et al. 2010) can induce a frequency offset of about, typically, Δν/50. Thus, a second free parameter, simply a frequency offset, or equivalently an offset of ε less than 0.02 (Eq. (1)), is useful for providing the best fit of the radial ridge. The location of the dipole ridge with respect to the radial ridge is given by the small separation d₀₁ (Eq. (1)), which is a function of the large separation (Mosser et al. 2011b).

The fit of the mixed-mode pattern is based on two free parameters: the period spacing ΔΠ₁ and the coupling constant q, as defined by Eq. (9) of Mosser et al. (2012c), which closely follows the formalism of mixed modes given by Unno et al. (1989). In order to determine ΔΠ₁ on the RGB, it is worthwhile to consider that this period is a function of the large separation. For the low-mass stars of the RGB with a degenerate helium core, a convenient proxy is given by the polynomial development (A.1)with Δν in μHz and ΔΠ₁ in s, according to Fig. 3 of Mosser et al. (2012c). When the rotational splitting is larger than half the mixed mode spacing at ν_max, this step cannot be done independent of the next one.

A.2. Rotational splittings

Great care must be taken to disentangle the splittings from the mixed mode spacings. Three major cases have to be considered for fitting the rotational splittings.

• If splittings are small and almost uniform with frequency, exceptthe modulation depicted by ℛ (Eq. (4)), then the estimate is straightforward. The unknown stellar inclination can be derived from the mode visibility, which depends on the azimuthal order m. According to the probability of having an inclination i proportional to sini, in most cases doublets with m = ± 1 are observed. Note that, even if the components m = −1 and  + 1 have the same visibility, they may in practice present different heights, due to the stochastic excitation of the modes. Such splittings smaller than the mixed-mode spacings are seen in the lower stages of the RGB and in the clump (Fig. A.1).

• If apparent splittings at ν_max seem to increase with increasing frequency, then the most plausible solution is that δν_rot is close to half the mixed-mode spacing at ν_max. These apparent splittings result in fact from a mixing of the splittings embedded with the spacings. Such a situation occurs when the apparent splittings are composed of the m = ± 1 component of the mixed mode order n_m and of the m = ∓ 1 component of the adjacent orders n_m ± 1. The true splittings, significantly larger than the apparent splittings, are almost uniform for g-m modes. This uniformity is used for iterating the solution. Such cases occur most often for RGB stars with Δν in the range [9–12 μHz] (Fig. A.4).

• If apparent splittings seem very irregular, then the most plausible solution is that δν_rot is much larger than half the mixed-mode spacing at ν_max. In fact, the apparent splittings are complex structures resulting from a mixing of components of two or three different mixed-mode orders. A careful visual inspection is necessary to disentangle them. The mixed-mode asymptotic expression and the empirical expression of the rotational splitting are accurate enough for resolving complex cases that occur for RGB stars with Δν ≤ 9 μHz (Fig. A.5).

We have used gravity échelle diagrams to represent the mixed modes (Bedding et al. 2011; Mosser et al. 2012c). Due to the complexity of the features caused by embedded splittings and mixed modes spacings, the échelle diagrams cannot be used to identify the rotational splittings, but are useful for improving the accuracy of the fit. In the examples shown (Fig. A.6), a 10-s shift between the periods of the observed and modeled peaks correspond to an accuracy in frequency of about Δν/100.

A.3. A dipole mode forest?

The complete fit of the rotational splittings is based on three parameters: the maximum splitting δν_rot and the two parameters λ and β entering the definition of ℛ. The best fit is provided by correlating the observed multiplets with synthetic multiplets.

Since the parameters λ and β are found to vary in narrow ranges, the solution for inferring δν_rot (and simultaneously ΔΠ₁ on the RGB with low Δν) is based on considering them as constants. As a result, five free parameters are enough for fitting the whole red giant oscillation spectrum. Variation of λ and β allows a better fit. The stellar inclination can be derived from the ratio of the visibility of the m = ± 1 components compared to the central component.

In a typical spectrum, more than 30 mixed-mode orders, representing about 60 to 120 individual modes with a height larger than eight times the background are simultaneously fitted. The typical accuracy of the fit, of about Δν/200 or better, is enough for avoiding any confusion in almost all cases, except for the most evolved RGB stars.

Finally, with the identification of the mixed mode spacings and of the rotational splittings, the dipole mode forest becomes a well-organized garden à la française.

Appendix B: Two-layer model

B.1. Core and surface contributions

In order to estimate the contribution of the core and surface rotation, we simplify the stellar stratification to a 2-layer model. We denote by δν_c and δν_s the rotational frequency of the core and at the surface, respectively, and δν_g/2 and δν_p the measured splitting on g and p modes, respectively. The factor 1/2 in δν_g/2 accounts for the Ledoux coefficient. The contributions of the surface and of the core are written (B.1)The coefficient x_p and x_g are derived from the rotational kernels. From the solution (B.2)and from the observation of the splitting of p-m modes indicating δν_p ≃ δν_g/4 (a factor of about 1/2 comes from 1 − λ in Eq. (4), an another factor of 1/2 comes from the Ledoux coefficient), one derives that the measure of δν_g is an indicator of the core rotation (B.3)For an RGB star at the bump with Δν = 5 μHz, the values x_p and x_g derived from the kernels give η = 1.06 ± 0.04, very close to unity. A less evolved star, as considered by Beck et al. (2012), whose mixed modes correspond to much smaller radial gravity orders, has . Deheuvels et al. (2012a) derived a similar result for a giant with Δν ≃ 29 μHz at the bottom of the RGB. This shows that δν_rot is less dominated by the core rotation for early RGB stars. One also derives that, in all cases, the surface rotation δν_s is small, and that measuring it precisely from the g-m mode-splitting is not possible.

B.2. Link to the eigenfunction properties

The value of the coefficients x_g and x_p introduced in Eq. (B.1) can be approximated by the expression of the rotational splitting (Eqs. (8) and (9)). Basically, the integration of the wave function has a contribution varying as the number of nodes in the core and in the envelope, respectively. As a consequence, x_g ∝ n_g and x_p ∝ n_p. In order to more precisely account for the complex form of the wave function, we suppose: (B.4)\newpage\noindentwith γ_g < 0 to account for the negative value of n_g. The validity of this development implicitly assumes that γ_p and  | γ_g |  are constant not so far from unity. Hence, neglecting δν_s in Eq. (B.1), we derive: (B.5)The radial order n_p and n_g have to be estimated at the frequency ν_max where the oscillation amplitude is maximum. Then, we can derive that η = δν_c/δν_g is related to the global seismic parameters Δν and ΔΠ₁, where ΔΠ₁ is the period spacing of gravity modes: (B.6)The fit of the integrated kernels calculated at different evolutionary stages gives γ ≃ 0.65. This phenomenological result based on a simple two-layer model is to be considered as a proxy only.

© ESO, 2012