Free Access
Volume 553, May 2013
Article Number A97
Number of page(s) 25
Section Stellar structure and evolution
Published online 16 May 2013

Online material

Appendix A: Period fit and mode identification from the optimal model of PG 1336–018

The period match and mode identification obtained for the best-fit model of PG 1336-018 are given in Table A.1. The relative and absolute differences for each pair of the 25 observed/theoretical modes are provided, in period and frequency, ΔX/X (in %), ΔP (in s), and Δν (in μHz). On average, the relative dispersion between the observed and theoretical periods is 0.18%, which is similar to the best-fit model of second generation uncovered by Charpinet et al. (2008b) that led to 0.17%.

Independent constraints on the mode identification in PG 1336−018 exist from time-resolved spectroscopy (Vučković et al. 2009). Their detailed line-profile analysis excluded the (ℓ, |m|) =  (3,3), (1,0), (2,1), or (2,0) modes for the 5435 μHz pulsation (f5 in Table A.1), although data were too noisy to uniquely identify this pulsation with the radial or sectoral dipole/quadrupole modes. The 5435 μHz f5 pulsation is identified as the fundamental radial mode (k,ℓ,m) = (0,0,0) in Table A.1, which is consistent with the results of Vučković et al. 2009 (see in particular their Fig. 13).

It is also interesting to look back at the three uncertain periods of Kilkenny et al. (2003) that were not included in the

seismic search, 184.04 s (f10), 178.96 s (f13), and 173.59 s (f8). The periods f10 and f8 all have an acceptable counterpart in the best-fit model theoretical spectrum (see Table A.1). The period f10 can be associated with the (l,k,m) = (2,0, − 1) mode for a relative dispersion ΔX/X = −0.33% and f8 finds a counterpart at ΔX/X = −0.26% with the (l,k,m) = (4,0, − 2) mode. These matches do not significantly affect the overall quality of the fit. A l = 3 mode (not shown here) is needed to account for the last uncertain period, 178.96 s (f13). We recall, however, that the existence of these periods needs to be confirmed by additional observations.

The mode identification of the best-fit model respects quite well, qualitatively, the expected amplitude hierarchy. On average, one may expect that modes of higher l would have lower apparent amplitudes even though this argument can hardly be applied to individual identifications, since intrinsic amplitudes are not known and may vary significantly from one mode to another. According to the mode identification in Table A.1, the observed average amplitudes are 0.145% (for l =  0 modes), 0.230%, 0.183%, and 0.081%. This significant drop for l = 4 modes corresponds well to the theoretical expectation that modes with l = 4 have a much lower visibility than the l ≤ 2 modes (Randall et al. 2005, 2007).

Table A.1

Mode identification and details of the frequency fit obtained for the optimal solution.

Appendix B: Probability density functions of stellar parameters with known model uncertainties

thumbnail Fig. B.1

Probability density functions derived from asteroseismology using models with a uniform distribution of iron in solar proportions. The parameters inferred for the sdB component of PG 1336–018 are, from the upper-left panel to the lower-right panel, the mass M, log g, the radius R, loq q(H), log q(core), Xcore(C+O), and L/L. The red-hatched regions between the two vertical solid red lines shows the 1σ range containing 68.3% of the distribution. In the panel showing the mass, the filled circles with the dotted lines are the three orbital solutions for the mass of the sdB component proposed by Vučković et al. (2007) with their 1σ uncertainties. In the panel showing log g, The filled circle with the solid line and dotted line indicate the value derived from spectroscopy with its 1σ and 3σ uncertainties, respectively. Finally, in the panel showing the radius, the filled circle with the solid horizontal line indicates the 1σ error associated with the relevant orbital solution from Vučković et al. (2007). In each panel, the blue vertical dashed and dotted lines show the seismic solution and its 1σ uncertainty obtained with our standard 3G models.

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thumbnail Fig. B.2

Same as Fig. B.1, but using models with a nonuniform iron abundance profile reduced by a factor of four relative to the value expected at diffusive equilibrium.

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thumbnail Fig. B.3

Same as Fig. B.1, but using models with a nonuniform iron abundance profile reduced by a factor of two relative to the value expected at diffusive equilibrium.

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thumbnail Fig. B.4

Same as Fig. B.1, but using models with a smooth He/H transition.

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thumbnail Fig. B.5

Same as Fig. B.1, but using models with modified rates for the triple-α and 12C(α,γ)16O nuclear reactions.

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© ESO, 2013

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