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Appendix A: Models with different input physics or optimization details

Appendix A.1: Optimization of set A models with alternate prescriptions

In the following, we present other optimization models, all based on the reference physics of set A (Table 2). These models were optimized following Table 3, but with different approaches or choices of free parameters, as described below and listed in Table A.1. The results of the models are listed in Tables A.2 and A.3.

• 1.

  Cases 1− Y/2 − YAs explained in the main text, in cases 1 and 2a, b, and c, the initial helium content Y could not be adjusted because of the lack of observational constraints. Here we investigate the consequence of not deriving Y from the ΔY/ ΔZ = 2 enrichment law (as was done in cases 1 and 2 of Table 3). We sought a solution with the lowest possible initial helium content, never lower than the primordial abundance. These choices have an important impact on the results and are therefore discussed in the main text. We point out that in case A1-Y it was possible to find a solution with a primordial helium abundance. This is because no seismic constraints were used in this case. On the other hand, in cases A2a− Y, b− Y, and c− Y, we had to increase the helium abundance above the primordial one because of the seismic constraints introduced, to find a solution that agreed with both the seismic and the classical observational constraints (we found that models with low Y either have a too low temperature or a too high luminosity).

• 2.

  Cases 1-α0.550, 1-α0.826 In case 1, α_conv could not be adjusted because of the lack of observational constraints and, as often seen in papers, the solar value α_{conv, ⊙} was used. Here, we investigated the impact of other choices and we sought a solution for acceptable, extreme values of α_conv. Numerical 2D simulations of convection suggest that α_conv might differ by a few tenths of dex from the solar value (see e.g. Ludwig et al. 1999). We investigated changes of α_conv of 20 per cent around the solar value that correspond to α_conv,min = 0.550 and α_conv,max = 0.826. These choices have an important impact on the results and are therefore discussed in the main text.

• 3.

  Cases 1-ov30 No overshooting was assumed in the reference models of case 1. Here, to estimate its impact, we chose a rather high value, i.e. α_ov = 0.30. The impact on case 1 results is discussed in the main text.

• 4.

  Cases 1, 2, 7-νrad In these models, the impact of mixing that results from the radiative diffusivity associated with the kinematic radiative viscosity is investigated. Following Morel & Thévenin (2002), we added an extra mixing diffusion coefficient d_rad = D_R × ν_rad with D_R = 1 that limits gravitational settling in the outer stellar layers of stars with thin convective envelopes. As can be seen in Tables A.2 and A.3, the impact is weak and is not discussed further.

• 5.

  Case 5-allfreq In the list of frequencies extracted by Ballot et al. (2011), 31 frequencies were given, of which 28 were flagged as secure. In this model, we considered the 31 values and found that the impact is weak and that the results remain inside the uncertainty range we gave in the main text.

• 6.

  Case 6-YM, 7-YM For each optimization case, there are a range of initial helium-mass doublets (Y₀, M) that provide seismically equivalent optimized models. We investigated the Y₀ − M degeneracy by searching for optimized models with different values of Y₀ and M. As discussed in the main text, the impact on age is weak but a range of possible masses of HD 52265 is found.

  Table A.2

  Same as Table 4, but for different optimization options (see Sect. 3 and Tables A.1 and 3).

  Table A.3

  Same as Table 5, but for different optimization options (see Sect. 3 and Tables A.1 and 3)

• 7.

  Case 6-nocorrel As explained in the main text, in case 6, we took into account the correlations between the frequency separation ratios and calculated the χ² from Eq. (13). In this model the correlations are not considered and the χ² is evaluated from Eq. (14). The impact is weak and is not discussed further.

• 8.

  Case 6-interr Roxburgh & Vorontsov (2013) recently claimed that model-fitting by searching for a best fit of observed and model separation ratios at the same radial orders n is incorrect, and that the correct procedure is to compare the model ratios interpolated to the observed frequencies. We followed this recommendation here. The impact is weak and the results remain in the uncertainty range we gave in the main text.

• 9.

  Case 7-rot As explained in the main text, rotation and its effects on the transport of angular momentum and chemicals was treated as in Marques et al. (2013), and we tuned the K_w coefficient that enters the treatment of magnetic braking by winds to match the observed rotation period. The impact on age and mass is weak for this rather evolved star. A thorough study of the effects of rotation on HD 52265 will be presented in a forthcoming paper.

• 10.

  Case 7-noSE and case 7-bSE4.9. As explained in the main text, in case 7, we corrected the individual frequencies for the surface effects with the Kjeldsen et al. (2008) empirical prescription and calibrated the b_SE parameter in Eq. (12) to achieve the best match between observed and modelled frequencies. In model 7-bSE4.9, we used the solar value of b_SE calibrated by Kjeldsen et al. (2008) and found that the impact on the results is weak. In model 7-noSE, we did not correct frequencies for the surface effects. The impact is strong, in particular on age. This is discussed in the main text.

• 11.

  Case 7-pms. The main models were calculated by starting the computation at the zero-age main sequence. This model was evolved from the pre-main sequence. The impact is weak.

• 12.

  Cases 6 and 7-ov. In the models presented in the main text, overshooting was either neglected (sets A − H, K) or fixed (sets I, J). Since there are enough observational constraints to add overshooting and convective penetration as additional free parameters in cases 6 and 7, we considered this possibility here. We therefore also optimized the values of α_ov and ξ_PC. As also discussed in the main text, we found that low values of α_ov are favoured (range 0.00− 0.04) and rather high values of ξ_PC (range 0.90−1.25). This latter result, related to the oscillatory behaviour of the frequencies, close to the convective envelope agrees with the conclusions of Lebreton & Goupil (2012).

Appendix A.2: Optimization with different input physics

In the following, we present optimization models based on the different choices of input physics listed in Table 2. These models were optimized following Table 3. The results are listed in Tables A.4 and A.5. Discussions are found in the main text.

© ESO, 2014