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Appendix A: Power-spectrum modelling and optimisation

Appendix A.1: Modelling the power spectrum

To describe the observed power spectral density of a mode peak in the frequency power spectrum, we use a standard Lorentzian function (see, e.g., Anderson et al. 1990; Gizon & Solanki 2003) given by (A.1)The use of a Lorentzian function for the mode line profile comes from the nature of solar-like p-modes; the modes are stochastically driven by turbulent convection in the outer envelope after which they are intrinsically damped (see, e.g., Goldreich et al. 1994). In this equation, H_nlm is the mode height, ν_nlm is the resonance frequency of the mode, while Γ_nl is a measure of the damping rate of the mode and gives the FWHM of L_nlm(ν).

For slow stellar rotation the star is generally assumed to rotate as a rigid body and the modes will to first order be split as (Ledoux 1951) (A.2)Here m is the azimuthal order of the mode, Ω is the angular rotation rate of the star, and C_nl is a dimensionless constant that describes the effect of the Coriolis force (the Ledoux constant). For high-order low-degree solar-like oscillations, like the ones we wish to analyse, this quantity is of the order C_nl< 10^-2, and is therefore neglected. In this way we see that the splitting due to rotation between adjacent components of a multiplet will to a good approximation be given by ν_s = Ω/2π.

In assuming equipartition of power between the components of a multiplet (i.e. no assumed preference in the excitation for prograde over retrograde propagating modes), it is possible to calculate the geometrical modulation of the relative visibility between the 2l + 1 multiplet components as a function of i_⋆ as (see, e.g., Dziembowski 1977; Gizon & Solanki 2003) (A.3)where are the associated Legendre functions.

With this, the limit spectrum (noise-free) to be fit to the power spectrum is as expressed in Eq. (1) (see Fig. 2). By comparing Eqs. (A.1) and (1), we see that the height is given by ℰ_lm(i)S_nl. By assuming equipartition of energy between different radial orders, this can be written as (A.4)The factor is a measure of the relative visibility in power (primarily set by partial cancellation) between non-radial and radial (l = 0) modes, while α_{l = 0}(ν) represents a (mainly) frequency-dependent height modulation for the radial modes, generally represented by a Gaussian centred on the frequency of maximum oscillation power, ν_max.

Appendix A.2: Optimisation procedure

The fitting of Eq. (1) to the power spectrum is made in a Bayesian manner by mapping the posterior probability: (A.5)Here p(Θ | I) is the prior probability assigned to the parameters Θ from any prior information I, and p(D | Θ,I) is the likelihood of the observed data D given the parameters Θ. The posterior is approximated using the affine invariant MCMC sampler emcee (Foreman-Mackey et al. 2013; see also Hou et al. 2012). Using the emcee routine, the posterior distribution is mapped, after which parameter estimates are evaluated as the median of the respective marginalised distributions (see Sect. 4). In the sampling we enable the parallel tempering scheme of emcee and use five temperatures with tempering parameters set as β_i = 1.2^{1 − i} (Benomar et al. 2009; Handberg & Campante 2011). The affine invariant character of the emcee sampler ensures that it works efficiently in spite of linear parameter correlations, which are a problem for many MCMC algorithms. In our optimisation we make both a fit to a large (full fit) and a small (small fit) frequency range (see Sect. 4 for details). We employ 1500 (full fit) and 2000 (small fit) walkers, all initiated from a sampling of the prior distributions. Each walker is stopped after 10 000 steps, after which we thin the chains by a factor of 10 (full fit) or 5 (small fit). We cut away a burn-in part of each chain based on the Geweke⁷ statistics (Geweke 1992), and check for good mixing using the autocorrelation time of the chains and by performing a visual inspection of the traces of walkers in parameter space. We refer to Handberg & Campante (2011) and references therein for further details on the MCMC nomenclature and Foreman-Mackey et al. (2013) for the specifics of the emcee sampler. To ensure better numerical stability we map the logarithm of the posterior with the description of the log-likelihood function from Anderson et al. (1990) and Toutain & Appourchaux (1994). With regard to priors, we use top-hat priors for location parameters (e.g., ν_nl) and scale invariant modified Jeffryes’ priors for scale parameters (e.g., S_{n 0}). To decrease the computation time the limit-spectrum (Eq. (1)) was only fit to the frequency range that includes the identified oscillation modes (see Sect. 4.1 for further details). To better constrain the stellar noise-background in the relatively narrow range occupied by the oscillation modes, Eq. (2) was first fit to the power spectrum in the frequency range from 100 − 8496 μHz (the upper limit is the approximate Nyquist frequency of SC data) and included either one or two characteristic time-scales corresponding to the contributions from granulation only or granulation and faculae (this lower-limit frequency ensures that the activity component can be omitted). We also added a Gaussian function to Eq. (2) to account for the power excess from solar-like oscillations seen in HAT-P-7. Using the deviance information criterion (DIC; Spiegelhalter et al. 2002), we found that only one component is needed to describe of the background. The medians of the posteriors from this fit were then used to fix the background in the fit of Eq. (1).

Appendix B: Line widths and visibilities

Appendix B.1: Line widths

The line widths can have a strong impact on the estimated splitting because a small splitting could be equally well fit by a slightly larger line width. This is especially a problem when the splitting is smaller than the line width. We show in Fig. B.1 our fitted line widths for the radial (l = 0) modes and their associated uncertainties. As a sanity check, we may first compare the mode line width at the frequency of maximum power, ν_max, with the estimate from Eq. (2) of Appourchaux et al. (2012) and using combined values from their Table 2. Using the spectroscopic temperature of T_eff = 6350 ± 126 K, yields Γ ≈ 3.5 ± 1.1 μHz. From the three l = 0 modes closest to ν_max we find (central) values of Γ between 5.5 and 6.4 μHz (see Fig. B.1), which is higher than expected from the Appourchaux et al. (2012) formulae.

Fig. B.1 Measured line widths for radial-order modes are given by the open symbols as a function of frequency (left axis), and plotted with associated errors. Theoretical linear damping rates, multiplied by two, are shown by the solid curve (right axis). The dotted curve shows smoothed damping rates, multiplied by a factor of 2.2. Inputs to the theoretical model were taken from the best-fit GARSTEC model, e.g. the radius at the base of the surface convection zone (Rbcz/R⋆ = 0.862).

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Additionally, we estimated linear damping rates, η, which we assumed to be approximately equal to half of the observed line widths, i.e. Γ ≃ ηπ^-1, if η is in units of angular frequency. The outcome is shown in Fig. B.1. The computations included a full non-adiabatic treatment of the pulsations and convection dynamics. Both the convective heat and momentum (turbulent pressure) fluxes were treated consistently in the equilibrium and pulsation computations using the nonlocal generalisation of the time-dependent convection model by Gough (1977a,b). The computations were carried out as described by Houdek et al. (1999) and Houdek & Gough (2002). For the non-local convection parameters we adopted the values a² = 900 and b² = 2000, and the mixing-length paramter was calibrated to obtain the same depth of the surface convection zone as in the best GARSTEC model. For the anisotropy parameter, Φ (see Houdek & Gough 2002), the value 2.50 was adopted.

After applying a median smoothing filter to ηπ^-1, with a width in frequency corresponding to five radial modes, the result of 2.8 μHz at ν_max of the oscillation heights lies within the error bars of the observational scaling relation by Appourchaux et al. (2012). To fit the observations in Fig. B.1 we multiplied the median-smoothed estimates by a factor of 2.2 (dotted curve in Fig. B.1), which agrees with previous comparisons between line width observations and model estimates for hotter solar-like stars (see, e.g., Houdek 2006, 2012).

Appendix B.2: Visibilities

For the visibilities (see Appendix A), we estimate and from the small fit. These agree reasonably well with the theoretical values of and estimated from the tables of Ballot et al. (2011). We do note, however, that this agreement is no guarantee for correct values because some stars deviate from the simple theoretical estimates (see, e.g., Lund et al. 2014). If we calculate the visibilities using the method described in Ballot et al. (2011), but adopt a quadratic limb-darkening (LD) law and measured LD-parameters from fits to the planetary transit by Van Eylen et al. (2013) and Morris et al. (2013), we obtain values that are slightly lower than those from theoretical LD parameters: and . These values agree within the errors with the fit values, which is encouraging given the very simplified assumptions adopted in the Ballot et al. (2011) calculation, where for instance all non-adiabatic effects are neglected.

Appendix C: Determining (B – V)₀

We determined the photometric stellar parameters for HAT-P-7 by combining asteroseismic results with the Infrared flux method (IRFM) (see Silva Aguirre et al. 2011b, 2012). We adopted our seismic log g and the spectroscopic metallicity from Huber et al. (2013b, 2014) and used the IRFM implementation described in Casagrande et al. (2014), where different three-dimensional reddening maps are used to constrain extinction. At a distance of 320 pc (approximate distance to HAT-P-7 determined by Pál et al. 2008), reddening varies between 0.02 <E⁽B − V⁾< 0.03.

Unfortunately, optical measurements of HAT-P-7 are quite uncertain, and depending on whether the Tycho2 (Høg et al. 2000) or APASS (Henden et al. 2009) photometry is used, the resulting T_eff will vary anywhere between 6350 and 6650 K. At the magnitude of our star, Tycho2 photometry becomes increasingly uncertain (Høg et al. 2000) (although its T_eff would be in overall good agreement with the spectroscopic estimate of T_eff = 6350 ± 126 K), while the APASS (B − V)₀ index (i.e., after correcting it for reddening) is almost as red as the solar one (Ramírez et al. 2012), thus suggesting a T_eff close to solar. The higher T_eff is, however, confirmed by the (reddening corrected) J − K_s index of HAT-P-7 (indeed bluer than the solar one; Casagrande et al. 2012). From these considerations we thus discard the APASS (B − V)₀ as faulty, and adopt a photometric T_eff = 6500 ± 150 K, where the generous errors account for the discussed uncertainties. With this T_eff and [Fe/H] = 0.26, we can invert the colour-T_eff-[Fe/H] relation of Casagrande et al. (2010), which returns an intrinsic (i.e., unreddened) colour of (B − V)₀ = 0.455 ± 0.040 mag. For spectroscopic T_eff and uncertainty, its (B − V)₀ = 0.495 ± 0.022.

Synthetic photometry offers an alternative way of assessing the (B − V)₀ colour for HAT-P-7 (all synthetic quantities are obtained by interpolating at the “known” physical parameters of the star, and are thus unaffected by reddening). We use the large grid of MARCS (Gustafsson et al. 2008) synthetic colours and interpolation routines provided by Casagrande & VandenBerg (2014) to infer the (B − V)₀ index of HAT-P-7, for the spectroscopic parameters and asteroseismic log g.

To estimate the uncertainty in the synthetic (B − V)₀, we also compute all possible T_eff and [Fe/H] combinations allowed by the spectroscopic uncertainties (while the seismic log g is so precisely known that changing it makes no difference). With this procedure, we obtain (B − V)₀ = 0.498 ± 0.020, in excellent agreement with the estimate from the empirical colour-T_eff-[Fe/H] relation when using the spectroscopic T_eff.

For the gyrochronology calculation we used the value (B − V)₀ = 0.495 ± 0.022 from the IRFM when using the spectroscopic T_eff.

Appendix D: Peak-bagging results

© ESO, 2014