Issue 
A&A
Volume 570, October 2014



Article Number  A54  
Number of page(s)  16  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201424326  
Published online  15 October 2014 
Online material
Appendix A: Powerspectrum 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 solarlike pmodes; 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 highorder lowdegree solarlike 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 (noisefree) 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 nonradial and radial (l = 0) modes, while α_{l = 0}(ν) represents a (mainly) frequencydependent 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 (ForemanMackey 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 burnin part of each chain based on the Geweke^{7} 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 ForemanMackey 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 loglikelihood function from Anderson et al. (1990) and Toutain & Appourchaux (1994). With regard to priors, we use tophat 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 limitspectrum (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 noisebackground 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 timescales corresponding to the contributions from granulation only or granulation and faculae (this lowerlimit 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 solarlike oscillations seen in HATP7. 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 radialorder 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 bestfit GARSTEC model, e.g. the radius at the base of the surface convection zone (R_{bcz}/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 nonadiabatic 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 timedependent convection model by Gough (1977a,b). The computations were carried out as described by Houdek et al. (1999) and Houdek & Gough (2002). For the nonlocal convection parameters we adopted the values a^{2} = 900 and b^{2} = 2000, and the mixinglength 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 mediansmoothed 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 solarlike 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 limbdarkening (LD) law and measured LDparameters 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 nonadiabatic effects are neglected.
Appendix C: Determining (B – V)_{0}
We determined the photometric stellar parameters for HATP7 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 threedimensional reddening maps are used to constrain extinction. At a distance of 320 pc (approximate distance to HATP7 determined by Pál et al. 2008), reddening varies between 0.02 <E^{(}B − V^{)}< 0.03.
Unfortunately, optical measurements of HATP7 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 HATP7 (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 colourT_{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 HATP7 (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 HATP7, 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 colourT_{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: Peakbagging results
Frequencies extracted from the MCMC peakbagging.
© ESO, 2014
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