Online material

Appendix A: Details on the orbital fit

Appendix A.1: Errors on the radial velocities

The RV of β Pictoris A measured within the same day are extremely variable because of the activity of the star. During the HARPS monitoring of the star, β Pic A was either observed multiple times during a night to evaluate and average the stellar activity, or at a single time (Borgniet et al., in prep.). We averaged the data over one day to estimate a daily RV mean. To estimate the error on the RV corresponding to a night that properly account for the stellar noise, we assumed that the intrinsic RV variability is sinusoidal: A × sin(ω × t + φ) + C(t), with A the amplitude of the variability, ω the angular frequency, t the time, φ the phase, and C an offset velocity. RV measurements performed over one single day can be regarded as successive values of a random variable following this law with random t. The resulting random RV has the following probability function: (A.1)The variance of this law is A²/ 2. Taking the arithmetic mean of n independent measurements over one night gives an estimate of the offset velocity C with A/ as error. We now need to estimate A. We assume that C varies with time, but A does not. For a given day with the highest number of measurements N, the statistical variance SN of these data is calculated. An accurate, unbiased estimator of A² is 2 × (N/ (N − 1)) × SN, so that for any other day with n measurements the error can be estimated to be , where s is the mean of the given HARPS RV measurment errors of the day. This way, errors are reduced for a day with many measurements and kept large for days with one or two measurements.

Appendix A.2: Choices of the priors

Priors on the orbital parameters are identical to those used in Chauvin et al. (2012) when only the system astrometry is accounted for in the orbital fit. Changes to them appear to have little influence on the posterior distributions of orbital parameters. In contrast, this is not the case for the mass determination because of the weak constraint provided by the RV data. The most straightforward prior we can assume for the amplitude of the RV curve of β Pic A K is linear, but a logarithmic prior (linear in lnK) is also worth considering because K is proportional to P^{− 1/3} (where P is the orbital period), and a logarithmic prior for P was already assumed. Figure 1 shows the posterior mass determination for both priors. Because of the activity of the star, the data are compatible with planet masses down to virtually 0. But a lower cutoff at 2 M_Jup was assumed to remain compatible with the observed luminosity of the planet. The linear prior nevertheless appears to favor larger masses than the logarithmic prior. Then the major difference resides in the shape of the posterior distribution. The linear prior exhibits a clear peak around 6 M_Jup. This difference illustrates the difficulty in deriving a relevant fit of the mass of β Pic b. Obviously, the RV data are too noisy to allow a clear determination, but i) a conservative upper limit is confirmed; and ii) the peak around 6 M_Jup needs to be confirmed with future data.

Appendix B: Samples of comparison spectra

For the purpose of the empirical analysis, we used four samples of spectra of ultracool MLT dwarfs found in the literature. The SpecXPrism library⁴ represents sample 1. Sample 2 is composed of spectra of M and L dwarfs with features indicative of low surface gravity (Allers & Liu 2013; Manjavacas et al. 2014; Liu et al. 2013; Schneider et al. 2014). The third sample is made of spectra of members of 1−150 Myr old young moving groups and clusters (Lodieu et al. 2008; Rice et al. 2010; Bonnefoy et al. 2014; Gagné et al. 2014). The fourth sample is composed of spectra of young MLT companions (Patience et al. 2010; Lafrenière et al. 2010; Wahhaj et al. 2011; Bonnefoy et al. 2010, 2014).

Appendix C: Description of the LESIA model grids

Baudino et al. (2014) developed a radiative-convective equilibrium model for young giant exoplanets in the context of direct imaging. The input parameters are the planet surface gravity (log g), effective temperature (T_eff), and elemental composition. Under the additional assumption of thermochemical equilibrium, the model predicts the equilibrium-temperature profile and mixing-ratio profiles of the most important gases. Opacity sources include the H₂-He collision-induced absorption and molecular lines from H₂O, CO, CH₄, NH₃, VO, TiO, Na, and K. Line opacity is modeled using k-correlated coefficients pre-calculated over a fixed pressure-temperature grid. Absorption by iron and silicate cloud particles is added above the expected condensation levels with a fixed scale height and a given optical depth at some reference wavelength. To study β Pic b, we built five grids of models with T_eff between 700 and 2100 K (100 K increments), log g between 2.1 and 5.5 dex (0.1 dex increments), and solar system abundances (Lodders 2010). One model grid was created without clouds (hereafter set I). We added three grids with cloud particles located between condensation level and a 100 times lower pressure, with a particle radius of 30 μm (τ = 0.1, 1, 3; hereafter set II, III, IV), a scale height equal to the gas scale height, and optical depths (τ_cloud) of 1τ and 0.15τ at 1.2 μm for Fe and Mg₂SiO₄, respectively (assuming the same column density for both clouds). We used an additional grid (hereafter set V) with a particle radius of 3 μm and τ_cloud of 1 and 0.018. The grid properties are summarized in Table C.1.

Appendix D: Fit of the spectral energy distribution

The planet SED was built from the Y_s and CH_4S,1% band photometry reported reported in Males et al. (2014), J,H, L′ and M′ band photometry Bonnefoy et al. (2013), K_s-band photometry from Currie et al. (2013), and NB_4.04 band magnitude from Quanz et al. (2010). The SED and spectral-fitting procedures are described in Bonnefoy et al. (2013) and Bonnefoy et al. (2014), respectively.

	Fig. D.1 Comparison of β Pic b SED to best-fitting synthetic spectra (solid line) and fluxes (horizontal lines) from the BT-SETTL, DRIFT-PHOENIX, and LESIA atmospheric models grids.
Open with DEXTER

© ESO, 2014