Using our stellar parameters, we derived an estimate of the spectroscopic distances of XO-2N and XO-2S by means of the following procedure. We generated Monte Carlo (MC) normal distributions for each spectroscopic parameter Teff, [Fe/H], and log  g, composed of 10 000 random values and centred on the best estimates (Table 2). By keeping the stellar radii fixed to the values listed in Table 2 (for XO-2N we used the most accurate estimate), for each MC simulation we first determined the stellar bolometric luminosity L∗ (in solar units) from the Stefan-Boltzmann law, and then we derived the absolute bolometric magnitude Mbol from the relation Mbol = 4.75 − 2.5·log (L∗). By estimating the appropriate bolometric correction (BC), a value for the absolute magnitude in V-band MV was then obtained. The BC term was evaluated using the code provided by Casagrande & VandenBerg (2014). An additional input is the colour excess E(B − V) of the star, which we derived through the relation E(B − V) = AV(s)/3.1, where AV(s) is the interstellar dust extinction in V-band integrated at the distance s of the star (in pc) and measured along the line of sight. We derived AV(s) by adopting a simplified model of the local distribution of the interstellar dust density (Drimmel & Spergel 2001), expressed by the relation ρ = ρ0·sech2(z/hs), where z is the height of the star above the Galactic plane and hs is the scale-height of the dust, for which we adopted the value of 190 pc. The term z is related to the distance s and the Galactic latitude of the star b by the formula z = s·sinb. From this model we obtained the relation AV(s) = AV(tot)·sinh(s·sin b/hs)/cosh (s·sin b/hs), where AV(tot) is the interstellar extinction in V band along the line of sight integrated through the Galaxy, and can be estimated from 2D Galactic maps. For this purpose we used the value AV (tot) = 0.16 mag derived from the maps of Schlafly & Finkbeiner (2011)6. By assuming s = 150 pc as a prior distance of the stars (Burke et al. 2007), we obtained AV (150 pc) ~ 0.06 mag, corresponding to E(B − V) = 0.019 mag. This is the value used as input to the code of Casagrande & VandenBerg (2014) to obtain a first guess of the BC in V-band. This in turn was used in the distance modulus formula V − (Mbol − BCV) = 5·log (s) −5 − AV(s), to obtain a new value for the stellar distance s. The new distance was used to repeat the procedure iteratively, by determining at each step a new value of AV(s) and BCV(s), and finally another estimate of s. When the absolute difference between the last and previous calculated values of s was below 0.1 pc, the iterative process was interrupted and the last derived value for s was assumed as the distance of the star for the Nth Monte Carlo simulation. The adopted estimates for the distance of the XO-2 components are the median of the distributions of the 10 000 MC values, and the asymmetric error bars defined as the 15.85th and 68.3th percentile (see Table 1). Because they are model-dependent, we do not argue here whether the difference of ~2.5 pc between the XO-2S and XO-2S distances is real. We only note that the two values are compatible within the uncertainties and that our best estimate for the distance of XO-2N locates the star a few parsec closer than reported by Burke et al. (2007).
Citation: Drimmel & Spergel 2001
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