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Appendix A: Atmospheric model

First, we describe opacity models for the water vapor atmosphere. We define the Planck-type (κ^P) and the Rosseland-type mean opacities (κ^r) as where ν is the frequency; κ_ν the monochromatic opacity at a given ν; T_⋆ the stellar effective temperature; T_atm the atmospheric temperature of the planet; and B_ν the Planck function. The subscripts, “th” and “v”, mean opacities in the thermal and visible wavelengths, respectively. In this study, we assume T_⋆ = 5780 K. We adopt HITRAN opacity data for water (Rothman et al. 2009) and calculate mean opacities for 1000 K, 2000 K, and 3000 K at 1, 10, 100 bar. Mean opacities are fitted to power-law functions of P and T, using the least squares method; where P is the pressure and T the temperature.

In this study, we basically follow the prescription developed by Guillot (2010) except for the treatment of the opacity. We consider a static, plane-parallel atmosphere in local thermodynamic equilibrium. We assume that the atmosphere is in radiative equilibrium between an incoming visible flux from the star and an outgoing infrared flux from the planet. Thus, the radiation energy equation and radiation momentum equation are written as and the atmosphere in radiative equilibrium satisfies (A.13)where J_v (J_th), H_v (H_th), and K_v (K_th) are, respectively, the zeroth-, first-, and second-order moments of radiation intensity in the visible (thermal) wavelengths, m the atmospheric mass coordinate, dm = ρdz, where z is the altitude from the bottom of the atmosphere, ρ the density, and B the frequency-integrated Planck function, (A.14)where σ is the Stefan-Boltzmann constant. We assume here that thermal emission from the atmosphere at visible wavelengths are negligible, so that B_ν ~ 0 in the visible region. The six moments of the radiation field are defined as where J_ν is the mean intensity, 4πH_ν the radiation flux, and 4πK_ν/c the radiation pressure (c is the speed of light).

We integrate three moments of specific intensity, J_ν,H_ν and K_ν, over all the frequencies: where I_ν,μ is the specific intensity and θ the angle of a intensity with respect to the z-axis, μ = cosθ. The energy conservation of the total flux implies (A.20)where T_irr is the irradiation temperature given by (A.21)where R_⋆ is the radius of the host star and a the semi-major axis.

For the closure relations, we use the Eddington approximation (e.g. Chandrasekhar 1960), namely, For an isotropic case of both the incoming and outgoing radiation fields, we find boundary conditions of the moment equations as follows (see also Guillot 2010, for details): Thus, we integrate Eqs. (A.9)–(A.13) over m numerically, using mean opacities of (A.5)–(A.8) and boundary conditions of (A.24)–(A.26), and then determine a T–P profile of the water vapor atmosphere. We assume that the boundary is at P₀ = 1 × 10^-5 bar. The choice of P₀ (≤1 × 10^-5bar) has little effect on the atmospheric temperature-pressure structure. T₀ is determined in an iterative fashion until abs(T₀ −  [πB(m = 0,P₀,T₀)/σ] ^1/4) ≤ 0.01 is fulfilled. Then we integrate Eqs. (A.9)–(A.13) over m by the 4th-order Runge-Kutta method, until we find the point where dlnT/dlnP ≥ ∇_ad. The pressure and temperature, P_ad and T_ad, are the boundary conditions for the convective-interior structure (see Sect. 2.1).

In Fig. A.1, we show the P–T profile for the solar-composition atmosphere with g = 980 cm s^-2, T_int = 300 K, and T_irr = 1500 K (dotted line). In this calculation, we take and as functions of P and T from Freedman et al. (2008) and calculate and , for P = 1 × 10^-3,0.1,1,10 bar and T = 1500 K from HITRAN and HITEMP data that include H₂, He, H₂O, CO, CH₄, Na, and K for the solar abundance respectively as (A.27)by use of (A.2). The thin and thick parts of the dotted line represent the radiative and convective zones, respectively.

In addition, we test our atmosphere model by comparing it with the P–T profile derived by Guillot (2010) with γ = κ_v/κ_th = 0.4 (solid line), which reproduces more detailed atmosphere models by Fortney et al. (2005) and Iro et al. (2005, see Fig. 6 of Guillot 2010). As seen in Fig. A.1, our atmospheric model yields a P–T profile similar to that from Guillot (2010). In our model, temperatures are relatively low compared with the Guillot (2010) model at P ≲ 40 bar, which is due to difference in opacity. In our model, deep regions of P ≳ 40 bar are convective, while there is no convective region in the Guillot (2010) model because of constant opacity. We have compared our P–T profile with the Fortney et al. (2005)’s and Iro et al. (2005)’s profiles, which are shown in Fig. 6 of Guillot (2010) and confirmed that our P–T profile in the convective region is almost equal to their profiles. Of special interest in this study is the entropy at the radiative/convective boundary, because it governs the thermal evolution of the planet. In this sense, it is fair to say that our atmospheric model yields appropriate boundary conditions for the structure of the convective interior.

	Fig. A.1 Temperature–pressure profiles for a solar-composition atmosphere (see the details in text). The solid (red) and dotted (green) lines represent both Guillot (2010)’s (γ = 0.4) and our model’s, respectively. The thin and thick parts of the dotted line represent the radiative and convective regions, respectively. We have assumed g = 980 cm s-2, Tint = 300 K, and Tirr = 1500 K.
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Finally, we describe an analytical expression for our atmospheric model. We basically follow the prescription developed by Heng et al. (2012), except for the treatment of the opacity. As Heng et al. (2012) mentioned, it would be a challenging task without assumption of constant and to obtain analytical solutions for J_v and H_v. Here we assume and are constant throughout the atmosphere. We differentiate (A.9) and (A.10) by m and obtain where μ² = K_v/J_v. Assuming J_v = H_v = 0 as m → ∞, we obtain (A.30)where and J_v,0 and H_v,0 are the values of J_v and H_v evaluated at m = 0, respectively. In general, the heat transportation, such as circulation, produces a specific luminosity of heat. Heng et al. (2012) introduced the specific luminosity as Q, which has units of erg s^-1 g^-1. Q can be related to the moments of the specific intensity and we obtain (A.31)We integrate Eq. (A.31) and obtain (A.32)where H_∞ is the value of H evaluated at m → ∞ and (A.33)To obtain H_th and J_th, we substitute Eq. (A.31) in Eqs. (A.11) and (A.12) and integrate by m. Then we obtain where f_Kth = K_th/J_th, f_Hth = H_th/J_th, and (A.36)That is, we obtain (A.37)where and . In our conditions, we assume , f_Kth = 1/3, f_Hth = 1/2 and Q = 0. Consequently, we obtain

the temperature profile as (A.41)where(A.42)If we assume and , Eq. (A.41) agrees with Eq. (27) of Heng et al. (2012).

© ESO, 2014