Open Access
Volume 562, February 2014
Article Number A80
Number of page(s) 14
Section Planets and planetary systems
Published online 10 February 2014

Online material

Appendix A: Atmospheric model

First, we describe opacity models for the water vapor atmosphere. We define the Planck-type (κP) and the Rosseland-type mean opacities (κr) as where ν is the frequency; κν the monochromatic opacity at a given ν; T the stellar effective temperature; Tatm the atmospheric temperature of the planet; and Bν 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 Jv (Jth), Hv (Hth), and Kv (Kth) 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 Bν ~ 0 in the visible region. The six moments of the radiation field are defined as where Jν is the mean intensity, 4πHν the radiation flux, and 4πKν/c the radiation pressure (c is the speed of light).

We integrate three moments of specific intensity, Jν,Hν and Kν, over all the frequencies: where Iν,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 Tirr 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 TP 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)/σ1/4) ≤ 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, Pad and Tad, are the boundary conditions for the convective-interior structure (see Sect. 2.1).

In Fig. A.1, we show the PT profile for the solar-composition atmosphere with g = 980 cm s-2, Tint = 300 K, and Tirr = 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 PT profile derived by Guillot (2010) with γ = κv/κth = 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 PT 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 PT 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 PT 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.

thumbnail 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 Jv and Hv. Here we assume and are constant throughout the atmosphere. We differentiate (A.9) and (A.10) by m and obtain where μ2 = Kv/Jv. Assuming Jv = Hv = 0 as m → ∞, we obtain (A.30)where and Jv,0 and Hv,0 are the values of Jv and Hv 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 Hth and Jth, we substitute Eq. (A.31) in Eqs. (A.11) and (A.12) and integrate by m. Then we obtain where fKth = Kth/Jth, fHth = Hth/Jth, and (A.36)That is, we obtain (A.37)where and . In our conditions, we assume , fKth = 1/3, fHth = 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

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