tokens: are computed by integrating the hydrostatic equations with the equation of state given by where p is the pressure, ρ is the mass density, G is the gravitational constant, p b is the pressure at the base of the radiative envelope, and and a is the isothermal sound velocity, and R J is the radius of Jupiter. In the convective core where p ≫ p b , we obtain corresponding to a polytrope of the index n = 1, and hence for Γ 1 ≡ (∂ln p /∂ln ρ) S ≈ 2 the square of the Brunt–Väisälä frequency where On the other hand, in the radiative envelope where p ≪ p b , we have and and c denotes the adiabatic sound speed. Since p /ρ ≈ a 2 ∝ T for an ideal gas, the temperature T is constant for a constant a , suggesting an isothermal atmosphere. In this paper, following Auclair-Desrotour Leconte ( 2018 ), for the Jovian planet model of mass M = 0.7 M J with M J being the Jovian mass, we use p b = 100 bar = 10 8 dyn cm −2 and set the outer boundary R e at p = 0.01dyn cm −2 and define the planet’s radius R = R e /1.01 = 9.31 × 10 9 cm. We also assume Γ 1 = 2 in the interior. Fig. 1 is the propagation diagram where N 2 and with l = 2 are plotted as a function of r / R where L l is the Lamb frequency. Figure 1. Squared characteristic frequencies N 2 (solid lines) and (dashed lines) plotted as a function of r / R (left-hand panel) and of log p (right-hand panel), where N and L l are, respectively, the Brunt–Väisälä frequency and Lamb frequency and δ A = 10 −4 is assumed for N in the convective core. Note that the squared frequencies are normalized by GM / R 3 where M and R are the mass and radius of the planet, G is the gravitational constant and l = 2 is used for L l . The dotted line plots log 10 | N 2 | in the convective core where N 2 0. 2.2 Perturbed basic equations The perturbed basic equations for a uniformly rotating planet may be given by where ρ, p, s, T , and c p are, respectively, the mass density, pressure, specific entropy, temperature, and specific heat at constant pressure, is the energy flux vector, is the displacement vector, α T = −(∂ln ρ/∂ln T ) P , and (′) and δ indicate, respectively, the Eulerian and Lagrangian perturbation of quantities. Note that we have applied the Cowling approximation neglecting the Euler perturbation of the gravitational potential due to self-gravity, and that we have ignored the effects of rotational deformation. The vector denotes the angular velocity vector of rotation and is assumed to be constant for uniformly rotating planets. Here, we have assumed that the perturbations are proportional to the factor e iω t + i m φ , where ω = σ + m Ω with is the oscillation frequency in the co-rotating frame of the planet and σ is the frequency observed in the inertial frame, and m denotes the azimuthal wavenumber around the axis of rotation. Since evolutionary models of irradiated Jovian planets computed with appropriate equations of state and opacities are not available to us, we employ an approximate treatment to estimate the
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