Overstable Convective Modes of Rotating Hot Jupiters
Umin Lee



































#1
#1#1


#1

≳
≲



    #1 #1 iRotating Hot JupitersU.Lee
        lee@astr.tohoku.ac.jpAstronomical Institute, Tohoku University, Sendai, Miyagi 980-8578, Japan
        We calculate overstable convective modes of uniformly rotating hot Jupiters, which
        have a convective core and a thin radiative envelope.
        Convective modes in rotating planets have complex frequency ω and 
        are stabilized by rapid rotation so that their growth rates ∝ω_ I= Im(ω) are much smaller than those for the non-rotating planets.
        The stabilized convective modes excite low frequency gravity waves in the radiative envelope by frequency resonance between them.
        We find that the convective modes that excite envelope gravity waves remain unstable even in the presence of non-adiabatic dissipations in the envelope.
        We calculate the heating rates due to non-adiabatic dissipations of the oscillation energy of the unstable convective modes and find that the magnitudes of the heating rates cannot be large enough to inflate hot Jupiters sufficiently so long as the oscillation amplitudes remain in the linear regime.
        
        
        
        
        
        § INTRODUCTION
         
        
        
        
        Calculating adiabatic convective modes of uniformly rotating massive main sequence stars, Lee & Saio (1986) found 
        that the convective modes of rotating stars become overstable and are stabilized by rapid rotation
        so that their growth rates are much smaller than those of the non-rotating stars.
        They also found that the stabilized convective modes excite high radial order g-modes in the envelope of the stars when the convective modes are in frequency resonance with envelope g-modes.
        Carrying out non-adiabatic calculations, Lee & Saio (1987b) also showed that the unstable convective modes that excite envelope g-modes remain unstable even in the presence of non-adiabatic dissipations in the envelope.
        Using the traditional approximation (e.g., Lee & Saio 1987a, 1997), it was shown that the rotationally stabilized convective modes have a branch of negative oscillation energy (Lee & Saio 1990b). 
        The excitation of envelope g-modes by the convective modes is caused by
        linear mode coupling between the convective modes having negative energy of oscillation and
        g-modes, which have positive oscillation energy (Lee & Saio 1989).
        
        
        
        It has been suggested that hot Jupiters orbiting close to the host stars
        have larger radii compared with those of Jovian planets of similar masses and ages, orbiting far from their host stars.
        The strong irradiation by the host star, however, is not necessarily effective to inflate the planets to the radii observationally estimated (e.g., Baraffe et al. 2003).
        To explain the larger radii of the hot Jupiters, tidal heating in the interior of the planets has been suggested.
        However, it is well known that gravitational tides are likely to cause synchronization between the 
        orbital motion and spin of the planet in timescales much shorter than the ages of the planets
        (e.g., Bodenheimer et al 2001).
        Some authors have proposed thermal tides as a mechanism that keeps the spin asynchronous with the orbital motion of the planets
        so that tidal heating in the interior can be operative
        (e.g., Arras & Socrates 2010; Auclair-Desrotour et al 2017; Auclair-Desrotour & Leconte 2018).
        
        
        
        We compute overstable convective modes of rotating hot Jupiters.
        Overstable convective modes in rotating Jovian planets could produce various low frequency phenomena in the planets.
        Lee & Saio (1990a) proposed that g-modes excited in the Jovian atmosphere by 
        overstable convective modes could be a cause of low frequency phenomena observed in Jupiter (e.g., Deeming et al 1989; Magalhães et al 1989).
        In this paper, as a heating mechanism, 
        we estimate the amount of non-adiabatic dissipation of the oscillation energy of
        unstable convective modes of hot Jupiters.
        Here, we consider unstable convective modes in the core which excite gravity waves in the envelope,
        expecting that
        this excitation of g-modes could enhance thermal dissipation in the envelope.
        To estimate the dissipation rates of the oscillation energy, we need non-adiabatic
        treatments of the oscillations of rotating planets.
        For this purpose we employ an approximate treatment of non-adiabatic oscillations, 
        following Auclair-Desrotour & Leconte (2018).
        Method of solution for the non-adiabatic calculations is given in  2 and the numerical results
        and conclusions are presented in  3 and  4, respectively.
        We give a brief account of the traditional approximation applied to the stabilized convective modes in the Appendix.
        
        
        
        
        
        
        
        § METHOD OF SOLUTIONS
        
        
         §.§ Equilibrium Model
        
        
        Following Arras & Socrates (2010) and Auclair-Desrotour & Leconte (2018), 
        we employ for our modal analyses of hot Jupiters simple static models, 
        consisting of a thin radiative envelope and a convective core.
        In this paper, we ignore the deformation of rotating planets and hence the planet models are 
        assumed to be spherical symmetric.
        Such Jovian models are computed by integrating the hydrostatic equations
        dp dr=-ρGM_r r^2,
        dM_r dr=4π r^2ρ,
        
        with the equation of state given by
        ρ(p)=e^-p/p_bp a^2+(1-e^-p/p_b)√(p K_c),
        
        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
        
        K_c=GR_J^2,    a^2=√(p_bK_c),
        
        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
        
        p≈ K_cρ^Γ   with  Γ=2,
        
        corresponding to a polytrope of the index n=1, and hence for Γ_1≡(∂ln p/∂lnρ)_S≈ 2 the square of the Brunt-Väisälä frequency 
        
        N^2≡ -gA≈ 0,
        
        where 
        
        g=-1ρdp dr=GM_r r^2,
           A≡dlnρ dr-1Γ_1dln p dr. 
        
        On the other hand, in the radiative envelope where p≪ p_b, we have
        
        p≈ a^2ρ,
        
        and
        
        N^2≈ 
        (Γ_1-1)g^2 c^2,    c^2=Γ_1pρ,
        
        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=0.7M_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=0.01 dyn/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.
        Figure 1 is the propagation diagram where N^2 and L_l^2≡ l(l+1)c^2/r^2 with l=2 are plotted as a function of r/R where L_l is 
        the Lamb frequency.
        
        
        
        
        
        
         §.§ Perturbed Basic Equations
        
        
        The perturbed basic equations for a uniformly rotating planet may be given by
        
        -ρω^2ξ+2ρωΩ×ξ-ρ'ρ∇ p+∇ p'=0,
        ρ'+∇·(ρξ)=0,
        ωρ Tδ s=-∇·F',
        δ s c_p=1α_T(1Γ_1δ p p-δρρ),
        
        where
        ρ, p, s, T, and c_p are respectively the mass density, pressure, specific entropy,
        temperature, and specific heat at constant pressure, F is the energy flux vector, ξ is the displacement vector, 
        α_T=-(∂lnρ/∂ln T)_P,
        and (') and δ indicate resepctively 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 Ω 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^ω t+ mϕ, 
        where ω=σ+mΩ with Ω=|Ω| 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 non-adiabatic effects δ s,
        following Auclair-Desrotour & Leconte (2018).
        We write the term on the right-hand-side of equation (<ref>) as
        ∇·F'
        ∼ω_Dρ c_p TT' T
        =ω_D ρ c_p T(δ T T+∇ Vξ_r r)=
        ω_Dρ c_p T(δ s c_p+∇_adδ p p+∇ Vξ_r r),
        
        where we have used
        δ s/c_p=δ T/T-∇_adδ p/p with ∇_ad=(∂ln T/∂ln p)_S
        and ∇=dln T/ dln p, 
        and  
        ω_D≡2π/ τ_0.
        Here, τ_0 is the thermal timescale given by
        τ_0=τ_*×1 2[(p p_*)^1/2+(p p_*)^2],
        
        where p_*(≪ p_b) is the pressure 
        at the base of the heated layer by the irradiation of the host stars
        and is set equal to p_*=10^6 dyne/cm^2 (e.g., Auclair-Desrotour & Leconte 2018), and τ_* is a parameter to specify the efficiency
        of the radiative cooling
        and τ_*=1 day is assumed (Auclair-Desrotour & Leconte 2018; see also Iro et al 2005).
        From equation (<ref>) 
        we obtain
        δ s c_p=-ω_Dω+ω_D(∇_adδ p p+∇ Vξ_r r).
        
        When we take the limit of short thermal timescales τ_0→ 0 (i.e.,
        ω_D→∞),  
        we obtain isothermal perturbations T'/T=0.
        On the other hand, in the limit of long thermal timescale τ_0→∞
        so that ω_D→ 0, we have δ s→ 0, which corresponds to adiabatic perturbations, expected in the deep interior
        of the planet.
        
        
        
        
        
         §.§ Oscillation Equations
        
        
        
        To describe the perturbations of rotating planets, we employ a series expansion in terms of spherical harmonic functions
        Y_l^m(θ,ϕ) for a given m with different ls, assuming the planet is axisymmetric.
        The pressure perturbation, for example, is given as
        
        p'(r,θ,ϕ,t)=∑_lp'_l(r)Y_l^m(θ,ϕ)e^ω t, 
        
        and the displacement vector ξ is
        ξ_r(r,θ,ϕ,t)=r∑_lS_l(r)Y_l^m(θ,ϕ)e^ω t,
        ξ_θ(r,θ,ϕ,t)=r∑_l,l'[H_l(r)∂∂θY_l^m(θ,ϕ)+T_l'1sinθ∂∂ϕY_l'^m(θ,ϕ)]e^ω t,
        ξ_ϕ(r,θ,ϕ,t)=r∑_l,l'[H_l(r)1sinθ∂∂ϕY_l^m(θ,ϕ)-T_l'∂∂θY_l'^m(θ,ϕ)]e^ω t.
        
        where l=|m|+2(j-1) and l'=l+1 for even modes and l=|m|+2j-1 and l'=l-1 for odd modes and
        j=1, 2, ⋯, j_ max (see, e.g., Lee & Saio 1986).
        
        
        Substituting the expansions into the perturbed basic equations we obtain a set of ordinary differential equations
        for the expansion coefficients such as S_l(r) and p'_l(r).
        Using the dependent variables defined as
        y_1=(S_l),   y_2=(p'_lρ g r),   y_6=(δ s_l c_p),   h=(H_l),   t=(T_l'),
        
        we write the set of differential equations as
        
        rdy_1 dr=(VΓ_1-3)y_1-VΓ_1y_2+Λ_0h+α_Ty_6,
        
        rdy_2 dr=(c_1ω̅^2+rA)y_1+(1-U-rA)y_2-2c_1ω̅Ω̅(mh+C_0t)
        +α_Ty_6,
        
        -M_0h+L_1t=-νKy_1,
        L_0h-M_1t=1 c_1ω̅^2y_2+mνΛ_0^-1y_2,
        y_6=-ω_Dω+ω_D V[∇_ady_2-(∇_ad-∇)y_1 ],
        
        where 
        ν=2Ωω,   ω̅=ωσ_0,   Ω̅=Ωσ_0,   σ_0=√(GM R^3),
        
        V=-dln p dln r,    U=dln M_r dln r,    c_1=(r/R)^3 M_r/M, 
        
        and M and R denote respectively the mass and radius of the planet, and
        the temperature T and the specific heat c_p are assumed to be those for an ideal gas, that is,
        
        T=μ Rpρ,    c_p=5 2 Rμ,
        
        and R is the gas constant, and μ is the mean molecular weight, for which we use μ=1.3, 
        and the matrices Λ_0, C_0, L_0, L_1, M_0, M_1, K
        are defined in Lee & Saio (1990).
        
        
        
        Eliminating the variables h, t, and y_6 between equations from (<ref>) to (<ref>), we obtain the oscillation equations given by
        
        rdy_1 dr
        =[(VΓ_1-3+α_T (∇_ad-∇) Vω/ω_D+1)1+νWO]y_1
        +[W c_1ω̅^2-(VΓ_1+α_T∇_adVω/ω_D+1)1]y_2,
        
        rdy_2 dr   =   [(c_1ω̅^2+rA+α_T(∇_ad-∇) Vω/ω_D+1)1-4c_1Ω̅^2G]y_1
              +[(1-U-rA-α_T∇_ad Vω/ω_D+1)1-νO^TW]y_2,
        
        where 1 is the unit matrix, 
        and the matrices W, O, and G are
        defined by
        W=Λ_0(L_0-M_1L_1^-1M_0)^-1,   O=mΛ_0^-1-M_1L_1^-1K,   G=O^TWO-C_0L_1^-1K.
         
        Note that by simply setting ω_D=0, we restore the oscillation equations for adiabatic modes.
        Because we assume the perturbations are proportional to e^ω t, modes with negative
        ω_ I= Im(ω) are unstable modes.
        
        The boundary condition at the centre is the regularity condition of the functions ry_1
        and ry_2.
        We use the surface boundary conditions given by δ p=0 at the surface.
        
        
        
        § NUMERICAL RESULTS
        
        
        
        
        The Jovian models computed in  2.1 have the structure of a polytrope of the index n=1 in the core
        and of an isothermal atmosphere in the envelope.
        For Γ_1=2, the quantity |Γ_1^-1-dlnρ/dln p| in the core
        is very small except in the layers close to the boundary between the radiative envelop and convective core.
        Since Γ_1^-1-dlnρ/dln p=α_T(∇-∇_ ad),
        Γ_1^-1-dlnρ/dln p≈ 0 means that the convective core
        has an almost adiabatic structure.
        If the planets are rapidly rotating and have magnetic fields, however,
        finite negative values of N^2 in the convective core are more plausible (e.g., Stevenson 1979).
        In this paper, assuming rapidly rotating Jovian planets, we expect Γ_1^-1-dlnρ/dln p to have a finite positive value in the convective core, that is,
        1Γ_1-dlnρ dln p=δ_A,
        
        where δ_A is a positive constant.
        Since α_T=1 for an ideal gas, δ_A is equal to the superadiabatic temperature gradient ∇-∇_ad.
        For a positive value for δ_A, we obtain
        
        N^2=-g rVδ_A<0
        
        in the convective core.
        In the isothermal radiative envelope, we simply assume ∇_ad=0.5 and ∇=0.
        
        
        
        In Figure 2, we plot several sequences of the complex eigenfrequency ω̅=ω̅_ R+ω̅_ I for the m=-1 and m=-2 convective modes of even parity, assuming
        δ_A=10^-4 in the convective core, where
        we have used j_ max=12 for the expansion of the perturbations.
        Here, we are interested in prograde (ω̅_ R>0) and unstable (ω̅_ I<0)
        convective modes of rotating Jovian planets (see Lee & Saio 1986).
        Note that convective modes at Ω̅=0 have pure imaginary eigenvalue ω̅ such that ω̅^2<0,
        which is proportional to l(l+1) with l being the harmonic degree of the modes.
        Each of the mode sequences, as a function of Ω̅, has a sharp peak of ω̅_ R, at which
        the convective mode is stabilized in the sense that |ω̅_ I|≪ω̅_ R.
        The height of the peaks for m=-2 is higher that that for m=-1.
        When the convective modes reach the peaks, they are resonantly coupled with envelope g-modes and
        ω̅_ R of the modes
        starts decreasing rapidly with increasing Ω̅, keeping ω̅_ I<0, that is,
        the convective modes stay unstable even in the 
        presence of strong dissipations associated with g-modes in the envelope.
        We have carried out similar calculations assuming δ_A=10^-2 to find that the frequency ω̅ of
        the convective modes scales as ω̅∝√(δ_A).
        
        We find that convective modes in the rapidly rotating planets are separated into two types, which we may call type A and type B.
        For the convective modes belonging to type A, the complex frequency ω̅
        is approximately given by equation (<ref>), which is derived based on the WKB analysis discussed in the Appendix. 
        We can therefore assign a pair of indices (k,n) to each of the mode sequences, where
        the index k is an integer used to label the eigenvalue λ_k,m of the Laplace tidal
        equation and the index n is also an integer corresponding to the number of radial nodes of the dominant
        expansion coefficient S_l(r) of the eigenfunction (see Appendix).
        For the stabilized convective modes, the integer index k is negative and λ_k,m becomes negative
        for prograde modes of ν≡ 2Ω/ω_ R>0 (see Fig. <ref>).
        The examples of such assignment of the indices (k,n) to type A convective modes are given in Fig. 3.
        Note that we found only convective modes that could be associated with k=-2, -6, -10, ⋯.
        For the convective modes of type A, the angular part of the eigenfunction is approximately represented by
        the Hugh function Θ_k,m(cosθ), which is an eigenfunction of the Laplace tidal equation associated with the eigenvalue λ_k,m
        and can be represented by a linear combination of the associated Legendre functions P_l^m(cosθ) (e.g., Lee & Saio 1997).
        When |ω̅_ I|≪ω̅_ R, the stabilized convective modes are likely to
        interact with gravity waves propagating in the envelope.
        The interactions affect the complex frequency ω̅ and may produce wiggles in ω_ I as a function of Ω̅.
        
        
        
        
        
        
        Let Ω̅_k,m,n signify the rotation frequency at which type A convective modes
        are stabilized making a peak of ω̅_ R.
        From Fig. 3, we find Ω̅_-2,-2,0>Ω̅_-2,-1,0.
        We also find that Ω̅_-2,m,n<Ω̅_-6,m,n<Ω̅_-10,m,n<⋯ for given m and n, and
        the number of nodes in the θ direction increases with increasing |k| for negative k.
        This indicates that as |k| and |m| increase, the horizontal wavelengths become shorter and
        the rotation rate necessary to stabilize convective modes becomes higher.
        This trend is opposite to that for the wavelengths in the radial direction, that is,
        for given k and m, Ω̅_k,m,n decreases as the wavelengths in the radial direction decrease
        with increasing n.
        
        
        
        For the convective modes of type B, on the other hand, 
        the first component S_l of l=|m| has dominant amplitudes, particularly
        before their stabilization.
        To classify the type B convective modes, we simply use the number of radial nodes of the eigenfunction such that
        B_n.
        To stabilize the type B convective modes, we generally need much higher rotation rates Ω̅ than
        for the type A convective modes.
        The rotation rate Ω̅
        necessary to stabilize the type B convective modes increases as the number of radial nodes increases, which trend is
        opposite to that found for the type A convective modes.
        
        
        
        
        
        
        
        Because of strong non-adiabatic effects in the envelope, 
        the kinetic energy of unstable convective modes dissipates and can be used to heat up the thin envelope.
        The normalized energy dissipation rate, D, averaged over the oscillation period, may be given by
        
        D   ≡   -1 L_eqP∫_0^PdE dt dt
           ≈   ω̅_ R 2∫_0^R Im(∑_lδ T^*_l Tδ s_l c_p)
        ρ Tc_pr^3σ_0 L_ eqdr r
           =   -ω̅_ R 2∫_0^R Im[1ω/ω_D+1∑_l∇_adδ p_l^* p(∇_adδ p_l p+∇ VS_l)]
        ρ Tc_pr^3σ_0 L_ eqdr r,
        
        where P=2π/ω_ R, L_ eq=4π R^2 F_ eq, x=r/R,
        δ T/T=∇_adδ p/p+δ s/c_p, and we have assumed |ω_ I|≪|ω_ R|. 
        Assuming that the host star has the effective temperature T_ eff,*=5.8×10^3K, the radius R_*=R_⊙,
        and the distance d between the planet and the host star is d=0.05AU, we estimate L_eq=4π R^2σ_ SBT^4_ eff,*(R_*/d)^2≈ 6×10^29 erg/s
        where σ_ SB is the Stefan Boltzmann constant.
        Note also that if the spin of the planet is in synchronization with the orbital motion around the host star
        so that Ω≈√(GM_*/d^3),
        we have Ω/σ_0≈ 0.05 for the mass M_*≈ M_⊙ of the host star.
        
        
        Assuming that the oscillation energy of the convective modes is approximately constant and
        independent of Ω, that is,
        ω_ R^2∫_0^Rρξ·ξ^*dV=σ_0^2f_0^2∫_0^Rρ r^2 dV,
        
        where f_0 is the amplitude parameter, we define the non-dimensional oscillation amplitudes A_c as
        
        A_c=√((∫_0^Rρ r^2dr)^-1∫_0^R|ξ/r|^2ρ r^2 dr).
        
        The normalization (<ref>) implies that A_c= f_0/|ω̅_ R| and hence
        f_0/|ω̅_ R|≪ 1 for linear pulsations.
        Replacing R by r in equation (<ref>), we may define the function D(r),
        which is plotted in Fig. 4 for the stabilized convective modes of m=-1 and m=-2 where we assume f_0=1.
        Note that D(r) and D are proportional to f_0^2.
        This figure shows that the strong heating takes place at the bottom of the radiative envelope.
        In Figure 5, we plot the quantity D as a function of Ω̅ along
        the convective mode sequences of m=-1 and m=-2 for δ_A=10^-4 where f_0=1 is used.
        The magnitudes of D for the type A convective modes are at most of order of 10 and
        D changes erratically as a function of Ω̅ after their stabilization.
        These erratic behaviors of D after stabilization are caused by interaction between
        the convective modes in the core and gravity waves in the envelope.
        For the type B convective modes, on the other hand, the magnitudes of D after their stabilization
        can be as large as D≳ 10^2.
        This rapid increase of D indicates that energy dissipations associated with envelope g-modes 
        become significant when g-modes are resonantly excited by convective modes after their stabilization.
        Note that D of the type B convective modes does not strongly suffer erratic changes even after
        their stabilization.
        
        
        
        § CONCLUSIONS
        
        
        We calculate overstable convective modes in rotating hot Jupiters that
        have the convective core and thin radiative envelope, by
        taking account of the non-adiabatic effects.
        As the rotation rate increases, the convective modes in the core are stabilized to attain |ω̅_ I|≪ω̅_ I^0, where ω̅^0_ I is the value of ω̅ at Ω̅=0.
        For the stabilized convective modes we also have |ω̅_ I|≪|ω̅_ R|.
        We find that the stabilized convective modes are separated into two types, A and B, depending on the angular dependence of the eigenfunctions.
        The qualitative behavior of type A convective modes are well understood by the WKB analysis
        under the traditional approximation for the oscillation modes of rotating stars.
        The stabilized convective modes in the core excite gravity waves in the envelope and remain unstable even in the presence of strong non-adiabatic dissipations in the envelope.
        This mechanism can be an excitation mechanism of low frequency prograde g modes of
        rotating stars having a convective core and a radiative envelope, as originally suggested for 
        massive main sequence stars (Lee & Saio 1986; Lee 1988).
        
        
        Computing the non-adiabatic dissipation rate D of the oscillation energy of the convective modes,
        we showed that the dissipative heating most strongly occurs at the bottom of the envelope and that
        D becomes comparable to or even greater than the irradiated luminosity due to the host star
        for the oscillation amplitudes A_c∼ f_0/ω̅_ R for f_0=1.
        Baraffe et al (2003) carried out evolution calculation of irradiated Jovian planets
        to obtain inflated hot Jupiter models.
        They showed that if there exists an extra heat source of the magnitudes of order of L_extra∼10^27-5×10^27  erg/s in the deep interior, the planet models could inflate
        by an enough amount to attain the radii consistent with the observationally estimated radius of the hot Jupiter HD 209458b.
        For L_eq∼ 5× 10^29  erg/s and hence L_extra/L_eq∼ 10^-2,
        to produce the heating rate L_extra∼ D_1f_0^2L_eq where D_1 is the value of D for f_0=1 and
        D_1∼ 1-10^3 as suggested by Fig. 5,
        we can use f_0∼ 10^-2-10^-1, for which
        A_c 1 for the convective mode frequency ω̅_ R∼ 10^-3.
        In this case the linearity condition A_c≪ 1 could not be satisfied.
        This suggests that the heating by the overstable convective modes could not be large enough to inflate the hot Jupiters so long as the amplitudes are limited to the linear regime.
        
        The conclusion given above concerning the magnitudes of D has been derived based on a very approximate treatment of non-adiabatic effects on the oscillation modes.
        For the non-adiabatic analyses of oscillation modes of hot Jupiters, it is desirable to use
        evolutionary models computed with appropriate opacities and equations of state.
        Only with such Jovian models, we are able to reach a definite conclusion concerning the efficiency of the heating mechanism we discussed in this paper.
        
        
        
        
        The frequency ω̅ of convective modes and the rotation frequency Ω̅_k,m,n
        at which the convective modes are stabilized are approximately proportional to 
        √(δ_A) assumed for the convective core.
        The value of δ_A=10^-4, which we use in this paper, 
        may be large for the interior of rotating Jovian planets as suggested by Stevense (1976).
        Even for a much smaller value of δ_A, convective modes with large |k| and |m| (type A)
        or with large n and |m| (type B) could survive rotational stabilization.
        The presence of a magnetic field may change the conclusion about the rotational stabilization of convective modes.
        It is an important problem to numerically examine the effects of rotation and 
        magnetic field on the stabilization of convective modes of rotating stars.
        
        
        
        
        
        § WKB ANALYSIS IN THE TRADITIONAL APPROXIMATION
        
        
        
        
        
        WKB analysis is a useful tool to understand the numerical results presented in  3.
        We employ the traditional approximation to discuss the oscillations of uniformly rotating stars,
        neglecting the terms proportional to Ω_H=-Ωsinθ in the perturbed equations of motion.
        In the traditional approximation, the θ dependence of the 
        perturbations such as ξ_r and p' can be represented by a single function Θ_k,m(θ) so that
        ξ_r(r,θ,ϕ,t)=ξ_r(r)Θ_k,m(θ) e^(mϕ+σ t), where
        Θ_k,m(θ) is called the Hough function, which is an eigenfunction of the Laplace's tidal equation associated with the eigenvalue λ_k,m and k is an integer introduced to classify the oscillation modes
         (see, e.g., Lee & Saio 1997).
        The eigenvalue λ_k,m depends on ν=2Ω/ω for given (k,m).
        In Figure <ref>, λ_k,m and its derivative dlnλ_k,m/dlnν
        are plotted as a function of ν=2Ω/ω for m=-1, where positive (negative) ν
        corresponds to prograde (retrograde) modes for negative m.
        For non-negative integers k, 
        λ_k,m→ l_k(l_k+1) with l_k=|m|+k when ν→ 0, and 
        λ_k,m∝ν^2 for large |ν| except for λ_k=0,m on the prograde side (ν>0).
        For negative integers k, λ_k,m on the retrograde side vanishes at ν=l'_k(l'_k+1)/m with l'_k=|m|+|k|-1 and positive λ_k,ms on the retrograde side are associated with 
        r-modes.
        Similar figures for m=-2 are given in Lee & Saio (1997).
        
        
        
        
        
        
        
        As discussed in Lee & Saio (1989), the coupling between rotationally stabilized convective modes
        of k_c<0 in the core and low frequency g-modes of k_g≥0 in the envelope of a massive main sequence star may be described by a dispersion relation
        tanΦ_c(ω)tanΦ_g(ω)=ϵ,
        
        where ϵ is the coupling coefficient between the two modes, and 
        Φ_c(ω)=∫_0^x_c√(-λ_k_c,mrA/c_1)ω̅dx x-α_k_cπ,   Φ_g(ω)=∫_x_c^1√(-λ_k_g,mrA/c_1)ω̅dx x-α_k_gπ,
        
        where x=r/R and x_c represents the boundary between the convective core and radiative envelope,
        and α_k is a constant which may depend of k and m.
        Note that for m<0 and ν>0, λ_k_c,m for the convective modes is negative and 
        λ_k_g,m for the g-modes is positive as shown by Figure <ref>.
        Assuming that gravity waves in the envelope are totally dissipated, we replace
        tanΦ_g=(e^Φ_g-e^-Φ_g)/(e^Φ_g+e^-Φ_g) for g-modes by - to obtain
        tanΦ_c(ω)=ϵ.
        
        For complex frequency ω=ω_ R+ω_ I, assuming |ω_ I|≪|ω_ R|, we obtain
        tanΦ_c(ω_ R)+
        ω_ I∂∂ω_ RtanΦ(ω_ R)
        =ϵ,
        
        the real part of which gives
        tanΦ_c(ω_ R)=0,
        
        and determines the frequency as
        ω̅_ R=1(α_k_c+n)π∫_0^x_c√(-λ_k_c,mrA c_1)dx x,
        
        where n is an integer that satisfies Φ_c(ω_ R)=nπ.
        On the other hand, the imaginary part of equation (<ref>) leads to
        ω_ I∂∂ω_ RtanΦ_c(ω_ R)
        =-ω_ Iω_ R(∫_0^x_c√(-λ_k_c,mrA/ c_1)ω̅_ Rdx x)F_k_c,m=ϵ
        and hence
        ω_ I=-ω_ Rϵ(α_k_c+n)π F_k_c,m
        where
        
        F_k,m=1+1 2dln|λ_k,m| dlnν.
        
        We let ν^0 satisfy F_k,m(ν^0)=0, which occurs only for 
        prograde modes with negative k.
        Since
        
        F_k,m∂ω̅_ R∂Ω̅=dln|λ_k,m| dν,
        
        we have ∂ω̅_ R/∂Ω̅→±∞ when
        ν→ν^0.
        Using WKB analysis, Lee & Saio (1989, 1990, 1997) discussed that
        the energy of the oscillation modes of rotating stars
        is proportional to λ_k,m F_k,m and the energy can be negative for rotationally stabilized convective modes associated with λ_k,m<0 if F_k,m>0.
        As shown by Figure <ref>, F_k,m for the convective modes becomes positive
        for ν 5.
        The convective modes appear as prograde modes of ω_ R>0 for 
        m<0
        and hence have ω_ I<0 for F_k,m>0, that is, the convective modes with
        negative energy become overstable when the oscillation energy leaks out of the core.
        
        
        
        
        
        
        At ν=ν^0_k,m the real part of the frequency ω_ R^0 is given by equation 
        (<ref>) and hence the rotation rate Ω^0=ν^0ω_ R^0/2.
        We thus have for convective modes of negative λ_k,m and kω̅^0_ RΩ̅^0=ν^0(ω̅^0_ R)^2/2=1 2π^2-λ^0_k,mν^0_k,m
        (α_k+n)^2(∫_0^x_c√(rA c_1)dx x)^2,
        
        where λ^0_k,m=λ_k,m(ν^0_k,m).
        Assuming
        1 2π^2-λ^0_k,mν^0_k,m
        (α_k+n)^2=c^2
        
        with c being a constant, we obtain for n=0α_k=1√(2)π c√(-λ^0_k,mν^0_k,m),
        
        and the value of -λ^0_k,mν^0_k,m is tabulated in Table 1.
        
        We numerically solve the dispersion relation (<ref>) and pick up the convective modes
        that satisfies ω̅_ R>0, ω̅_ I<0, |ω̅_ I/ω̅_ R|<1/2, and
        n≥0, where we use ϵ=10^-3 and δ_A=10^-4.
        In Figure A2, we plot the complex frequency ω̅ for m=-1 convective modes of even parity.
        We find a reasonable agreement between
        the WKB results and the full numerical calculations shown in Figure 2 for type A convective modes, which probes
        that the WKB analysis is useful to classify the convective modes 
        numerically obtained.
        Note however that the WKB equation (<ref>) cannot necessarily be used to explain the type B convective modes plotted in Fig. 2.
        
        
        [Auclair-Desrotour P. et al(2017)]2017A   A..603..A108 Auclair-Desrotour P. et al, 2017, A&A, 603, A108
        [Auclair-Desrotour P., Leconte J.(2018)]2018A   A..613..A45 Auclair-Desrotour P., Leconte J., 2018, A&A, 613, A45
        [Arras P., Bildsten L.(2006)]2006ApJ..650..394 Arras P., Bildsten L., 2006, ApJ, 650, 394
        [Arras P., Socrates A.(2010)]2010ApJ..714..1 Arras P., Socrates A., 2010, ApJ, 714, 1
        [Baraffe I.etal(2003)]2003A   A..402..701 Baraffe I., Chabrier G., Barman T.S., Allard F., Hauschildt P.H., 2003, A&A, 402, 701
        [Deming D.etal(1989)]1989ApJ..343..456 Deming D., Mumma M.J., Espenak F., Jennings D.E., Kostiuk T., Wiedemann G.,
        Loewenstein R., Piscitelli J, 1989, ApJ, 343, 456
        [Iro N.etal(2005)]2005A   A..436..719 Iro N., Bézard B., Guillot T., 2005, A&A, 436, 719
        
        [Lee U., Saio H.(1986)]1986MNRAS..221..365 Lee U., Saio H., 1986, MNRAS, 221, 365
        [Lee U., Saio H.(1987a)]1987aMNRAS..224..513 Lee U., Saio H., 1987a, MNRAS, 224, 513
        [Lee U., Saio H.(1987b)]1987bMNRAS..225..643 Lee U., Saio H., 1987b, MNRAS, 225, 643
        [Lee U., Saio H.(1989)]1989MNRAS..237..875 Lee U., Saio H., 1989, MNRAS, 237, 875
        [Lee U., Saio H.(1990a)]1990aApJ..359..29 Lee U., Saio H., 1990a, ApJ, 359, 29
        [Lee U., Saio H.(1990b)]1990bApJ..360..590 Lee U., Saio H., 1990b, ApJ, 360, 590
        [Lee U., Saio H.(1997)]1997ApJ..491..839 Lee U., Saio H., 1997, ApJ, 491, 839
        
        [Magalhães J.A.etal(1989)]1989Nature..337..444 Magalhães J.A., Weir A.L., Conrath B.J., Gierasch P.J., Leroy S.S., 1989, Nature, 337, 444
        
        [Stevenson D.J.(1979)]Geophys.Astrophy.Fluid.Dynamics1979..12..139 Stevenson D.J., Geophys. Astrophy. Fluid. Dynamics, 1979, 12, 139
        [Stevenson D.J.(1979a)]1977ApJS..35..221 Stevenson D.J., Salpeter E.E., 1977, ApJS, 35, 221
        [Stevenson D.J.(1979b)]1977ApJS..35..239 Stevenson D.J., Salpeter E.E., 1977, ApJS, 35, 239
