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Table D.1

Notations used along this work and their designations in order of appearance in the text.

Symbol Definition Reference
e Planet eccentricity Sect. 1
eAR Critical eccentricity for asynchronous rotation Sect. 1
Mp Planet mass Sect. 2
Rp Planet radius Sect. 2
     
Hoc Ocean depth Sect. 2
ρoc Ocean density Sect. 2
n⋆ Planet mean motion Sect. 2
n⋆ Orbital angular momentum of the planet Sect. 2
Planetocentric reference frame Sect. 2
Reference frame co-rotating with the planet Sect. 2
Ω Spin vector Sect. 2
Ω Rotation rate of the planet Sect. 2
t Time Sect. 2
r Radial coordinate (spherical coordinates) Sect. 2
θ Colatitude (spherical coordinates) Sect. 2
φ Longitude (spherical coordinates) Sect. 2
Spherical unit-vector basis Sect. 2
g Planet surface gravity Sect. 2
ρp Planet mean density Sect. 2
     
Moc Mass of the ocean Sect. 2
Mc Mass of the solid part Sect. 2
Rc Radius of the solid part Sect. 2
ρc Mean density of the solid part Sect. 2
σR Rayleigh drag frequency Sect. 2
τR Typical timescale of core-ocean coupling by viscous friction Sect. 2
r⋆ Star-planet distance Eq. (1)
M⋆ Mass of the host star Eq. (1)
Û⋆ Gravitational potential of the host star Eq. (1)
^ Symbol used to highlight real quantities Eq. (1)
˜ Symbol used to highlight complex quantities Eq. (1)
ÛT Real tidal gravitational potential Eq. (2)
ℜ Real part of a complex number Eq. (3)
ℑ Imaginary part of a complex number Eq. (3)
i Imaginary number Eq. (3)
UT Complex tidal gravitational potential Eq. (3)
l Latitudinal degree (spherical harmonics) Eq. (3)
m Longitudinal degree (spherical harmonics) Eq. (3)
s Eccentricity degree Eq. (3)
σ Forcing frequency Eq. (3)
σm,s Forcing tidal frequency of the mode associated with the doublet Eq. (3)
Normalised associated Legendre functions Eq. (3)
     
-component of the forcing tidal gravitational potential at the planet surface Eq. (3)
a Semi-major axis of the planet Eq. (4)
Al,m,s Dimensionless eccentricity functions Eq. (4)
δl,k Kronecker symbol Eq. (5)
δs<0 Coefficient equal to 1 for s < 0, 0 otherwise Eq. (5)
     
Unnormalised associated Legendre functions Eq. (5)
     
Hansen coefficients Eq. (5)
σ2,s Forcing frequency of the degree-s eccentricity term Eq. (6)
Degree-s component of the perturbing tidal potential Eq. (7)
     
Semidiurnal component of the perturbing tidal potential Eq. (8)
     
Degree-1 eccentricity component of the perturbing tidal potential Eq. (9)
     
σ2,1 Forcing frequency of the degree-1 eccentricity component Eq. (9)
Degree-3 eccentricity component of the perturbing tidal potential Eq. (10)
     
σ2,3 Forcing frequency of the degree-3 eccentricity component Eq. (10)
     
etrans Transition eccentricity Sect. 3.1
     
μ Effective unrelaxed shear modulus of the solid core Sect. 3.2
     
Stress tensor Eq. (12)
     
Strain tensor Eq. (12)
     
Complex shear modulus of the solid part Eq. (13)
     
Γ Gamma function Eq. (14)
     
α Rheological exponent of the material in the Andrade model Eq. (14)
     
τM Maxwell relaxation time of the material Eq. (14)
     
η Viscosity of the material Eq. (14)
     
τA Andrade time of the material in the Andrade model Eq. (14)
     
PT Tidal period Sect. 3.2
     
β Andrade parameter Sect. 3.2
     
Gravitational Love number of the degree-l mode Eq. (15)
     
Displacement Love number of the degree-l mode Eq. (15)
     
Gravitational load Love number of the degree-l mode Eq. (15)
     
Displacement load Love number of the degree-l mode Eq. (15)
     
Dimensionless effective rigidity Eq. (16)
     
Al Dimensionless rigidity coefficient associated with the degree-l component Eq. (17)
     
ξc Vertical displacement of the oceanic floor Sect. 3.3
     
V Velocity vector of the horizontal component of tidal flows Sect. 3.3
     
Vθ Latitudinal component of the velocity field Sect. 3.3
     
Vφ Longitudinal component of the velocity field Sect. 3.3
     
ξoc Vertical displacement of the ocean surface Sect. 3.3
     
η Variation of the ocean thickness Sect. 3.3
     
∂X Partial derivative with respect to X Eq. (18)
     
Ψ Perturbed potential encompassing the gravitational forcing and coupling with the solid part Eq. (18)
     
∇h Horizontal gradient operator Eq. (20)
     
∇h ⋅ Horizontal divergence operator Eq. (21)
     
Ψm,σ Fourier component of the perturbed potential Eq. (22)
     
Vm,σ Fourier component of the velocity vector Eq. (22)
     
ηm,σ Fourier component of the vertical variation of the ocean depth Eq. (22)
     
Degree-l component of Ψm,σ expanded in series of associated Legendre functions Eq. (23)
     
Degree-l component of Vm,σ expanded in series of associated Legendre functions Eq. (23)
     
Degree-l component of ηm,σ expanded in series of associated Legendre functions Eq. (23)
     
Complex tidal frequency Eq. (24)
     
Complex spin parameter Eq. (24)
     
Laplace’s tidal operator Eq. (27)
     
n Degrees of Hough functions Eq. (29)
     
Degree-n component of Ψm,σ expanded in series of Hough functions Eq. (29)
     
Degree-n Hough function Eq. (29)
     
Eigenvalue associated with the degree-n Hough function Eq. (30)
     
Equivalent depth of the degree-n Hough mode Eq. (31)
     
Coefficients of Hough functions expanded in series of the Eq. (32)
     
Coefficients of the expanded in series of Hough functions Eq. (33)
     
Overlap coefficients weighting the degree-n component of the oceanic tidal torque Eq. (34)
     
Tilt factor associated with the tidal gravitational forcing of the perturber Eq. (35)
     
Tilt factor associated with the distortion of the oceanic layer Eq. (36)
     
σn,k Complex characteristic frequencies of the planet Eq. (41)
     
kh ;n Horizontal wavenumber of the degree-n Hough mode Eq. (41)
     
Component of the force vector inducing the tidal perturbation Eq. (42)
     
Tidal torque exerted on the planet Eq. (44)
     
Quadrupolar component of the effective gravitational Love number of the planet Eq. (45)
     
Tidal torque exerted on the solid part in absence of oceanic layer Eq. (46)
     
Quadrupolar tidal Love number of the solid part in absence of oceanic layer Eq. (47)
     
A2 Dimensionless rigidity coefficient associated with the l = 2 component Eq. (48)
Tidal torque exerted on the ocean in case of solid part of infinite rigidity Eq. (49)
     
Complex eigenfrequency of the degree-n Hough mode in the positive-frequency range Eq. (50)
     
Complex eigenfrequency of the degree-n Hough mode in the negative-frequency range Eq. (50)
     
σn Characteristic frequency of the degree-n surface gravity mode Eq. (54)
     
σmax ;n Frequency at which the resonant peak of a mode reaches a maximum Eq. (56)
     
Maximum of reached by the resonance peak Eq. (58)
     
Degree-s eccentricity frequency associated with a peak Sect. 4.3
     
Ωeq ;s Rotation rate corresponding to the degree-s spin-orbit resonance Sect. 4.3
     
P⋆ Orbital period of the planet Sect. 4.4
     
M⊙ Mass of the Sun Sect. 4.4
     
Spin parameter associated with the semidiurnal tidal component Eq. (62)
     
γs Parameter controlling the existence of asynchronous states in the case of pure oceanic tide Eq. (65)
     
X Normalised frequency Eq. (69)
     
Dimensionless constant Eq. (69)
     
uα Maximum of the polynomial function defined by Eq. (69) Eq. (71)
     
Prot Rotation period of the planet Sect. 6.1
     
Hoc ;s Ocean depth associated with the resonance of the degree-s eccentricity component Sect. 6.1
     
Constant factor in the expression of the solid tidal torque in the non-resonant regime Eq. (73)
     
Tidal torque exerted on the planet Eq. (75)
     
Torque due to triaxiality Eq. (75)
     
∙ Time derivative Eq. (75)
     
A, B, C Principal moments of inertia of the planet (in increasing magnitude) Eq. (76)
     
γ2,s angle associated with the degree-s perturbation component Eq. (76)
     
ϑ Planet rotation angle Eq. (76)
     
M Planet mean anomaly Eq. (76)
     
.. Second order time derivative Eq. (78)
     
Δ Width of the resonance Eq. (79)
     
Even component of the tidal torque Eq. (81)
     
Odd component of the tidal torque Eq. (82)
     
Capture probability of the s∕2 spin-orbit resonance Eq. (83)
     
ΔP Normalised width of the 1:1 spin-orbit resonance Eq. (84)
     
γ Logarithm of the ocean depth Sect. 8
     
Č Normalised moment of inertia Sect. 8
     
τev Evolution timescale of the planet spin rotation Eq. (89)
     
Eccentricity and mean anomaly function Eq. (B.1)
     
v True anomaly Eq. (B.1)
     
N Order of the fast Fourier transform (FFT) in the calculation of Hansen coefficients Appendix B
     
K Truncation index of the series of Hansen coefficients calculated using the FFT method Appendix B
     
Complex triaxial torque Eq. (C.2)

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