The rotation of planets hosting atmospheric tides: from Venus to habitable superEarths
^{1} IMCCE, Observatoire de Paris, CNRS UMR 8028, PSL, 77 Avenue DenfertRochereau, 75014 Paris, France
email: pierre.auclairdesrotour@ubordeaux.fr
^{2} Laboratoire AIM ParisSaclay, CEA/DRF – CNRS – Université Paris Diderot, IRFU/SAp Centre de Saclay, 91191 GifsurYvette Cedex, France
email: stephane.mathis@cea.fr
^{3} LESIA, Observatoire de Paris, PSL Research University, CNRS, Sorbonne Universités, UPMC Univ. Paris 06, Univ. Paris Diderot, Sorbonne Paris Cité, 5 place Jules Janssen, 92195 Meudon, France
^{4} CIDMA, Departamento de Física, Universidade de Aveiro, Campus de Santiago, 3810193 Aveiro, Portugal
email: correia@ua.pt
Received: 12 April 2016
Accepted: 17 November 2016
The competition between the torques induced by solid and thermal tides drives the rotational dynamics of Venuslike planets and superEarths orbiting in the habitable zone of lowmass stars. The resulting torque determines the possible equilibrium states of the planet’s spin. Here we have computed an analytic expression for the total tidal torque exerted on a Venuslike planet. This expression is used to characterize the equilibrium rotation of the body. Close to the star, the solid tide dominates. Far from it, the thermal tide drives the rotational dynamics of the planet. The transition regime corresponds to the habitable zone, where prograde and retrograde equilibrium states appear. We demonstrate the strong impact of the atmospheric properties and of the rheology of the solid part on the rotational dynamics of Venuslike planets, highlighting the key role played by dissipative mechanisms in the stability of equilibrium configurations.
Key words: planetstar interactions / planets and satellites: dynamical evolution and stability / planets and satellites: atmospheres / celestial mechanics
© ESO, 2017
1. Introduction
Twenty years after the discovery of the first exoplanet around a Solartype star (Mayor & Queloz 1995), the population of detected lowmass extrasolar planets has grown rapidly. Their number is now considerable and will keep increasing. This generates a strong need for theoretical modelling and predictions. Because the mass of such planets is not large enough to allow them to accrete a voluminous gaseous envelope, they are mainly composed of a telluric core. In many cases (e.g. 55 Cnc e, see Demory et al. 2016), this core is covered by a thin atmospheric layer as is observed on the Earth, Venus and Mars. The rotation of these bodies strongly affects their surface temperature equilibrium and atmospheric global circulation (Forget & Leconte 2014). Therefore, it is a key quantity to understand their climate, particularly in the socalled habitable zone (Kasting et al. 1993) where some of them have been found (Kopparapu et al. 2014).
The rotational dynamics of superVenus and Venuslike planets is driven by the tidal torques exerted both on the rocky and atmospheric layers (see Correia et al. 2008). The solid torque, which is induced by the gravitational tidal forcing of the host star, tends to despin the planet to pull it back to synchronization. The atmospheric torque is the sum of two contributions. The first one, caused by the gravitational tidal potential of the star, acts on the spin in a similar way to the solid tide. The second one, called the “thermal tide”, results from the perturbation due to the heating of the atmosphere by the stellar bolometric flux. The torque induced by this component is in opposition to those gravitationally generated. Therefore, it pushes the angular velocity of the planet away from synchronization. Although the mass of the atmosphere often represents a negligible fraction of the mass of the planet (denoting f_{A} this fraction, we have f_{A} ~ 10^{4} for Venus), thermal tides can be of the same order of magnitude and even stronger than solid tides (Dobrovolskis & Ingersoll 1980).
This competition naturally gives rise to prograde and retrograde rotation equilibria in the semimajor axis range defined by the habitable zone, in which Venuslike planets are submitted to gravitational and thermal tides of comparable intensities (Gold & Soter 1969; Dobrovolskis & Ingersoll 1980; Correia & Laskar 2001). Early studies of this effect were based on analytical models developed for the Earth (e.g. Chapman & Lindzen 1970) that show a singularity at synchronisation, while Correia & Laskar (2001) avoid this weak point by a smooth interpolation annulling the torque at synchronization. Only recently, the atmospheric tidal perturbation has been computed numerically with global circulation models (GCM; Leconte et al. 2015), and analytically by AuclairDesrotour et al. (2017, hereafter referred to as P1), who generalized the reference work of Chapman & Lindzen (1970) by including the dissipative processes (radiation, thermal diffusion) that regularize the behaviour of the atmospheric tidal torque at the synchronization.
Here, we revisit the equilibrium rotation of superEarth planets based on the atmospheric tides model presented in P1. For the solid torque, we use the simplest physical prescription, a Maxwell model (Remus et al. 2012; Correia et al. 2014), because the rheology of these planets is unknown.
2. Tidal torques
2.1. Physical setup
For simplicity, we considered a planet in a circular orbit of radius a and mean motion n around a star of mass M_{∗} and luminosity L_{∗} exerting on the planet thermal and gravitational tidal forcing (Fig. 1). The planet, of mass M_{P} and spin Ω, has zero obliquity so that the tidal frequency is σ = 2ω, where ω = Ω−n. It is composed of a telluric core of radius R covered by a thin atmospheric layer of mass M_{A} = f_{A}M_{P} and pressure height scale H such that H ≪ R. This fluid layer is assumed to be homogeneous in composition, of specific gas constant ℛ_{A} = ℛ_{GP}/m (ℛ_{GP} and m being the perfect gas constant and the mean molar mass respectively), in hydrostatic equilibrium and subject to convective instability, that is, N ≈ 0, with N designating the BruntVäisälä frequency, as observed on the surface of Venus (Seiff et al. 1980). Hence, the pressure height scale is (1)where T_{0} is the equilibrium surface temperature of the atmosphere and g the surface gravity which are related to the equilibrium radial distributions of density ρ_{0} and pressure p_{0} as p_{0} = ρ_{0}gH. We introduce the first adiabatic exponent of the gas Γ_{1} = (∂lnp_{0}/∂lnρ_{0})_{S} (the subscript S being the specific macroscopic entropy) and the parameter κ = 1−1/Γ_{1}. The radiative losses of the atmosphere, treated as a Newtonian cooling, lead us to define a radiative frequency σ_{0}, given by (2)where C_{p} = ℛ_{A}/κ is the thermal capacity of the atmosphere per unit mass and J_{rad} is the radiated power per unit mass caused by the temperature variation δT around the equilibrium state. We note here that the Newtonian cooling describes the radiative losses of optically thin atmospheres, where the radiative transfers between layers can be ignored. We applied this modelling to optically thick atmosphere, such as Venus’, because the numerical simulations by Leconte et al. (2015) show that it can also describe well tidal dissipation in these cases, with an effective Newtonian cooling frequency. For more details about this physical setup, we refer the reader to P1.
2.2. Atmospheric tidal torque
For null obliquity and eccentricity, the tidal gravitational potential is reduced to the quadrupolar term of the multipole expansion (Kaula 1964), , where t is the time, ϕ the longitude, θ the colatitude, r the radial coordinate, the normalized associated Legendre polynomial of order ^{(}l,m^{)} = ^{(}2,2^{)} and U_{2} its radial function. Since H ≪ R, U_{2} is assumed to be constant. The thick atmosphere of a Venuslike planet absorbs most of the stellar flux in its upper regions, which are, as a consequence, strongly thermally forced; only 3% of the flux reaches the surface (Pollack & Young 1975). However the tidal effects resulting from the heating by the ground determine the tidal mass redistribution since the atmosphere is far denser near the surface than in upper regions (Dobrovolskis & Ingersoll 1980; Shen & Zhang 1990). These layers can also be considered in solid rotation with the solid part as a first approximation because the velocity of horizontal winds is less than 5 m s^{1} below an altitude of 10 km (Marov et al. 1973). So, introducing the mean power per unit mass J_{2}, we choose for the thermal forcing the heating at the ground distribution , where τ_{J} ≫ 1 represents the damping rate of the heating with altitude and depends on the vertical thermal diffusion of the atmosphere at the surface (e.g. Chapman & Lindzen 1970). This allows us to establish for the atmospheric tidal torque the expression (see P1, Eq. (174)) (3)where is the density at the ground. This function of σ is of the same form as that given by Ingersoll & Dobrovolskis (1978, Eq. (4)). The quadrupolar tidal gravitational potential and heat power are given by (P1, Eq. (74)) (4)where designates the gravitational constant, ε the effective fraction of power absorbed by the atmosphere and α a shape factor depending on the spatial distribution of tidal heat sources. Denoting the distribution of tidal heat sources as a function of the stellar zenith angle Ψ, the parameter α is defined as , where is the normalized Legendre polynomial of order n = 2. If we assume a heat source of the form if , and else, we get the shape factor . The tidal torque exerted on the atmosphere is partly transmitted to the telluric core. The efficiency of this dynamical (viscous) coupling between the two layers is weighted by a parameter β (0 ≤ β ≤ 1). Hence, with ω_{0} = σ_{0}/ 2, the transmitted torque (Eq. (3)) becomes (5)
Fig. 1 Tidal elongation of a Venuslike rotating planet, composed of a solid core (brown) and a gaseous atmosphere (blue), and submitted to gravitational and thermal forcings. 

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2.3. Solid tidal torque
For simplicity and because of the large variety of possible rheologies, we have assumed that the telluric core behaves like a damped oscillator. In this framework, called the Maxwell model, the tidal torque exerted on an homogeneous body can be expressed (e.g. Remus et al. 2012) (6)the imaginary part of the second order Love number (k_{2}), K a nondimensional rheological parameter, σ_{M} the relaxation frequency of the material composing the body, with (7)where G and η are the effective shear modulus and viscosity of the telluric core. Finally, introducing the frequency ω_{M} = σ_{M}/^{(}2 + 2K^{)}, we obtain (8)
3. Rotational equilibrium states
3.1. Theory
The total torque exerted on the planet is the sum of the two previous contributions: . When the atmospheric and solid torques are of the same order of magnitude, several equilibria can exist, corresponding to (Fig. 2). The synchronization is given by Ω_{0} = n and nonsynchronized retrograde and prograde states of equilibrium, denoted Ω_{−} and Ω_{+} respectively, are expressed as functions of a (Eqs. (5) and (8)) (9)where the difference to synchronization, ω_{eq}, is given by (10)
Fig. 2 Normalized total tidal torque exerted on a Venuslike planet (in red) and its solid (in green) and atmospheric (in blue) components as functions of the forcing frequency ω/n with ω_{0}/n = 2 and ω_{M}/n = 6. 

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As a first approximation, we consider that parameters A, ω_{0} and ω_{M} do not vary with the starplanet distance. In this framework, the equilibrium of synchronization is defined for all orbital radii, while those associated with Eq. (10) only exist in a zone delimited by a_{inf}<a<a_{sup}, the boundaries being (11)with the ratio . In the particular case where ω_{0} = ω_{M}, the distance a_{eq} = Aω_{0} corresponds to an orbit for which the atmospheric and solid tidal torques counterbalance each other whatever the angular velocity (Ω) of the planet. Studying the derivative of ω_{eq} (Eq. (10)), we note that when the starplanet distance increases, the pseudosynchronized states of equilibrium get closer to the synchronization (ω_{M}<ω_{0}) or move away from it (ω_{M}>ω_{0}), depending on the hierarchy of frequencies associated with dissipative processes. To determine the stability of the identified equilibria, we introduce the first order variation δω such that ω = ω_{eq} + δω and study the sign of in the vicinity of an equilibrium position for a given a. We first treat the synchronization, for which (12)We note that a_{eq} = a_{inf} if ω_{0}<ω_{M} and a_{eq} = a_{sup} otherwise. In the vicinity of nonsynchronized equilibria, we have (13)Therefore, within the interval , the stability of states of equilibrium also depends on the hierarchy of frequencies. If ω_{M}<ω_{0} (ω_{M}>ω_{0}), he synchronized state of equilibrium is stable (unstable) and the pseudosynchronized ones are unstable (stable). For a<a_{inf} the gravitational tide predominates and the equilibrium at Ω = n is stable. It becomes unstable for a>a_{sup}, because the torque is driven by the thermal tide.
Fig. 3 Top: solid (left panel), atmospheric (middle panel) and total tidal torque (right panel) exerted on a Venuslike planet as functions of the reduced forcing frequency ω/n (horizontal axis) and orbital radius a in logarithmic scale (vertical axis). The colour level corresponds to the torque in logarithmic scale with isolines at . Bottom: sign of the solid (left panel), atmospheric (middle panel) and total (right panel) tidal torque as functions of the same parameters: white (blue) areas are associated with positive (negative) torques. Stable (unstable) states of equilibrium are designated by blue (red) lines (with A = 1.88 × 10^{19} m s). The pink band corresponds to the habitable zone for the black body equilibrium temperature T_{eq} = 288 K ± 20% for a 1 M_{⊙} Solartype star at the age of the Sun. 

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3.2. Comparison with previous works
In many early studies dealing with the spin rotation of Venus, the effect of the gravitational component on the atmosphere is disregarded so that the atmospheric tide corresponds to a pure thermal tide (e.g. Dobrovolskis & Ingersoll 1980; Correia & Laskar 2001; Correia et al. 2003, 2008; Cunha et al. 2015). Moreover, the torque induced by the tidal elongation of the solid core is generally assumed to be linear in these works. This amounts to considering that ω ≪ ω_{M} and ω_{0} ≪ ω_{M}. Hence, the relation Eq. (10) giving the positions of nonsynchronized equilibria can be simplified: (14)Following Correia et al. (2008), we take ω_{0} = 0. We then obtain ω_{eq}∝ while Correia et al. (2008) found ω_{eq}∝a. This difference can be explained by the linear approximation of the sine of the phase lag done in this previous work. Given that the condition ω_{0}<ω_{M} is satisfied, the nonsynchronized states of equilibrium are stable in this case.
By using a GCM, Leconte et al. (2015) obtained numerically for the atmospheric tidal torque an expression similar to the one given by Eq. (3). They computed with this expression the possible states of equilibrium of Venuslike planets by applying to the telluric core the Andrade and constantQ models and showed that for both models, up to five equilibrium positions could appear. This is not the case when the Maxwell model is used, as shown by the present work. Hence, although these studies agree well on the existence of several nonsynchronized states of equilibrium, they also remind us that the theoretical number and positions of these equilibria depend on the models used to compute tidal torques and more specifically on the rheology chosen for the solid part.
Numerical values used in the case of Venuslike planets.
4. The case of Venuslike planets
We now illustrate the previous results for Venuslike planets (Table 1). The frequency ω_{M} is adjusted so that the present angular velocity of Venus corresponds to the retrograde state of equilibrium identified in the case where the condition ω_{M}>ω_{0} is satisfied (Ω_{−} in Eq. (9)).In Fig. 3, we plot the resulting tidal torque and its components, as well as their signs, as functions of the tidal frequency and orbital radius. These maps show that the torque varies over a very large range, particularly the solid component (∝a^{6} while ∝a^{5}). The combination of the solid and atmospheric responses generates the nonsynchronized states of equilibrium observed on the bottom left panel, which are located in the interval (see Eq. (11)) and move away from the synchronization when a increases (see Eq. (10)), as predicted analytically.
For illustration, in Fig. 4, we show the outcome of a different value of ω_{M}, with ω_{0}>ω_{M}, contrary to Fig. 3 where ω_{M}>ω_{0}. We observe the behaviour predicted analytically in Sect. 3. In Fig. 3, the nonsynchronized equilibria are stable and move away from the synchronization when a increases, but they are unstable in Fig. 4.
We note that the value of the solid Maxwell frequency obtained for stable nonsynchronized states of equilibrium, ω_{M} = 1.075 × 10^{4} s^{1}, is far higher than those of typical solid bodies (ω_{M} ~ 10^{10} s^{1}, see Efroimsky 2012). The Maxwell model, because of its decreasing rate as a function of ω (i.e. ∝ ω^{1}), underestimates tidal torque for tidal frequencies that are greater than ω_{M}, which leads to overestimate the Maxwell frequency when equalizing atmospheric and solid torques. Using the Andrade model for the solid part could give more realistic values of ω_{M}, as proved numerically by Leconte et al. (2015), because the decreasing rate of the torque is lower in the Andrade model than in the Maxwell model (i.e. ∝ ω^{− α} with α = 0.2−0.3).
Finally, we must discuss the assumption we made when we supposed that the parameters of the system did not depend on the starplanet distance. The surface temperature of the planet and the radiative frequency actually vary with a. If we consider that T_{0} is determined by the balance between the heating of the star and the black body radiation of the planet, then T_{0}∝a^{− 1/2}. As σ_{0}∝, we have σ_{0}∝a^{− 3/2}. These new dependences modify neither the expressions of ω_{eq} (Eq. (10)), nor the stability conditions of the states of equilibrium (Eqs. (12) and (13)). However, they have repercussions on the boundaries of the region where nonsynchronized states exist. This changes are illustrated by Fig. 4 (bottom panel), which shows the stability diagram of Fig. 3 (bottom left panel) computed with the functions T_{0}(a) = T_{0;♀}(a/a_{♀})^{−1/2} and σ_{0}(a) = σ_{0;♀}(a/a_{♀})^{−3/2} (a_{♀}, T_{0;♀} and σ_{0;♀} being the semimajor axis of Venus and the constant temperature and radiative frequency of Table 1 respectively).
Fig. 4 Sign of the total tidal torque as a function of the tidal frequency ω/n (horizontal axis) and orbital radius (a) (vertical axis) for two different cases. Top: with ω_{0} = 3.77 × 10^{7} s^{1} and ω_{M} = 3.7 × 10^{8} s^{1} (ω_{0}>ω_{M}). Bottom: with T_{0} and ω_{0} depending on the starplanet distance and ω_{M}>ω_{0}. White (blue) areas correspond to positive (negative) torque. Blue (red) lines designate stable (unstable) states of equilibrium. 

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5. Discussion
A physical model for solid and atmospheric tides allows us to determine analytically the possible rotation equilibria of Venuslike planets and their stability. Two regimes exist depending on the hierarchy of the characteristic frequencies associated with dissipative processes (viscous friction in the solid layer and thermal radiation in the atmosphere). These dissipative mechanisms have an impact on the stability of nonsynchronized equilibria, which can appear in the habitable zone since they are caused by solid and atmospheric tidal torques of comparable intensities. This study can be used to constrain the equilibrium rotation of observed superEarths and therefore to infer the possible climates of such planets. Also here the Maxwell rheology wasused. This work can be directly applied to alternative rheologies such as the Andrade model (Efroimsky 2012). The modelling used here is based on an approach where the atmosphere is assumed to rotate uniformly with the solid part of the planet (AuclairDesrotour et al. 2017). As general circulation is likely to play an important role in the atmospheric tidal response, we envisage examining the effect of differential rotation and corresponding zonal winds of the tidal torque in future studies.
Acknowledgments
P. AuclairDesrotour and S. Mathis acknowledge funding by the European Research Council through ERC grant SPIRE 647383. This work was also supported by the “exoplanètes” action from Paris Observatory and by the CoRoT/Kepler and PLATO CNES grant at CEASaclay. A. C. M. Correia acknowledges support from CIDMA strategic project UID/MAT/04106/2013.
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All Tables
All Figures
Fig. 1 Tidal elongation of a Venuslike rotating planet, composed of a solid core (brown) and a gaseous atmosphere (blue), and submitted to gravitational and thermal forcings. 

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In the text 
Fig. 2 Normalized total tidal torque exerted on a Venuslike planet (in red) and its solid (in green) and atmospheric (in blue) components as functions of the forcing frequency ω/n with ω_{0}/n = 2 and ω_{M}/n = 6. 

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In the text 
Fig. 3 Top: solid (left panel), atmospheric (middle panel) and total tidal torque (right panel) exerted on a Venuslike planet as functions of the reduced forcing frequency ω/n (horizontal axis) and orbital radius a in logarithmic scale (vertical axis). The colour level corresponds to the torque in logarithmic scale with isolines at . Bottom: sign of the solid (left panel), atmospheric (middle panel) and total (right panel) tidal torque as functions of the same parameters: white (blue) areas are associated with positive (negative) torques. Stable (unstable) states of equilibrium are designated by blue (red) lines (with A = 1.88 × 10^{19} m s). The pink band corresponds to the habitable zone for the black body equilibrium temperature T_{eq} = 288 K ± 20% for a 1 M_{⊙} Solartype star at the age of the Sun. 

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In the text 
Fig. 4 Sign of the total tidal torque as a function of the tidal frequency ω/n (horizontal axis) and orbital radius (a) (vertical axis) for two different cases. Top: with ω_{0} = 3.77 × 10^{7} s^{1} and ω_{M} = 3.7 × 10^{8} s^{1} (ω_{0}>ω_{M}). Bottom: with T_{0} and ω_{0} depending on the starplanet distance and ω_{M}>ω_{0}. White (blue) areas correspond to positive (negative) torque. Blue (red) lines designate stable (unstable) states of equilibrium. 

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In the text 