Tidal Evolution of Exoplanets
aa r X i v : . [ a s t r o - ph . E P ] S e p Tidal evolution of exoplanets
Alexandre C.M. Correia
University of Aveiro
Jacques Laskar
Paris ObservatoryTidal effects arise from differential and inelastic deformation of a planet by a perturbingbody. The continuous action of tides modify the rotation of the planet together with its orbituntil an equilibrium situation is reached. It is often believed that synchronous motion is the mostprobable outcome of the tidal evolution process, since synchronous rotation is observed for themajority of the satellites in the Solar System. However, in the XIX th century, Schiaparelli alsoassumed synchronous motion for the rotations of Mercury and Venus, and was later shown tobe wrong. Rather, for planets in eccentric orbits synchronous rotation is very unlikely. Therotation period and axial tilt of exoplanets is still unknown, but a large number of planets havebeen detected close to the parent star and should have evolved to a final equilibrium situation.Therefore, based on the Solar System well studied cases, we can make some predictions forexoplanets. Here we describe in detail the main tidal effects that modify the secular evolutionof the spin and the orbit of a planet. We then apply our knowledge acquired from SolarSystem situations to exoplanet cases. In particular, we will focus on two classes of planets,“Hot-Jupiters” (fluid) and “Super-Earths” (rocky with atmosphere).
1. INTRODUCTION
The occurrence, on most open ocean coasts, of high seatide at about the time of Moon’s passage across the merid-ian, early prompted the idea that Earth’s satellite exerts anattraction on the water. The occurrence of a second hightide when the Moon is on the opposite meridian was a greatpuzzle, but the correct explanation of the tidal phenomenawas given by Newton in “
Philosophiæ Naturalis PrincipiaMathematica ”. Tides are a consequence of the lunar andsolar gravitational forces acting in accordance with laws ofmechanics. Newton realized that the tidal forces also mustaffect the atmosphere, but he assumed that the atmospherictides would be too small to be detected, because changesin weather would introduce large irregular variations uponbarometric measurements.However, the semi-diurnal oscillations of the atmo-spheric surface pressure has proven to be one of the mostregular of all meteorological phenomena. It is readily de-tectable by harmonic analysis at any station over the world(e.g. Chapman and Lindzen 1970). The main difference inrespect to ocean tides is that atmospheric tides follow theSun and not the Moon, as the atmosphere is essentially ex-cited by the Solar heat. Even though tides of gravitationalorigin are present in the atmosphere, the thermal tides aremore important as the pressure variations on the groundare more sensitive to the temperature gradients than to thegravitational ones.The inner planets of the Solar System as well as themajority of the main satellites present today a spin dif-ferent from what is believed to have been the initial one (e.g. Goldreich and Soter 1966; Goldreich and Peale 1968).Planets and satellites are supposed to rotate much faster inthe beginning and any orientation of the spin axis may beallowed (e.g. Dones and Tremaine 1993; Kokubo and Ida2007). However, tidal dissipation within the internal lay-ers give rise to secular evolution of planetary spins and or-bits. In the case of the satellites, spin and orbital evolutionis mainly driven by tidal interactions with the central planet,whereas for the inner planets the main source of tidal dis-sipation is the Sun (in the case of the Earth, tides raised bythe Moon are also important).Orbital and spin evolution cannot be dissociated becausethe total angular momentum must be conserved. As a con-sequence a reduction in the rotation rate of a body impliesan increment of the orbit semi-major axis and vice-versa.For instance, the Earth’s rotation period is increasing about2 ms/century (e.g. Williams 1990), and the Moon is conse-quently moving away about 3.8 cm/year (e.g. Dickey et al.1994). On the other hand, Neptune’s moon, Triton, andthe Martian moon, Phobos, are spiraling down into theplanet, clearly indicating that the present orbits are not pri-mordial, and may have undergone a long evolving processfrom a previous capture from an heliocentric orbit (e.g.Mignard 1981; Goldreich et al. 1989; Correia 2009). Boththe Earth’s Moon and Pluto’s moon, Charon, have a sig-nificant fraction of the mass of their systems, and there-fore they could be classified as double-planets rather thanas satellites. The proto-planetary disk is unlikely to pro-duce double-planet systems, whose origin seems to be dueto a catastrophic impact of the initial planet with a bodyof comparable dimensions (e.g. Canup and Asphaug 2001;1anup 2005). The resulting orbits after collision are mostlikely eccentric, but the present orbits are almost circularsuggesting that tidal evolution subsequently occurred.The ultimate stage for tidal evolution corresponds tothe synchronous rotation, a configuration where the ro-tation rate coincides with the orbital mean motion, sincesynchronous equilibrium corresponds to the minimum ofdissipation of energy. However, when the eccentricityis different from zero some other configurations are pos-sible, such as the 3/2 spin-orbit resonance observed forplanet Mercury (Colombo 1965; Goldreich and Peale 1966;Correia and Laskar 2004) or the chaotic rotation of Hype-rion (Wisdom et al. 1984). When a dense atmosphere ispresent, thermal atmospheric tides may also counterbalancethe gravitational tidal effect and non-resonant equilibriumconfigurations are also possible, as it is illustrated by the ret-rograde rotation of Venus (Correia and Laskar 2001). Ad-ditional effects may also contribute to the final evolutionof the spin, such as planetary perturbations or core-mantlefriction.Despite the proximity of Mercury and Venus to theEarth, the determination of their rotational periods has onlybeen achieved in the second half of the XX th century,when it became possible to use radar ranging on the plan-ets (Pettengill and Dyce 1965; Goldstein 1964; Carpenter1964). We thus do not expect that it will be easy to observethe rotation of the recently discovered exoplanets. Never-theless, many of the exoplanets are close to their host star,and we can assume that exoplanets’ spin and orbit havealready undergone enough dissipation and evolved into afinal equilibrium possibility. An identical assumption hasbeen done before by Schiaparelli for Mercury and Venus(1889), who made predictions for their rotations based onDarwin’ work (1880). Schiaparelli’s predictions were laterproved to be wrong, but were nevertheless much closer tothe true rotation periods than most values derived form ob-servations in the two previous centuries. As Schiaparelli,we may dare to establish predictions for the rotation peri-ods of some already known exoplanets. We hope that theadditional knowledge that we gained from a better under-standing of the rotation of Mercury and Venus will helpus to be at least as close to the reality as Schiaparelli was.Indeed, observations also show that many of the exoplan-ets have very eccentric orbits. In some cases eccentricitieslarger than 0.9 are found (e.g. Naef et al. 2001; Jones et al.2006; Tamuz et al. 2008), which opens a wide variety forfinal tidal equilibrium positions, different from what we ob-serve around the Sun.In this Chapter we will describe the tidal effects thatmodify the secular evolution of the spin and orbit of aplanet. We then apply our knowledge acquired from So-lar System situations to exoplanet cases. In particular, wewill focus on two classes of planets, “Hot-Jupiters” (fluid)and “Super-Earths” (rocky), which are close to the star andtherefore more susceptible of being arrived in a final equi-librium situation. γγ ε ε L X . K k . N . EquatorA M eanE cli p tic E c t fix plane Ec ψ θ Fig. 1.—
Andoyer’s canonical variables. L is the projection ofthe total rotational angular momentum vector L on the principalaxis of inertia K , and X the projection of the angular momentumvector on the normal to the orbit (or ecliptic) k . The angle betweenthe equinox of date γ and a fixed point of the equator A is the hourangle θ , and ψ = γ N + N γ is the general precession angle. Thedirection of γ is on a fixed plane E c , while γ is on the meanorbital (or ecliptic) E ct of date t .
2. MODEL DESCRIPTION
We will first omit the tidal effects, and describe the spinmotion of the planet in a conservative framework. The mo-tion equations will be obtained from an Hamiltonian for-malism (e.g. Goldstein 1950) of the total gravitational en-ergy of the planet (Sect. 2.1). Gravitational tides (Sect. 2.2)and thermal atmospheric tides (Sect. 2.3) will be describedlater. We also discuss the impact of spin-orbit resonances(Sect. 2.4) and planetary perturbations (Sect. 2.5).
The planet is considered here as a rigid body with mass m and moments of inertia A ≤ B < C , supported by thereference frame ( I , J , K ) , fixed with respect to the planet’sfigure. Let L be the total rotational angular momentum and ( i , j , k ) a reference frame linked to the orbital plane (where k is the normal to this plane). As we are interested in thelong-term behavior of the spin axis, we merge the axis offigure K with direction of the angular momentum L . In-deed, the average of K coincides with L /L up to J , where cos J = L · K (Bou´e and Laskar 2006). J is extremelysmall for large rocky planets ( J ≈ × − for the Earth),being even smaller for Jupiter-like planets that behave asfluids. The angle between K and k is the obliquity, ε , andthus, cos ε = k · K (Fig. 1).The Hamiltonian of the motion can be written usingcanonical Andoyer’s action variables ( L, X ) and their con-jugate angles ( θ, − ψ ) (Andoyer 1923; Kinoshita 1977). L = L · K = Cω is the projection of the angular mo-2entum on the C axis, with rotation rate ω = ˙ θ − ˙ ψ cos ε ,and X = L · k is the projection of the angular momentumon the normal to the ecliptic; θ is the hour angle betweenthe equinox of date and a fixed point of the equator, and ψ is the general precession angle, an angle that simultane-ously accounts for the precession of the spin axis and theorbit (Fig. 1). The gravitational potential V (energy per unit mass) gen-erated by the planet at a generic point of the space r , ex-panded in degree two of R/r , where R is the planet’s ra-dius, is given by (e.g. Tisserand 1891; Smart 1953): V ( r ) = − Gmr + G ( B − A ) r P ( ˆ r · J )+ G ( C − A ) r P ( ˆ r · K ) , (1)where ˆ r = r /r , G is the gravitational constant, and P ( x ) = (3 x − / are the Legendre polynomials of de-gree two. The potential energy U when orbiting a centralstar of mass m ⋆ is then: U = m ⋆ V ( r ) . (2)For a planet evolving in a non-perturbed keplerian orbit,we write: ˆ r = cos( ̟ + v ) i + sin( ̟ + v ) j , (3)where ̟ is the longitude of the periapse and v the trueanomaly (see Chapter 2: Keplerian Orbits and Dynamics ).Thus, transforming the body equatorial frame ( I , J , K ) intothe orbital frame ( i , j , k ) , we obtain (Fig. 1): ( ˆ r · J = − cos w sin θ + sin w cos θ cos ε , ˆ r · K = − sin w sin ε , (4)where w = ̟ + ψ + v is the true longitude of date. Theexpression for the potential energy (Eq. 2) becomes (e.g.Correia 2006): U = − Gmm ⋆ r + GCm ⋆ r E d P (sin w sin ε ) − Gm ⋆ r ( B − A ) F ( θ, w, ε ) , (5)where F ( θ, w, ε ) = 2 cos(2 θ − w ) cos (cid:18) ε (cid:19) +2 cos(2 θ + 2 w ) sin (cid:18) ε (cid:19) + cos(2 θ ) sin ε , (6)and E d = C − ( A + B ) C = k f R GC ω + δE d , (7) where E d is the dynamical ellipticity, and k f is the fluidLove number (pertaining to a perfectly fluid body with thesame mass distribution as the actual planet). The first part of E d (Eq. 7) corresponds to the flattening in hydrostatic equi-librium (Lambeck 1980), and δE d to the departure from thisequilibrium. Since we are only interested in the study of the long-term motion, we will average the potential energy U overthe rotation angle θ and the mean anomaly M : U = 14 π Z π Z π U dM dθ . (8)However, when the rotation frequency ω ≈ ˙ θ and the meanmotion n = ˙ M are close to resonance ( ω ≈ pn , for asemi-integer value p ), the terms with argument θ − pM ) vary slowly and must be retained in the expansions (e.g.Murray and Dermott 1999) cos(2 θ ) r = 1 a ∞ X p = −∞ G ( p, e ) cos 2( θ − pM ) , (9)and cos(2 θ − w ) r = 1 a ∞ X p = −∞ H ( p, e ) cos 2( θ − pM ) , (10)where a and e are the semi-major axis and the eccentricityof the planet’s orbit, respectively. The functions G ( p, e ) and H ( p, e ) can be expressed in power series in e (Table 1). Theaveraged non-constant part of the potential U becomes: UC = − α ω x − β " (1 − x ) G ( p, e ) cos 2( θ − pM )+ (1 + x ) H ( p, e ) cos 2( θ − pM − φ )+ (1 − x ) H ( − p, e ) cos 2( θ − pM + φ ) , (11)where x = X/L = cos ε , φ = ̟ + ψ , α = 3 Gm ⋆ a (1 − e ) / E d ω ≈ n ω (1 − e ) − / E d (12)is the “precession constant” and β = 3 Gm ⋆ a B − AC ≈ n B − AC . (13)For non-resonant motion, that is, when ( B − A ) /C ≈ (e.g. gaseous planets) or | ω | ≫ pn , we can simplify expres-sion (11) as: UC = − α ω x . (14)3 ABLE OEFFICIENTS OF G ( p, e ) AND H ( p, e ) TO e . p G ( p, e ) H ( p, e ) − e + 74 e e − / e + 2716 e e e + 158 e / e + 2716 e − e + 116 e e + 74 e − e + 1316 e / e e − e e e − e / e e The exact expression of the coefficients is given by G ( p, e ) = π R π (cid:0) ar (cid:1) exp(i 2 pM ) dM and H ( p, e ) = π R π (cid:0) ar (cid:1) exp(i 2 ν ) exp(i 2 pM ) dM . The Andoyer variables ( L , θ ) and ( X , − ψ ) are canon-ically conjugated and thus (e.g. Goldstein 1950; Kinoshita1977) dLdt = − ∂U∂θ , dXdt = ∂U∂ψ , dψdt = − ∂U∂X . (15)Andoyer’s variables do not give a clear view of the spinvariations, despite their practical use. Since ω = L/C and cos ε = x = X/L the spin variations can be obtained as: dωdt = − ∂∂θ (cid:18) UC (cid:19) , dψdt = − ω ∂∂x (cid:18) UC (cid:19) , (16)and dxdt = − L (cid:18) XL dLdt − dXdt (cid:19) = 1 ω (cid:20) x ∂∂θ + ∂∂ψ (cid:21) (cid:18) UC (cid:19) . (17)For non-resonant motion, we get from equation (14): dωdt = dxdt = 0 and dψdt = α x . (18)The spin motion reduces to the precession of the spin vectorabout the normal to the orbital plane with rate α x . Gravitational tides arise from differential and inelasticdeformations of the planet due to the gravitational effect ofa perturbing body (that can be the central star or a satel-lite). Tidal contributions to the planet evolution are basedon a very general formulation of the tidal potential, initi-ated by George H. Darwin (1880). The attraction of a bodywith mass m ⋆ at a distance r from the center of mass of theplanet can be expressed as the gradient of a scalar potential V ′ , which is a sum of Legendre polynomials (e.g. Kaula1964; Efroimsky and Williams 2009): V ′ = ∞ X l =2 V ′ l = − Gm ⋆ r ∞ X l =2 (cid:18) r ′ r (cid:19) l P l (cos S ) , (19)where r ′ is the radial distance from the planet’s center, and S the angle between r and r ′ . The distortion of the planetby the potential V ′ gives rise to a tidal potential, V g = ∞ X l =2 ( V g ) l , (20)where ( V g ) l = k l V ′ l at the planet’s surface and k l is theLove number for potential (Fig. 2). Typically, k ∼ . for Earth-like planets, and k ∼ . for giant planets(Yoder 1995). Since the tidal potential ( V g ) l is an l th de-gree harmonic, it is a solution of a Dirichlet problem, andexterior to the planet it must be proportional to r − l − (e.g.4bramowitz and Stegun 1972; Lambeck 1980). Further-more, as upon the surface r ′ = R ≪ r , we can retain inexpression (20) only the first term, l = 2 : V g = − k Gm ⋆ R (cid:18) Rr (cid:19) (cid:18) Rr ′ (cid:19) P (cos S ) . (21)In general, imperfect elasticity will cause the phase angleof V g to lag behind that of V ′ (Kaula 1964) by an angle δ g ( σ ) such that: δ g ( σ ) = σ ∆ t g ( σ ) , (22) ∆ t g ( σ ) being the time lag associated to the tidal frequency σ (a linear combination of the inertial rotation rate ω andthe mean orbital motion n ) (Fig. 3). Expressing the tidal potential given by expression (21)in terms of Andoyer angles ( θ, ψ ) , we can obtain the con-tribution to the spin evolution from expressions (15) using U g = m ′ V g at the place of U , where m ′ is the mass of theinteracting body. As we are interested here in the study ofthe secular evolution of the spin, we also average U g overthe periods of mean anomaly and longitude of the periapseof the orbit. When the interacting body is the same as theperturbing body ( m ′ = m ⋆ ) , we obtain: dωdt = − Gm ⋆ R Ca X σ b g ( σ )Ω gσ ( x, e ) , (23) dεdt = − Gm ⋆ R Ca sin εω X σ b g ( σ ) E gσ ( x, e ) , (24)where the coefficients Ω gσ ( x, e ) and E gσ ( x, e ) are polynomi-als in the eccentricity (Kaula 1964). When the eccentricityis small, we can neglect the terms in e , and we have: P σ b τ ( σ )Ω τσ = b τ ( ω ) x (cid:0) − x (cid:1) + b τ ( ω − n ) (1 + x ) (cid:0) − x (cid:1) + b τ ( ω + 2 n ) (1 − x ) (cid:0) − x (cid:1) + b τ (2 ω ) (cid:0) − x (cid:1) + b τ (2 ω − n ) (1 + x ) + b τ (2 ω + 2 n ) (1 − x ) , (25)and P σ b τ ( σ ) E τσ = b τ (2 n ) (cid:0) − x (cid:1) + b τ ( ω ) x − b τ ( ω − n ) (1 + x ) (2 − x )+ b τ ( ω + 2 n ) (1 − x ) (2 + x )+ b τ (2 ω ) x (cid:0) − x (cid:1) − b τ (2 ω − n ) (1 + x ) + b τ (2 ω + 2 n ) (1 − x ) . (26)The coefficients b τ ( σ ) are related to the dissipation ofthe mechanical energy of tides in the planet’s interior, re-sponsible for the time delay ∆ t g ( σ ) between the position of “maximal tide” and the sub-stellar point. They are re-lated to the phase lag δ g ( σ ) as: b g ( σ ) = k sin 2 δ g ( σ ) = k sin ( σ ∆ t g ( σ )) , (27)where τ ≡ g for gravitational tides. Dissipation equations(23) and (24) must be invariant under the change ( ω, x ) by ( − ω, − x ) which imposes that b ( σ ) = − b ( − σ ) , that is, b ( σ ) is an odd function of σ . Although mathematically equiva-lent, the couples ( ω, x ) and ( − ω, − x ) correspond to twodifferent physical situations (Correia and Laskar 2001).The tidal potential given by expression (21) can alsobe directly used to compute the orbital evolution due totides. Indeed, it can be seen as a perturbation of the grav-itational potential (Eq. 1), and the contributions to the or-bit are computed using Lagrange Planetary equations (e.g.Brouwer and Clemence 1961; Kaula 1964): dadt = 2 µna ∂U∂M , (28) dedt = √ − e µna e (cid:20)p − e ∂U∂M − ∂U∂̟ (cid:21) , (29)where µ = mm ⋆ / ( m + m ⋆ ) ≈ m is the reduced mass.We then find for the orbital evolution of the planet: dadt = − Gm ⋆ R µna X σ b g ( σ ) A gσ ( x, e ) , (30) dedt = − e Gm ⋆ R µna X σ b g ( σ ) E gσ ( x, e ) , (31)where the coefficients A gσ ( x, e ) and E gσ ( x, e ) are againpolynomials in the eccentricity. When the eccentricity issmall, we can neglect the terms in e , and we have: P σ b τ ( σ ) A τσ = b τ (2 n ) (1 − x ) − b τ ( ω − n ) (1 − x )(1 + x ) + b τ ( ω + 2 n ) (1 − x )(1 − x ) − b τ (2 ω − n ) (1 + x ) + b τ (2 ω + 2 n ) (1 − x ) , (32)and P σ b τ ( σ ) E τσ = b τ ( n ) (5 x − x − − b τ (2 n ) (1 − x ) + b τ (3 n ) (1 − x ) − b τ ( ω − n ) (5 x − x + 1)(1 − x )+ b τ ( ω + n ) (5 x + 1)(7 x − − x )+ b τ ( ω − n ) (1 − x )(1 + x ) − b τ ( ω + 2 n ) (1 − x )(1 − x ) − b τ ( ω − n ) (1 − x )(1 + x ) + b τ ( ω + 3 n ) (1 − x )(1 − x ) − b τ (2 ω − n ) (5 x − x − x ) + b τ (2 ω + n ) (5 x + 7)(7 x + 5)(1 − x ) + b τ (2 ω − n ) (1 + x ) − b τ (2 ω + 2 n ) (1 − x ) − b τ (2 ω − n ) (1 + x ) + b τ (2 ω + 3 n ) (1 − x ) . (33)5 mr CM . Fig. 2.—
Gravitational tides. The difference between the gravitational force exerted by the mass m on a point of the surface and thecenter of mass is schematized by the arrows. The planet will deform following the equipotential of all present forces. m g m δω CM n r . Fig. 3.—
Phase lag for gravitational tides. The tidal deformation takes a delay time ∆ t g to attain the equilibrium. During the time ∆ t g ,the planet turns by an angle ω ∆ t g and the star by n ∆ t g . For ε = 0 , the bulge phase lag is given by δ g ≈ ( ω − n )∆ t g . The dissipation of the mechanical energy of tides in theplanet’s interior is responsible for the phase lags δ ( σ ) . Acommonly used dimensionless measure of tidal damping isthe quality factor Q (Munk and MacDonald 1960), definedas the inverse of the “specific” dissipation and related to thephase lags by Q ( σ ) = 2 πE ∆ E = cot 2 δ ( σ ) , (34)where E is the total tidal energy stored in the planet, and ∆ E the energy dissipated per cycle. We can rewrite expres-sion (27) as: b g ( σ ) = k sign( σ ) p Q ( σ ) + 1 ≈ sign( σ ) k Q ( σ ) . (35)The present Q value for the planets in the Solar system canbe estimated from orbital measurements, but as rheology ofthe planets is badly known, the exact dependence of b τ ( σ ) on the tidal frequency σ is unknown. Many different au-thors have studied the problem and several models for b τ ( σ ) have been developed so far, from the simplest ones to themore complex (for a review see Efroimsky and Williams2009). The huge problem in validating one model betterthan the others is the difficulty to compare the theoreticalresults with the observations, as the effect of tides are very small and can only be detected efficiently after long peri-ods of time. Therefore, here we will only describe a fewsimplified models that are commonly used: The visco-elastic model
Darwin (1908) assumed that theplanet behaves like a Maxwell solid, that is, the planet re-sponds to stresses like a massless, damped harmonic oscil-lator. It is characterized by a rigidity (or shear modulus) µ e and by a viscosity υ e . A Maxwell solid behaves likean elastic solid over short time scales, but flows like a fluidover long periods of time. This behavior is also known aselasticoviscosity. For a constant density ρ , we have: b g ( σ ) = k f τ b − τ a τ b σ ) σ , (36)where k f is the fluid Love number (Eq. 7). τ a = υ e /µ e and τ b = τ a (1 + 19 µ e R/ Gmρ ) are time constants for thedamping of gravitational tides.The visco-elastic model is a realistic approximationof the planet’s deformation with the tidal frequency (e.g.Escribano et al. 2008). However, when replacing expres-sion (36) into the dynamical equations (23) and (24) weget an infinite sum of terms, which is not practical. As aconsequence, simplified versions of the visco-elastic modelfor specific values of the tidal frequency σ are often used.For instance, when σ is small, ( τ b σ ) can be neglected in6xpression (36) and b g ( σ ) becomes proportional to σ . The viscous or linear model
In the viscous model, it isassumed that the response time delay to the perturbation isindependent of the tidal frequency, i.e., the position of the“maximal tide” is shifted from the sub-stellar point by aconstant time lag ∆ t g (Mignard 1979, 1980). As usuallywe have σ ∆ t g ≪ , the viscous model becomes linear: b g ( σ ) = k sin( σ ∆ t g ) ≈ k σ ∆ t g . (37)The viscous model is a particular case of the visco-elasticmodel and is specially adapted to describe the behavior ofplanets in slow rotating regimes ( ω ∼ n ). The constant- Q model Since for the Earth, Q changes byless than an order of magnitude between the Chandler wob-ble period (about 440 days) and seismic periods of a fewseconds (Munk and MacDonald 1960), it is also commonto treat the specific dissipation as independent of frequency.Thus, b g ( σ ) ≈ sign( σ ) k /Q . (38)The constant- Q model can be used for periods of timewhere the tidal frequency does not change much, as is thecase for fast rotating planets. However, for long-term evo-lutions and slow rotating planets, the constant- Q model isnot appropriate as it gives rise to discontinuities for σ = 0 . Although both linear and constant models have somelimitations, for simplicity reasons they are the most widelyused in literature. The linear model has nevertheless an im-portant advantage over the constant model: it is appropriatedo describe the behavior of the planet near the equilibriumpositions, since the linear model closely follows the realis-tic visco-elastic model for slow rotation rates. The equa-tions of motion can also be expressed in an elegant way,so we will adopt the viscous model for the remaining ofthis Chapter, without loss of generality concerning the mainconsequences of tidal effects.Using the approximation (37) in expressions (23) and(24), we simplify the spin equations as (Correia and Laskar2010, Appendix B): ˙ ω = − KnC f ( e ) 1 + cos ε ωn − f ( e ) cos ε ! , (39)and ˙ ε ≈ KnCω sin ε (cid:16) f ( e ) cos ε ω n − f ( e ) (cid:17) , (40)where f ( e ) = 1 + 3 e + 3 e / − e ) / , (41) f ( e ) = 1 + 15 e / e / e / − e ) , (42) / n eccentricity ω e ε = 0 Fig. 4.—
Evolution of the equilibrium rotation rate ω e /n = f ( e ) /f ( e ) with the eccentricity when ε = 0 ◦ using the viscousmodel (Eq. 44). As the eccentricity increases, ω e also increases.The gravitational tides lead the planet to exact resonance when theeccentricity is respectively e / = 0 , e / = 0 . and e / = 0 . . and K = ∆ t k Gm ⋆ R a . (43)Because of the factor /ω in the magnitude of the obliq-uity variations (Eq. 40), for an initial fast rotating planet thetime-scale for the obliquity evolution will be longer than thetime-scale for the rotation rate evolution (Eq. 39). As a con-sequence, it is to be expected that the rotation rate reachesan equilibrium value earlier than the obliquity. For a givenobliquity and eccentricity, the equilibrium rotation rate, ob-tained when ˙ ω = 0 , is then attained for (Fig. 4): ω e n = f ( e ) f ( e ) 2 cos ε ε , (44)Replacing the previous equation in the expression for obliq-uity variations (Eq. 40), we find: ˙ ε ≈ − KnCω f ( e ) sin ε ε . (45)We then conclude that the obliquity can only decrease bytidal effect, since ˙ ε ≤ , and the final obliquity always tendsto zero. As for the spin, the semi-major axis and the eccentricityevolution can be obtained using the approximation (37) inexpressions (32) and (33), respectively (Correia 2009): ˙ a = 2 Kµa (cid:16) f ( e ) cos ε ωn − f ( e ) (cid:17) , (46)and ˙ e = 9 Kµa (cid:18) f ( e ) cos ε ωn − f ( e ) (cid:19) e , (47)where f ( e ) = 1 + 31 e / e / e /
16 + 25 e / − e ) / , (48)7 ( e ) = 1 + 3 e / e / − e ) , (49) f ( e ) = 1 + 15 e / e / e / − e ) / . (50)The ratio between orbital and spin evolution time-scalesis roughly given by C/ ( µa ) ≪ , meaning that the spinachieves an equilibrium position much faster than the orbit.Replacing the equilibrium rotation rate (Eq. 44) with ε = 0 (for simplicity) in equations (46) and (47), gives: ˙ a = − Kµa f ( e ) e , (51) ˙ e = − K µa f ( e )(1 − e ) e , (52)where f ( e ) = (1 + 45 e /
14 + 8 e + 685 e /
224 +255 e / e / − e ) − / / (1+3 e +3 e / . Thus, we always have ˙ a ≤ and ˙ e ≤ , and the final ec-centricity is zero. Another consequence is that the quan-tity a (1 − e ) is conserved (Eq. 101). The final equilibriumsemi-major axis is then given by a f = a (1 − e ) , (53)which is a natural consequence of the orbital angular mo-mentum conservation (since the rotational angular momen-tum of the planet is much smaller). Notice, however, thatonce the equilibrium semi-major axis a f is attained, thetidal effects on the star cannot be neglected, and they governthe future evolution of the planet’s orbit. The differential absorption of the Solar heat by theplanet’s atmosphere gives rise to local variations of tem-perature and consequently to pressure gradients. The massof the atmosphere is then permanently redistributed, ad-justing for an equilibrium position. More precisely, theparticles of the atmosphere move from the high tempera-ture zone (at the sub-stellar point) to the low temperatureareas. Indeed, observations on Earth show that the pressureredistribution is essentially a superposition of two pressurewaves (see Chapman and Lindzen 1970): a daily (or diur-nal) tide of small amplitude (the pressure is minimal at thesub-stellar point and maximal at the antipode) and a stronghalf-daily (semi-diurnal) tide (the pressure is minimal at thesub-stellar point and at the antipode) (Fig. 5).The gravitational potential generated by all of the parti-cles in the atmosphere at a generic point of the space r isgiven by: V a = − G Z ( M ) d M| r − r ′ | , (54)where r ′ = ( r ′ , θ ′ , ϕ ′ ) is the position of the atmospheremass element d M with density ρ a ( r ′ ) and d M = ρ a ( r ′ ) r ′ sin θ ′ dr ′ dθ ′ dϕ ′ . (55) Assuming that the radius of the planet is constant andthat the height of the atmosphere can be neglected, we ap-proximate expression (55) as: d M = R g p s ( θ ′ , ϕ ′ , t ) sin θ ′ dθ ′ dϕ ′ , (56)where g is the mean surface gravity acceleration, and p s thesurface pressure, which depends on the stellar insolation.Thus, p s depends on S , the angle between the direction ofthe Sun and the normal to the surface: p s ( θ ′ , ϕ ′ ) = p s ( S ) = + ∞ X l =0 ˜ p l P l (cos S ) , (57)where P l are the Legendre polynomials of order l and ˜ p l its coefficients. Developing also | r − r ′ | − in Legendrepolynomials we rewrite expression (54) as: V a = − ρ + ∞ X l =0 l + 1 ˜ p l (cid:18) Rr (cid:19) l +1 P l (cos S ) , (58)where ¯ ρ is the mean density of the planet. Since we areonly interested in pressure oscillations, we must subtractthe term of constant pressure ( l = 0 ) in order to obtain thetidal potential. We also eliminate the diurnal terms ( l =1 ) because they correspond to a displacement of the centerof mass of the atmosphere bulge which has no dynamicalimplications. Thus, since we usually have r ≫ R , retainingonly the semi-diurnal terms ( l = 2 ), we write: V a = −
35 ˜ p ¯ ρ (cid:18) Rr (cid:19) P (cos S ) . (59) Using the same methodology of previous sections, thecontributions of thermal atmospheric tides to the spin evo-lution are obtained from expressions (15) using U a = m ⋆ V a at the place of U : dωdt = − m ⋆ R C ¯ ρa X σ b a ( σ )Ω aσ ( x, e ) , (60) dεdt = − m ⋆ R C ¯ ρa sin εω X σ b a ( σ ) E aσ ( x, e ) , (61)where the terms Ω aσ ( x, e ) and E aσ ( x, e ) are also polynomi-als in the eccentricity, but different from their analogs forgravitational tides (Eqs. 23 and 24). Nevertheless, when ne-glecting the terms in e , they become equal and are givenby expressions (25) and (26), respectively (with τ = a ).For thermal atmospheric tides there is also a delay beforethe response of the atmosphere to the excitation (Fig. 6). Wename the time delay ∆ t a ( σ ) and the corresponding phaseangle δ a ( σ ) (Eq. 22). The dissipation factor b a ( σ ) is heregiven by: b a ( σ ) = ˜ p ( σ ) sin 2 δ a ( σ ) = ˜ p ( σ ) sin( σ ∆ t a ( σ )) . (62)8 P r CM m atmosphere hotcold STAR Fig. 5.—
Thermal atmospheric tides. The atmosphere’s heating decreases with the distance to the sub-stellar point P ⊙ . The atmosphericmass redistribution is essentially decomposed in a weak daily tide (round shape) and in a strong half-daily tide (oval shape). Siebert (1961) and Chapman and Lindzen (1970) haveshown that when | ˜ p ( σ ) | ≪ ˜ p , (63)the amplitudes of the pressure variations on the ground aregiven by: ˜ p ( σ ) = i γσ ˜ p (cid:18) ∇ · v σ − γ − γ J σ gH (cid:19) , (64)where γ = 7 / for a perfect gas, v is the velocity of tidalwinds, J σ the amount of heat absorbed or emitted by a unitmass of air per unit time, and H is the scale height at thesurface. We can rewrite expression (64) as, ˜ p ( σ ) = γ | σ | ˜ p ∇ · v σ − γ − γ J σ gH e ± i π (65) = | ˜ p ( σ ) | e ± i π , where the factor e ± i π can be seen as a supplementary phaselag of ± π/ : b a ( σ ) = | ˜ p ( σ ) | sin 2 (cid:16) δ a ( σ ) ± π (cid:17) = −| ˜ p ( σ ) | sin 2 δ a ( σ ) . (66)The minus sign above causes pressure variations to lead theSun whenever δ a ( σ ) < π/ (Chapman and Lindzen 1970;Dobrovolskis and Ingersoll 1980) (Fig. 6). Unfortunately, our knowledge of the atmosphere re-sponse to thermal excitation is still very incomplete. Asfor the gravitational tides, models are developed to dealwith the unknown part. Dobrovolskis and Ingersoll (1980)adopted a model called “ heating at the ground ”, where theysuppose that all the stellar flux absorbed by the ground F s isimmediately deposited in a thin layer of atmosphere at thesurface. The heating distributing may then be written as adelta-function just above the ground: J ( r ) = g ˜ p F s δ ( r − + ) . (67) Neglecting v over the thin heated layer, expression (64)simplifies: | ˜ p ( σ ) | = 516 γ − | σ | F s H = 516 γ | σ | gF s c p ¯ T s , (68)where the factor / represents the second-degree har-monic component of the insolation contribution (Dobrovolskis and Ingersoll1980) and c p is the specific heat at constant pressure.Nevertheless, according to expression (68), if σ = 0 ,the amplitude of the pressure variations ˜ p ( σ ) becomes in-finite. The amplitude cannot grow infinitely, as for a tidalfrequency equal to zero, a steady distribution is attained,and thus ˜ p (0) = 0 . Indeed, expression (64) is not validwhen σ ≈ because the condition (63) is no longer ver-ified. Using the typically accepted values for the Venu-sian atmosphere, c p ≈ − s − , ¯ T s ≈
730 K and F s ≈
100 Wm − (Avduevskii et al. 1976) we compute: | ˜ p ( σ ) | ≈ − ˜ p n | σ | , (69)which means that for σ ∼ n , the “ heating at the ground ”model of Dobrovolskis and Ingersoll (1980), can still beapplied. Since we are only interested in long-term be-haviors we can set ˜ p ( σ ) = 0 whenever | σ | ≪ n/ .Moreover, for tidal frequencies σ ∼ the dissipation lag sin( σ ∆ t a ( σ )) ≈ σ ∆ t a ( σ ) also goes to zero. We expectthat further studies about synchronous exoplanets’ atmo-spheres (e.g. Joshi et al. 1997; Arras and Socrates 2010),may provide a more accurate solution for the case σ ≈ .In presence of a dense atmosphere, another kind of tidescan arise: the atmosphere pressure upon the surface givesrise to a deformation, a pressure bulge, that will also beaffected by the stellar torque. At the same time, the atmo-sphere itself exerts a torque over the planet’s bulges (gravi-tational and pressure bulge). Nevertheless, we do not needto take into account additional tidal effects as their con-sequences upon the dynamical equations can be neglected(Hinderer et al. 1987; Correia and Laskar 2003a). A spin-orbit resonance occurs when there is a commen-surability between the rotation rate ω and the mean motion9 δ n ω r atmosphere CM a m Fig. 6.—
Phase lag for thermal atmospheric tides. During the time ∆ t a the planet turns by an angle ω ∆ t a and the star by n ∆ t a . For ε = 0 , the bulge phase lag is given by δ a ≈ ( ω − n )∆ t a . of the orbit n (Eqs. 9 and 10). The synchronous rotation ofthe Moon is the most common example. After the discov-ery of the 3/2 spin-orbit resonance of Mercury (Colombo1965), spin-orbit resonances were studied in great detail(Colombo and Shapiro 1966; Goldreich and Peale 1966;Counselman and Shapiro 1970; Correia and Laskar 2004,2009, 2010). When resonant motion is present we cannotneglect the terms in β in expression (11). Assuming forsimplicity a low obliquity ( x ≈ ) we obtain a non-zerocontribution for the rotation rate (Eq. 16): dωdt = − β H ( p, e ) sin 2 γ , (70)where γ = θ − pM − φ . The rotation of the planet will there-fore present oscillations around a mean value. The width ofthe corresponding resonance, centered at ω = pn , is: ∆ ω = p β H ( p, e ) . (71)Due to the tidal torque (Eq. 23), here denoted by T ,the mean rotation rate does not remain constant and maytherefore cross and be captured in a spin-orbit resonance.Goldreich and Peale (1966) computed a simple estimationof the capture probability P cap , and subsequent more de-tailed studies proved their expression to be essentially cor-rect (for a review, see Henrard 1993). Since the tidal torquescan usually be described by means of the torques consideredby Goldreich and Peale (1966), we will adopt here the samenotations. Let T = − K (cid:18) V + ˙ γn (cid:19) , (72)where K and V are positive constant torques, and ˙ γ = ω − pn . The probability of capture into resonance is then givenby (Goldreich and Peale 1966): P cap = 21 + πV n/ ∆ ω , (73) where ∆ ω is the resonance width (Eq. 71). In the slow rota-tion regime ( ω ∼ n ), where the spin encounters spin-orbitresonances and capture may occur, we compute for the vis-cous tidal model (Eq. 39): P cap = 2 " p − x x f ( e ) f ( e ) ! nπ ω − , (74) Like in the Solar System, many exoplanets are not alonein their orbits, but belong to multi-planet systems. Becauseof mutual planetary perturbations the orbital parameters ofthe planets do not remain constant and undergo secular vari-ations in time (Chapter 10:
Non-Keplerian Dynamics ). Animportant consequence for the spin of the planets is thatthe reference orbital plane (to which the obliquity and theprecession were defined) will also present variations. Wecan track the orbital plane variations by the inclination toan inertial reference plane, I , and by the longitude of theline of nodes, Ω . Under the assumption of principal axis ro-tation, the energy perturbation attached to an inertial framecan be written (Kinoshita 1977; N´eron de Surgy and Laskar1997): U pp = [ X (1 − cos I ) − L sin ε sin I cos ϕ ] d Ω dt + L sin ε sin ϕ dIdt , (75)where ϕ = − Ω − ψ .Although the Solar System motion is chaotic (Laskar1989, 1990), the motion can be approximated over severalmillion of years by quasi-periodic series. In particular forthe orbital elements that are involved in the precession driv-ing terms (Eq. 75), we have (Laskar and Robutel 1993): (cid:18) dIdt + i d Ω dt sin I (cid:19) e iΩ = X k J k e i( ν k t + φ k ) , (76)10nd (1 − cos I ) d Ω dt = X k L k cos( ν k t + ϕ k ) , (77)where ν k are secular frequencies of the orbital motion withamplitude J k and phase φ k , and i = √− . We may thenrewrite expression (75) as U pp = L X k h L k cos( ν k t + ϕ k ) x −J k p − x sin( ν k t + ψ + φ k ) i . (78)Assuming non-resonant motion, from equations (16) and(17) we get for the spin motion: dεdt = X k J k cos( ν k t + ψ + φ k ) , (79)and dψdt = α cos ε − X k L k cos( ν k t + ϕ k ) − cot ε X k J k sin( ν k t + ψ + φ k ) . (80)For planetary systems like the Solar System, the mu-tual inclinations remain small (Laskar 1990; Correia et al.2010), and it follows from expressions (76) and (77) thatthe amplitudes of J k and L k are bounded respectively by J k ∼ ν k I max , L k ∼ ν k I max / . (81)The term in L k in expression (80) for the precession varia-tions can then be neglected for small inclinations.From expression (79) it is clear that a resonance can oc-cur whenever the precession frequency is equal to the oppo-site of a secular frequency ν k (that is, ˙ ψ = − ν k ). Retainingonly the terms in k , the problem becomes integrable. Wecan search for the equilibrium positions by setting the obliq-uity variations equal to zero ( ˙ ε = 0 ). It follows then fromexpression (79) that ψ + ν k t + φ k = ± π/ , and replacingit in expression (80) with ˙ ψ = − ν k , we get α cos ε sin ε + ν k sin ε ≈ J k cos ε , (82)which gives the equilibrium positions for the spin ofthe planet, generally known as “Cassini states” (e.g.Henrard and Murigande 1987). Since J k /ν k ≪ (Eq. 81),the equilibrium positions for the obliquity are then: tan ε ≈ J k ν k ± α , cos ε ≈ − ν k α . (83)When | α/ν k | ≪ | α/ν | crit , the first expression gives states2 and 3, while the second expression has no real roots (states1 and 4 do not exist). When | α/ν k | ≫ | α/ν | crit , the first expression approximates Cassini states 1 and 3, while thesecond one gives states 2 and 4. States 1, 2 and 3 are sta-ble, while state 4 is unstable. Although gravitational tidesalways decrease the obliquity (Eq. 45), the ultimate stageof the obliquity evolution is to be captured into a Cassiniresonant state, similarly to the capture of the rotation in aspin-orbit resonance (Eq. 73).The complete system (Eqs. 79 and 80) is usually not in-tegrable as there are several terms in expression (76), butwe can look individually to the location of each resonance.When the resonances are far apart, the motion will behavelocally as in the integrable case, with just the addition ofsupplementary small oscillations. However, if several res-onances overlap, the motion is no longer regular and be-comes chaotic (Chirikov 1979; Laskar 1996). For instance,the present obliquity variations on Mars are chaotic and canvary from zero to nearly sixty degrees (Laskar and Robutel1993; Touma and Wisdom 1993; Laskar et al. 2004b).
3. APPLICATION TO THE PLANETS
The orbital parameters of exoplanets are reasonably welldetermined from radial velocity, transit or astrometry tech-niques, but exoplanets’ spins remain a mystery.The same applies to the primordial spins of the terrestrialplanets in the Solar System, since very little constraints canbe derived from the present planetary formation models. In-deed, a small number of large impacts at the end of the for-mation process will not average, and can change the spindirection. The angular velocities are also unpredictable, butthey are usually high, ω ≫ n (Dones and Tremaine 1993;Agnor et al. 1999; Kokubo and Ida 2007), although impactscan also form a slow rotating planet ( ω ∼ n ) if the size ofthe typical accreting bodies is much smaller than the proto-planet (e.g. Schlichting and Sari 2007). The critical angularvelocity for rotational instability is Kokubo and Ida (2007): ω cr ≈ . (cid:18) ρ − (cid:19) / hr − , (84)which sets a maximum initial rotation periods of about . h, for the inner planets of the Solar System.For the Jovian planets in the Solar System no impor-tant mechanism susceptible of altering the rotation rate isknown, but the orientation of the axis may also change bysecular resonance with the planets (e.g. Correia and Laskar2003b; Ward and Hamilton 2004; Bou´e and Laskar 2010).The fact that all Jovian planets rotate fast (Table 3) seems tobe in agreement with theoretical predictions (e.g. Takata and Stevenson1996).An empirical relation derived by MacDonald (1964)based on the present rotation rates of planets from Mars toNeptune (assumed almost unchanged) gives for the initialrotation rates ω ∝ m / R − . (85)Extrapolating for the remaining inner planets, we get initialrotation periods of about . h, . h, and . h for Mer-cury, Venus and the Earth, respectively, much faster than11oday’s values, which in agreement with the present forma-tion theories.The above considerations and expressions can also beextended to exoplanets. However, since many of the exo-planets are close to their host stars, it is believed that thespins have undergone significant tidal dissipation and even-tually reached some equilibrium positions, as it happens forMercury and Venus in the Solar System. Therefore, in thissection we will first review the rotation of the terrestrialplanets, and then look at the already known exoplanets. Inparticular, we will focus our attention in two classes of ex-oplanets, the “Hot-Jupiters” (fluid) and the “Super-Earths”(rocky with atmosphere), for which tidal effects may playan important role in orbital and spin evolution. The present spin of Mercury is very peculiar: the planetrotates three times around its axis in the same time asit completes two orbital revolutions (Pettengill and Dyce1965). Within a year of the discovery, the stability ofthis 3/2 spin-orbit resonance became understood as theresult of the solar torque on Mercury’s quadrupolar mo-ment of inertia combined with an eccentric orbit (Eq. 70)(Colombo and Shapiro 1966; Goldreich and Peale 1966).The way the planet evolved into the 3/2 configuration re-mained a mystery for long time, but can be explained as theresult of tidal evolution combined with the eccentricity vari-ations due to planetary perturbations (Correia and Laskar2004, 2009).Mercury has no atmosphere, and the spin evolution ofthe planet is therefore controlled by gravitational tidal in-teractions with the Sun. Tidal effects drive the final obliq-uity of Mercury close to zero (Eq. 45), and the averagedequation for the rotation motion near the p resonance canbe written putting expressions (70) and (39) together: dωdt = − β ′ sin 2 γ − K ′ (cid:20) ωn − f ( e ) f ( e ) (cid:21) , (86)where γ = θ − pM − φ , β ′ = βH ( p, e ) , and K ′ = Knf ( e ) /C . Note that near the p resonance a contributionfrom core-mantle friction may also be present, but we willneglect it here (for a full description see Correia and Laskar2009, 2010). The tidal equilibrium is achieved when dω/dt = 0 , that is, for a constant eccentricity e , when ω/n = f ( e ) /f ( e ) . In a circular orbit ( e = 0 ) thetidal equilibrium coincides with synchronization, whilethe equilibrium rotation rate ω/n = 3 / is achieved for e / = 0 . (Fig. 4).For the present value of Mercury’s eccentricity ( e ≈ . ), the capture probability in the 3/2 spin-orbit reso-nance is only about 7% (Eq. 74). However, as the eccentric-ity of Mercury suffers strong chaotic variations in time dueto planetary secular perturbations, the eccentricity can varyfrom nearly zero to more than 0.45, and thus reach valueshigher than the critical value e / = 0 . . Additional Fig. 7.— Simultaneous evolution of the rotation rate (a) and theeccentricity (b) of Mercury. In this example, there is no captureat the first encounter with the 3/2 resonance (at t ≈ − . Gyr). About 100 Myr later, as the mean eccentricity increases,additional crossing of the 3/2 resonance occurs, leading to capturewith damping of the libration (Correia and Laskar 2006). capture into resonance can then occur, at any time duringthe planet’s history (Correia and Laskar 2004).In order to check the past evolution of Mercury’s spin, itis not possible to use a single orbital solution, as due to thechaotic behavior the motion cannot be predicted preciselybeyond a few tens of millions of years. A statistical study ofthe past evolutions of Mercury’s orbit was then performed,with the integration of 1000 orbits over 4 Gyr in the past,starting with very close initial conditions, within the un-certainty of the present determinations (Correia and Laskar2004, 2009).For each of the 1000 orbital motion of Mercury, the ro-tational motion (Eq. 86) was integrated numerically withplanetary perturbations. As e ( t ) is not constant, ω ( t ) willtend towards a limit value ˜ ω ( t ) that is similar to an aver-aged value of ( f /f )( e ( t )) and capture into resonance cannow occur more often (Fig. 7). Globally, only 38.8% of thesolutions did not end into resonance, and the final captureprobability distribution was (Correia and Laskar 2004): P / = 2 . , P / = 55 . , P / = 3 . . With the consideration of the chaotic evolution of theeccentricity of Mercury, the present / resonant state be-comes the most probable outcome for the spin evolution.The largest unknown remains the dissipation factor k ∆ t in the expression of K (Eq. 43). A stronger dissipation in-creases the probability of capture into the / resonance, as ω/n would follow more closely f ( e ) /f ( e ) (Fig. 7), whilelower dissipation slightly decreases the capture probabil-ity. The inclusion of core-mantle friction also increases thechances of capture for all resonances (Correia and Laskar2009, 2010).12 ABLE OSSIBLE FINAL SPIN STATES OF V ENUS . state ε ω P (days) P s (days) F +0 ◦ n + ω s .
83 116 . F − ◦ n − ω s − . − . F + π ◦ − n − ω s − .
83 116 . F − π ◦ − n + ω s . − . There are two retrograde states ( F − and F − π ) and two prograde states ( F +0 and F + π ). In all cases the synodic period P s is the same(Correia and Laskar 2001). Venus an unique case in the Solar System: it presents aslow retrograde rotation, with an obliquity close to 180 de-grees and a 243-day period (Smith 1963; Goldstein 1964;Carpenter 1970). According to planetary formation theo-ries it is highly improbable that the present spin of Venus isprimordial, since we would expect a lower obliquity and afast rotating planet (Eq. 85).The present rotation of Venus is believed to representa steady state resulting from a balance between gravita-tional tides, which drives the planet toward synchronous ro-tation, and thermally driven atmospheric tides, which drivesthe rotation away (e.g. Gold and Soter 1969). The conju-gated effect of tides and core-mantle friction can tilt Venusdown during the planet past evolution, but requires high val-ues of the initial obliquity (e.g. Dobrovolskis 1980; Yoder1997). However, the crossing in the past of a large chaoticzone for the spin, resulting from secular planetary pertur-bations (Laskar and Robutel 1993), can lead Venus to thepresent retrograde configuration for most initial conditions(Correia and Laskar 2001, 2003b).Venus has a dense atmosphere and the planet is alsoenough close to the Sun to undergo significant tidal dissi-pation. Venus’ spin evolution is then controlled by tidaleffects (both gravitational and thermal). Tidal effects com-bined can drive the obliquity either to ε = 0 ◦ or ε = 180 ◦ (Correia et al. 2003). For the two final obliquity possibili-ties, the tidal components become very simplified, with (atsecond order in the planetary eccentricity) a single term oftidal frequency σ = 2 ω − n for ε = 0 and σ = 2 ω + 2 n for ε = π (Eq. 25). Combining expressions (23) and (60)we can write for the rotation rate: dωdt (cid:12)(cid:12)(cid:12)(cid:12) = −
32 [ K g b g (2 ω − n ) + K a b a (2 ω − n )] ,dωdt (cid:12)(cid:12)(cid:12)(cid:12) π = −
32 [ K g b g (2 ω + 2 n ) + K a b a (2 ω + 2 n )] , (87)where K g and K a are given by the constant part of expres- sions (23) and (60), respectively. Let f ( σ ) be defined as f ( σ ) = b a (2 σ ) b g (2 σ ) . (88)As b τ ( σ ) is an odd function of σ (Eq. 25), f ( σ ) is an evenfunction of σ of the form f ( | σ | ) . Thus, at equilibrium, with dω/dt = 0 , we obtain an equilibrium condition f ( | ω − xn | ) = − K g K a , (89)where x = +1 for ε = 0 and x = − for ε = π . Moreover,for all commonly used dissipation models f is monotonicand decreasing for slow rotation rates. There are thus onlyfour possible values for the final rotation rate ω f of Venus,given by | ω f − xn | = f − (cid:18) − K g K a (cid:19) = ω s . (90)Assuming that the present rotation of Venus correspondsto a stable retrograde rotation, since ω s > the only possi-bilities for the present rotation are ε = 0 and ω obs = n − ω s ,or ε = π and ω obs = ω s − n . In both cases, ω s = n + | ω obs | ( ω s is thus the synodic frequency). With ω obs = 2 π/ . n = 2 π/ .
701 day , (91)we have ω s = 2 π/ .
751 day . (92)We can then determine all four final states for Venus (Ta-ble 2). There are two retrograde states ( F − and F − π ) andtwo prograde states ( F +0 and F + π ). The two retrogradestates correspond to the observed present retrograde state ofVenus with period . days, while the two other stateshave a prograde rotation period of . days. Looking tothe present rotation state of the planet, it is impossible todistinguish between the two states with the same angularmomentum (Fig. 8).In order to obtain a global view of the possible finalevolutions of Venus’ spin, numerical integrations of theequations of motion for the dissipative effects (Eqs. 23, 24,13ig. 8.— Final states for a planet with strong atmospheric ther-mal tides. The original equilibrium point obtained at synchro-nization ( ω/n = 1 ) when considering uniquely the gravitationaltides, becomes unstable, and bifurcates at ε = 0 into two newstable fixed points F − and F +0 , and at ε = π into F − π and F + π (Correia and Laskar 2001, 2003b).
60 and 61), with the addition of planetary perturbations(Eqs. 79 and 80) were performed (Correia and Laskar 2001,2003b). In Figure 9 we show the possible final evolutionsfor a planet starting with an initial period ranging from 3 to12 days, with an increment of 0.25 day, and initial obliquityfrom ◦ to ◦ , with an increment of 2.5 degrees (rotationperiods faster than 3 days are excluded as they do not allowthe planet to reach a final rotation state within the age of theSolar System). Each color represents one of the possible fi-nal states. For high initial obliquities, the spin of Venus al-ways evolves into the retrograde final state F − π . It is essen-tially the same evolution as without planetary perturbations,since none of the trajectories encounters a chaotic zone forthe obliquity (Laskar and Robutel 1993). However, for evo-lutionary paths starting with low initial obliquities, we candistinguish two different zones: one zone corresponding toslow initial rotation periods ( P i > day) where the pro-grade rotation final state F − is prevailing, and another zonefor faster initial rotation periods ( P i < day), where wefind a mixture of the three attainable final states, F +0 , F − and F − π . To emphasize the chaotic behavior, we integrated F +0 F −(cid:13) F −(cid:13) π(cid:13) (a)(b)(c) (cid:13) Fig. 9.—
Final states of Venus’ spin with planetary perturbationsfor initial obliquity ( ε i ∈ [0 ◦ , ◦ ] ) and period ( P i ∈ [3 d ,
12 d] ).For high initial obliquities, the final evolution of Venus remainsessentially unchanged since none of the trajectories crossed thechaotic zone. The passage through the chaotic zone is reflectedby the scattering of the final states in the left side of the picture.To emphasize the chaotic behavior, in the bottom left corner ofpicture (a), additional integrations were done with the same initialconditions, but with a difference of − in the initial eccentricityof Mars (b) and Neptune (c) (Correia and Laskar 2003b). twice more the zone with P i < day, with a difference of − in the initial eccentricity of Mars (Fig. 9b), and witha difference of − in the initial eccentricity of Neptune(Fig. 9c). The passage through the chaotic zone is reflectedby the scattering of the final states in the left hand side ofthe picture. Contrary to Mercury and Venus, the Earth and Marsare not tidally evolved. For Mars, the tidal dissipationfrom the Sun is negligible. For the Earth, tidal dissi-pation is noticeable due to the presence of the Moon,but the Earth’s spin is still far from the equilibrium (e.g.N´eron de Surgy and Laskar 1997). Nevertheless, the spinaxis of both planets is subjected to planetary perturbationsand thus present some significant variations (Sect. 2.5).In the case of Mars, the presence of numerous secularresonances of the kind ˙ ψ = − ν k (Eq. 79) induce largechaotic variations in the obliquity, which can evolve be-tween 0 and degrees (Fig. 10). At present, the obliquityof Mars is very similar to the obliquity of the Earth, whichis a mere coincidence. Indeed, the obliquity of Mars hasmost certainly reached values larger than 45 degrees in thepast (Laskar et al. 2004a). The high obliquity periods led to14ig. 10.— Frequency analysis of Mars’ obliquity. (a) The fre-quency map is obtained by reporting in the ordinate the precessionfrequency value obtained for 1000 integrations over 56 Myr for thedifferent values of the initial obliquity (abscissa). A large chaoticzone is visible, ranging from ◦ to about ◦ , with two distinctchaotic zones, B and B . (b) Maximum and minimum valuesof the obliquity reached over 56 Myr. In (c), the power spectrumof the orbital forcing term (Eq. 76) is given in logarithmic scale,showing the correspondence of the chaotic zone with the main sec-ular frequencies s , s , s , s . (Laskar et al. 2004a). large climatic changes on Mars, with possible occurrence oflarge scale ice cycles where the polar caps are sublimatedduring high obliquity stages and the ice is deposited in theequatorial regions (Laskar et al. 2002; Levrard et al. 2004,2007b).In the case of the Earth, the precession frequency isnot in resonance with any orbital secular frequency ( ˙ ψ = ν k ). The obliquity of the Earth is then only subject tosmall oscillations of about 1.3 degrees around the meanvalue (23.3 degrees) with main periodicities around 40 000years (Laskar et al. 2004b). The small obliquity variationsare nevertheless sufficiently important to induce substantialchanges in the insolation received in summer in high lati-tude regions on the Earth, and they are imprinted in the ge-ological stratigraphic sequences (Hays et al. 1976; Imbrie1982).Due to tidal dissipation in the Earth-Moon system, theMoon is moving away from Earth at a 3.8 cm/yr rate(Dickey et al. 1994), and the rotation rate of the Earth isslowing down (Eqs. 46 and 39, respectively). As a con-sequence, the torque exerted on the equatorial bulge ofthe Earth decreases and thus also the Earth precession fre-quency (Eqs. 7 and 12). Using the present dissipation pa-rameters of the Earth, N´eron de Surgy and Laskar (1997)found that after 1.5 Gyr, the spin of the Earth will enter
10 202010304050 30 40 50 60 70 80 90 100 p r e c e ss i on c on s t an t ( " / y r) t i m e ( G y r) OBLIQUITY (degrees)
Fig. 11.—
Example of possible evolution of the Earth’s obliquityfor 5 Gyr in the future, due to tidal dissipation in the Earth-Moonsystem. The background of the figure is obtained as the result ofa stability analysis on about 250 000 numerical integrations of theobliquity of the Earth under planetary perturbations for 36 Myr,for various values of the initial obliquity of the Earth ( x -axis) andvarious values of the precession constant (left y -axis). We ob-serve a very large chaotic zone (with stripes on the figure) result-ing from overlap of orbital secular resonances (Laskar et al. 1993;Laskar and Robutel 1993). The numerical integration is then con-ducted over 5 Gyr years in the future for the obliquity of the Earth,including tidal dissipative effects. The two bold curves correspondto the minimum and maximum values reached by the obliquity andthe time scale is given in the right y -axis. As long as the orbitsstays in the regular region, the motion suffers only small (and reg-ular) variations. As soon as the orbit enters the chaotic zone, verystrong variations of the obliquity are observed, which wanders inall the chaotic zone, and very high values, close to degrees, arereached (N´eron de Surgy and Laskar 1997). a large chaotic zone of overlapping orbital secular reso-nances. From then, the Earth spin axis will evolve in awildly chaotic way, with a possible range from 0 to nearly90 degrees (Fig. 11).The main difference between the Earth and Mars is thusdue to the presence of the Moon, whose gravitational torqueon the equatorial bulge of the Earth prevents the spin axisto evolve in a largely chaotic state. In absence of the Moon,the behavior of the spin axis of the Earth and Mars wouldbe identical (Laskar et al. 1993; Laskar and Robutel 1993).Depending on the orbital configuration of the exoplane-tary systems, we thus expect to find planets that would beeither in a chaotic state as Mars or the moonless Earth, orin a regular state as the Earth with the Moon. It shouldbe stressed, however, the presence of a large satellite is notmandatory in order to stabilize the spin axis. Since the sta-bility of the axis is very important for the exoplanet climate,planetary perturbations should be taken into considerationwhen searching for another Earth-like environments.15 ABLE ONSTANTS FOR THE S OLAR S YSTEM OUTER PLANETS . quantity Jupiter Saturn Uranus Neptune P (h) 9.92 10.66 17.24 16.11 ρ (g/cm ) 1.33 0.69 1.32 1.64 C/MR k Q ( × ) ∼ ∼ ∼ ∼ One of the most surprising findings concerning exoplan-ets was the discovery of several giant planets with periodsdown to 3 days, that were designated by “Hot-Jupiters” (e.g.Santos et al. 2005). Many of the “Hot-Jupiters” were si-multaneously detected by transit method and radial Dopplershift, which allows the direct and accurate determination ofboth mass and radius of the exoplanet. Therefore, “Hot-Jupiters” are amongst the better characterized planets out-side the Solar System.Due to the proximity of the host star, “Hot-Jupiters” arealmost certainly tidally evolved. Given the large mass of“Hot-Jupiters”, they may essentially be composed of an ex-tensive Hydrogen atmosphere, similar to Jupiter or Saturn.As a consequence, despite the presence of an inner metalliccore, “Hot-Jupiters” can be treated as fluid planets, and wemay adopt a viscous model for the tidal dissipation (Eqs. 39,40). Thermal atmospheric tides may also be present (e.g.Arras and Socrates 2010, 2009; Goodman 2009), but we didnot take thermal tides into account, as gravitational tides areso strong for “Hot-Jupiters”, that they probably rule over allthe remaining effects (see Sect. 3.3).
The effect of gravitational tides over the obliquity is tostraighten the spin axis (Eq. 45), so we will adopt ε = 0 ◦ forthe obliquity. Assuming that the eccentricity and the semi-major axis of the planet are constant, we can derive fromexpression (39): ωn = f ( e ) f ( e ) + (cid:18) ω n − f ( e ) f ( e ) (cid:19) exp( − t/τ eq ) , (93)where τ − eq = Kf ( e ) /C is the characteristic time-scale forfully despinning the planet.As for Mercury, the final equilibrium rotation drivenby tides ( t → + ∞ ) is given by the equilibrium position ω e /n = f ( e ) /f ( e ) , which is different from synchronousrotation if the eccentricity is not zero (Fig. 4). Unlike Mer-cury, because “Hot-Jupiters” are assumed to be fluid, theyshould not present much irregularities in the internal struc-tures. Therefore ( B − A ) /C ≈ and we do not expect “Hot-Jupiters” to be captured in spin-orbit resonances. In-deed, determination of second-degree harmonics of Jupiterand Saturn’s gravity field from Pioneer and Voyager track-ing data (Campbell and Anderson 1989) provided a crudeestimate of the ( B − A ) /C value lower than ∼ − forSaturn and ∼ − for Jupiter. The ( B − A ) /C values forJupiter and Saturn are more than one order of magnitudesmaller than the Moon’s or Mercury’s value, leading to in-significant chances of capture. Furthermore, the detectionof an equatorial asymmetry is questionable. If an equatorialbulge originates from local mass inhomogeneities driven byconvection, it is probably not permanent and must have amore negligible effect if averaged spatially and temporally.The time required for dampening the rotation of theplanet depends on the dissipation factor k ∆ t (Eq. 43). As-suming that “Hot-Jupiters” are similar to the Solar Systemgiant planets, we can adopt k = 0 . , and a range for Q from to (Table 3). The Q -factor and the time lag ∆ t can be relied using expressions (22) and (34): Q − ≈ σ ∆ t . (94)Since we are using a viscous model, for which ∆ t is madeconstant, Q will be modified across the evolution as Q isinversely proportional to the tidal frequency σ . The Q -factor for the Solar System gaseous planets is measured fortheir present rotation states, which correspond to less thanone day (Table 3). We may then assume that exoplanetsshould present identical Q values when they were rotatingas rapidly as Jupiter, that is, Q − = ω ∆ t . For Q = 10 and ω = 2 π/ h, we compute a constant ∆ t ≈ . s.We have plotted in Figure 12, all known exoplan-ets, taken from The Extrasolar Planets Encyclopedia( http://exoplanet.eu/ ), that could have been tidally evolved.We consider that exoplanets are fully evolved if their rota-tion rate, starting with an initial period of 10 h, is dampenedto a value such that | ω/n − f ( e ) /f ( e ) | < . . The curvesrepresent the planets that are tidally evolved in a given timeinterval ranging from 0.001 Gyr to 10 Gyr . Figure 12 al-lows us to check whether the planet should be fully evolved.For a Solar type star, we can expect that all exoplanets thatare above the 1 Gyr curve have already reached the equi-librium rotation ω e . On the other hand, exoplanets that are16ig. 12.— Tidally evolved exospins with Q = 10 and initialrotation period P = 10 h. The labeled curves denote (in Gyr)the time needed by the rotation to reach the equilibrium (time-scales are linearly proportional to Q ). We assumed Jupiter’sgeophysical parameters for all planets (Table 3) (updated fromLaskar and Correia 2004). below the 10 Gyr curve are probably not yet fully tidallyevolved. As expected, all planets in circular orbits with a < . AU are tidally evolved. However, we are moreinterested on exoplanets further from the star with non zeroeccentricity which are tidally evolved, since the rotation pe-riod is not synchronous, but given by expression (44). Forinstance, for the planet around HD 80606, the orbital pe-riod is 111.7 days, but since e = 0 . we predict a rotationperiod of about 1.9 days. Until now we have assumed that the final obliquity of theplanet is zero degrees. However, Winn and Holman (2005)suggested that high obliquity values could be maintainedif the planet has been trapped in a Cassini state resonance(Eq. 82) since the early despinning process. For small am-plitude variations of the eccentricity and inclination, theequilibrium positions for the obliquity are given by expres-sion (83). Unless α = | ν k | , the state 1 is close to ◦ , andstate 3 is close to ◦ . We thus focus only on state 2 thatmay maintain a significant obliquity.To test the possibility of capture in the high obliqueCassini state 2, we can consider a simple scenario wherea “Hot-Jupiter” forms at a large orbital distance ( ∼ sev-eral AUs) and migrates inward to the current position( ∼ . AU). Before the planet reaches typically ∼ . AU,tidal effects do not affect the spin evolution, but the re-duction in the semi-major axis increases the precessionconstant (Eq. 12), so that the precession frequency ˙ ψ maybecome resonant with some orbital frequencies ν k (Eq. 80).The passage through resonance generally causes the obliq-uity to change (Ward 1975; Ward and Hamilton 2004;Hamilton and Ward 2004; Bou´e et al. 2009), rising the pos-sibility that the obliquity has a somewhat arbitrary value when the semi-major axis attains ∼ . AU.Tidal effects become efficient for a < . AU and drivethe obliquity to an equilibrium value cos ε ≈ n (1+6 e ) /ω (Eq. 40). For initial fast rotation rates ( ω ≫ n ), the equi-librium obliquity tends to ◦ . As the rotation rate is de-creased by tides, the equilibrium obliquity is reduced tozero degrees (Eq. 45). It is then possible that the obliquitycrosses several resonances (one for each frequency ν k ) inboth ways (increasing and decreasing obliquity), and thata capture occurs. Inside the resonance island, the restor-ing torque causes the obliquity to librate with amplitude(Correia and Laskar 2003b): cos ε ± ∆ cos ε ≈ − ν k α ± s J k α r − ν k α . (95)Using a linear approximation of the tidal torque (Eq. 40)around the resonant obliquity ε , the probability of cap-ture in the Cassini state 2 can be estimated from the an-alytical approach for spin-orbit resonances (Eq. 73), with(Levrard et al. 2007a) ∆ ωπV n = (cid:20) (1 − ε ) ωn + 2 cos ε π sin ε (2 − cos ε ω/n ) (cid:21) ∆ cos ε . (96)In Figure 13, we plotted the capture probabilities at 0.05,0.1 and 0.5 AU as a function of the rotation period for dif-ferent amplitudes ( J k ) and frequencies ( ν k ) characteristicof the Solar System (Laskar and Robutel 1993). As a rea-sonable example, we choose ν k = − ”/yr, but the resultsare not affected by changes on this value. Assuming an ini-tial rotation period of 12 h, the capture is possible and evenunavoidable if J k > < − π foreach initial obliquity. Statistics of capture were found to bein good agreement with previous theoretical estimates.To test the influence of migration on the capture stability,additional numerical simulations were performed for vari-ous initial obliquities and secular perturbations over typi-cally ∼ × yr. The migration process was simulated byan exponential decreasing of the semi-major axis towards0.05 AU with a − yr time scale. The obliquity libra-tions were found to be significantly shorter than spin-downand migration time scales so that the spin trajectory followsan “adiabatic invariant” in the phase space. Nevertheless,expression (40) indicates that the tidal torque dramaticallyincreases both with spin-down and inward migration pro-cesses ( dε/dt ∝ a − / ω − ). If the tidal torque exceedsthe maximum possible restoring torque (Eq. 79), the res-onant equilibrium is destroyed (the evolution is no longer17ig. 13.— Obliquity capture probabilities in resonance under the effect of gravitational tides, as a function of the rotation period at a)0.5 AU b) 0.1 AU c) 0.05 AU. J k is the amplitude of the secular orbital perturbations and ν k = − . ”/yr (Levrard et al. 2007a). adiabatic). For a given semi-major axis, the stability con-dition requires then that the rotation rate must always belarger than a threshold value ω crit. , which is always veri-fied if ω crit. < f ( e ) /f ( e ) . The stability condition can besimply written as (Levrard et al. 2007a) tan( ε ) < J k × τ eq , (97)where τ eq is the time scale of tidal despinning (Eq. 93). Itthen follows that the final obliquity of the planet cannot betoo large, otherwise the planet would quit the resonance.For instance, taking J k = 1 ”/yr at 0.05 AU (the highestvalue in Fig. 13), we need an obliquity ε < ◦ . For themore realistic amplitude J k = 0 . ”/yr, the resonant obliq-uity drops to ε < ◦ . Such a low resonant obliquity at0.05 AU is highly unlikely, because, according to expres-sion (83), the resonant state requires very high values ofthe orbital secular frequencies ( | ν k | > . × ”/yr). At0.5 and 0.1 AU, critical obliquity values are respectively83 ◦ and 41 ◦ and require more reasonable orbital secularfrequencies so that a stable capture is possible. In numer-ical simulation, the stability criteria for the final obliquity(Eq. 97) is empirically retrieved with an excellent agree-ment (Levrard et al. 2007a). When the obliquity leaves theresonance, the obliquity ultimately rapidly switches to theresonant stable Cassini state 1, which tends to ◦ (Eq. 83).We then conclude that locking a “Hot-Jupiter” in an obliqueCassini state seems to be a very unlikely scenario. Tidal energy is dissipated in the planet at the expenseof the rotational and orbital energy so that ˙ E = − Cω ˙ ω − ˙ a ( Gm ⋆ m ) / (2 a ) . Replacing the equilibrium rotation givenby equation (44) in expression (46) we obtain for the tidalenergy: ˙ E = Kn (cid:20) f ( e ) − f ( e ) f ( e ) 2 x x (cid:21) , (98) Eccentricity R a t e o f t i da l d i ss i pa t i on ( e r g / s ) Obliquity=90°Obliquity=90°, synchronous caseObliquity=45°Obliquiity=45°, synchronous case
Fig. 14.—
Rate of tidal dissipation within HD 209458 b as afunction of the eccentricity for ◦ (solid thin line) and ◦ (solidthick line) obliquity. The synchronous case is plotted with dashedlines for comparison. The dissipation factor Q /k is set to (Levrard et al. 2007a). or, at second order in eccentricity, ˙ E = Kn ε (cid:2) sin ε + e (cid:0) ε (cid:1)(cid:3) , (99)which is always larger than in the synchronous case (e.g.Wisdom 2004). In Figure 14 the rate of tidal heating withina non-synchronous and synchronous planet as a function ofthe eccentricity ( < e < . ) is compared for two differ-ent obliquities (Levrard et al. 2007a). The ratio between thetidal heating in the two situations is an increasing functionof both eccentricity and obliquity. For e ≈ , as observedfor “Hot-Jupiters”, the ratio may reach ∼ ◦ and ◦ obliquity, not being significantlymodified at larger eccentricity.We then conclude that planets in eccentric orbits and/orwith high obliquity dissipate more energy than planets insynchronous circular orbits. This may explain why some18lanets appear to be more inflated than initially expected(e.g. Knutson et al. 2007). A correct tidal energy balancemust then take into account the present spin and eccentricityof the orbit. In Sect. 2.2.4 we saw that under tidal friction the spin ofthe planet attains an equilibrium position faster than the or-bit. As a consequence, we can use the expression of theequilibrium rotation rate (Eq. 44) in the semi-major axisand eccentricity variations and find simplified expressions(Eqs. 51 and 52). Combining the two equations we get daa = 2 e de (1 − e ) , (100)whose solution is given by a = a f (1 − e ) − . (101)Replacing the above relation in expression (52) we find adifferential equation that rules the eccentricity evolution: ˙ e = − K f ( e )(1 − e ) e , (102)where K is a constant parameter: K = ∆ t k Gm ⋆ R µa f . (103)The solution of the above equation is given by F ( e ) = F ( e )e − K t , (104)where F ( e ) is an implicit function of e , which converges tozero as t → + ∞ . For small eccentricities, we can neglectterms in e and F ( e ) = e | − e | − / . The characteristictime-scale for fully dampening the eccentricity of the orbitis then τ orb ∼ /K , and the ratio between the spin andorbital time-scales τ eq τ orb ∼ (cid:18) Ra f (cid:19) . (105)Since a f = a (1 − e ) , for initial very eccentric orbits thetwo time-scales become comparable.We have plotted in Figure 15, all known exoplan-ets, taken from The Extrasolar Planets Encyclopedia( http://exoplanet.eu/ ), whose orbits could have beentidally evolved. We consider that they are fully evolvedif the eccentricity is dampened to a value e < . . Thecurves represent the time needed to damp the eccentric-ity starting with the present orbital parameters, for timeintervals ranging from 0.001 Gyr to 100 Gyr. Figure 15allows us to simultaneously check whether the planet istidally evolved, and the time needed to fully damp thepresent eccentricity. All planets in eccentric orbits expe-rience stronger tidal effects because the planet is close tothe star at the periapse. As a consequence, tidal friction Fig. 15.— Tidally evolved orbits of exoplanets with Q = 10 .The labeled curves denote (in Gyr) the time needed to circularizethe orbits of the planets ( e < . ) (time-scales are linearly pro-portional to Q ). We assumed Jupiter’s geophysical parametersfor all planets (Table 3). can, under certain conditions, be an important mechanismfor the formation of “Hot-Jupiters” (see Sect. 3.2.5).According to Figure 15, a significant fraction of exoplan-ets that are close to the host star ( a < . AU) still presenteccentricities up to 0.4, although tidal effects should havealready damped the eccentricity to zero. Observational er-rors and/or weaker tidal dissipation ( Q ≫ ) can bea possible explanation, but they will hardly justify all theobserved situations. A more plausible explanation is thatthe eccentricity of the exoplanet is being excited by grav-itational perturbations from an outer planetary companion(Sect. 2.5). Indeed, the eccentricity of a inner short-periodplanet can be excited as long as its (non-resonant) outercompanion’s eccentricity is non-zero. Mardling (2007) hasshown that the eccentricity of the outer planet will decayon a time-scale which depends on the structure of the in-ner planet, and that the eccentricities of both planets aredamped at the same rate, controlled by the outer planet(Fig. 16). The mechanism is so efficient that the outer planetmay be an Earth-mass planet in the “ habitable zones ” ofsome stars. As a consequence, the evolution time-scale forboth eccentricities can attain the Gyr instead of Myr, whichcould explain the current observations of non-zero eccen-tricity for some “Hot-Jupiters”. In current theories of planetary formation, the regionwithin 0.1 AU of a protostar is too hot and rarefied for aJupiter-mass planet to form, so “Hot-Jupiters” likely formfurther away and then migrate inward. A significant fractionof “Hot-Jupiters” has been found in systems of binary stars(e.g. Eggenberger et al. 2004), suggesting that the stellarcompanion may play an important role in the shrinkage ofthe planetary orbits. In addition, close binary star systems(separation comparable to the stellar radius) are also often19 p = 100 p = 100 Q Fig. 16.—
Tidal evolution of the eccentricities of planet HD 209458 b ( e p ) perturbed by a . M Jup companion at 0.4 AU ( e c ). Thedissipation factor Q p = 100 of the observed planet is set artificially low in order to illustrate the damping process (time-scales arelinearly proportional to Q p ). Top figures show the first stages of the evolution. The change in grayscale shows the transition of theeccentricity from the circulation phase to the libration phase. The diamond represents the moment where the eccentricity librations aredamped. Bottom figures show the final evolution of the eccentricities. The presence of a companion result that e p decays at the samerate of e c , while the dissipation rate for e c is controlled by Q p and not by Q c (Mardling 2007). accompanied by a third star. For instance, Tokovinin et al.(2006) found that 96% of a sample of spectroscopic bina-ries with periods less than 3 days has a tertiary component.Indeed, in some circumstances the distant companion en-hances tidal interactions in the inner binary, causing the bi-nary orbital period to shrink to the currently observed val-ues.Three-body systems can be stable for long-time scalesprovided that the system is hierarchical, that is, if the systemis formed by an inner binary (star and planet) in a nearlyKeplerian orbit with a semi-major axis a , and a outer staralso in a nearly Keplerian orbit about the center of mass ofthe inner system with semi-major a ′ ≫ a . An additionalrequirement is that the eccentricity e ′ of the outer orbit isnot too large, in order to prevent close encounters with theinner system. In this situation, perturbations on the innerplanetary orbit are weak, but can have important long-term effects (Chapter 10: Non-Keplerian Dynamics ).The most striking effect is known as the Lidov-Kozaimechanism (Kozai 1962; Lidov 1962), which allows the in-ner orbit to periodically exchange eccentricity with incli-nation. Even at large distances ( a ′ > I > . ◦ (that is, for cos I < (3 / / ). When I < . ◦ there is little variation in the planet’s inclinationand eccentricity. Secular effects of the Lidov-Kozai typecan then produce large cyclic variations in the planet’s ec-centricity e as a result of angular momentum exchange withthe companion orbit. Since the z -component of the planet’sangular momentum must be conserved and a is not modi-fied by the secular perturbations, the Kozai integral L K = (1 − e ) / cos I (106)20ig. 17.— Possible evolution of the planet HD 80606 b initially in an orbit with a = 5 AU, e = 0 . , and I = 85 . ◦ . The stellarcompanion is supposed to be a Sun-like star at a ′ = 1000 AU, and e ′ = 0 . . The diamonds mark the current position of HD 80606 balong this possible evolution (Wu and Murray 2003; Fabrycky and Tremaine 2007). is conserved during the oscillations (Lidov and Ziglin1976). Maxima in e occur with minima in I , and viceversa. If the inner orbit is initially circular, the maximumeccentricity achieved in a Kozai cycle is e max = (1 − (5 /
3) cos I ) / and the oscillation period of a cycle is ap-proximately P ′ /P (Kiseleva et al. 1998). The maximumeccentricity of the inner orbit in the Kozai cycle will remainfixed for different masses and distances of the outer star, butthe period of the Kozai cycle will grow with a ′ . Kozai cy-cles persist as long as the perturbation from the outer star isthe dominant cause of periapse precession in the planetaryorbit. However, small additional sources of periapse pre-cession such as the quadrupole moments, additional com-panions, general relativity or even tides can compensate theKozai precession and suppress the eccentricity/inclinationoscillations (e.g. Migaszewski and Go´zdziewski 2009).Because the Lidov-Kozai mechanism is able to inducelarge eccentricity excitations, a planet in an initial almostcircular orbit (for instance a Jupiter-like planet at 5 AUaround a Sun-like star) can experiment close approaches tothe host star at the periapse when the eccentricity increases to very high values. As a consequence, tidal effects increaseseveral orders of magnitude and according to expression(51) the semi-major axis of the orbit will decrease and theplanet migrate inward. At some point of the evolution, theperiapse precession will be dominated by other effects andthe eccentricity oscillations suppressed. From that momenton, the eccentricity is damped according to expression (52)and the final semi-major axis given by a f = a (1 − e ) .Ford and Rasio (2006) have derived that tidal evolution ofhigh eccentric orbits would end at a semi major axis a f equal to about twice the Roche limit R L . Indeed, at theclosest periapse distance, attained for e ≈ , we will have a (1 − e ) = R L , and thus a f = (1 + e ) R L ≈ R L .In Figure 17 we plot an example of combined Kozai-tidalmigration of the planet HD 80606 b. The planet is initiallyset in an orbit with a = 5 AU, e = 0 . , and I = 85 . ◦ .The stellar companion is supposed to be a Sun-like starat a ′ = 1000 AU, and e ′ = 0 . (Wu and Murray 2003;Fabrycky and Tremaine 2007). Prominent eccentricity os-cillations are seen from the very beginning and the energyin the planet’s spin is transferred to the orbit increasing the21emi-major axis for the first 0.1 Gyr (Eq. 39). As the equi-librium rotation is achieved (Eq. 44) the orbital evolutionis essentially controlled by equations (51) and (52), whosecontributions are enhanced when the eccentricity reacheshigh values. The semi-major axis evolution is executed byapparent “discontinuous” transitions precisely because thetidal dissipation is only efficient during periods of high ec-centricity. As dissipation shrinks the semi-major axis, peri-apse precession becomes gradually dominated by relativityrather than by the third body, and the periapse starts cir-culating as the eccentricity passes close to 0 at 0.7 Gyr.Tidal evolution stops when the orbit is completely circular-ized. The present semi-major axis and eccentricity of planetHD 80606 b are a = 0 . AU and e = 0 . , respectively,meaning that the tidal evolution on HD 80606 b is still un-der way (Fig. 15). The final semi-major axis is estimated toabout a f = 0 . AU, which corresponds to a regular “Hot-Jupiter”.
After a significant number of discoveries of gaseousgiant exoplanets, a new barrier has been passed with thedetections of several exoplanets in the Neptune and evenEarth-mass ( M ⊕ ) regime: − M ⊕ (Rivera et al. 2005;Lovis et al. 2006; Udry et al. 2007; Bonfils et al. 2007),that are commonly designated by “Super-Earths”. If thecommonly accepted core-accretion model can account forthe formation of “Super-Earths”, resulting in a mainlyicy/rocky composition, the fraction of the residual He-H atmospheric envelope accreted during the planet migrationis not tightly constrained for planets more massive than theEarth (e.g. Alibert et al. 2006). A minimum mass of below10 M ⊕ is usually considered to be the boundary betweenterrestrial and giant planets, but Rafikov (2006) found thatplanets more massive than 6 M ⊕ could have retained morethan 1 M ⊕ of the He-H gaseous envelope. For compari-son, masses of Earth’s and Venus’ atmosphere are respec-tively ∼ − and − times the planet’s mass. Despitesignificant uncertainties, the discoveries of “Super-Earths”provide an opportunity to test some properties that could besimilar to those of the more familiar terrestrial planets ofthe Solar System.Because some of the “Super-Earths” are potentially inthe “ habitable zone ” (Udry et al. 2007; Selsis et al. 2007),the present spin state is an important factor to constrain theclimates. As for Venus, thermal atmospheric tides may havea profound influence on the spin of “Super-Earths”. How-ever, the small eccentricity approximation done for Venus(Eq. 87) may no longer be adequate for “Super-Earths”,which exhibit a wide range of eccentricities, orbital dis-tances, or central star types. Although our knowledge of“Super-Earths” is restricted to their orbital parameters andminimum masses, we can attempt to place new constraintson the surface rotation rate, assuming that “Super-Earths”have a dense atmosphere.As for Venus, the combined effect of tides is to set the Fig. 18.— Evolution of ˙ ω (Eq. 107) with ω/n for different at-mospheric strengths ( ω s /n = 1 . , . , . ) and eccentricities( e = 0 . , . , . ). The top picture with e = 0 is the same as Fig-ure 8 for Venus. The equilibrium rotation rates are given by ˙ ω = 0 and the arrows indicate whether the equilibrium position is stableor unstable. For ω s /n > , we have two equilibrium possibili-ties, ω ± , one of which corresponds to a retrograde rotation (as forVenus). For ω s /n < , retrograde states are not possible, but wecan still observe final rotation rates ω − < n . For eccentric orbits,because of the harmonics in σ = 2 ω − n and σ = 2 ω − n , wemay have at most four different final possibilities (Eq. 110). When ω s /n becomes extremely small, which is the case for the presentobserved exoplanets with some eccentricity (Table 4), a single fi-nal equilibrium is possible for ω +1 (Correia et al. 2008). final obliquity at ◦ or ◦ (Correia et al. 2003). Adoptinga viscous dissipation model for tidal effects (Eq. 37) and the“ heating at the ground ” model (Dobrovolskis and Ingersoll1980) for surface pressure variations (Eq. 68), the averageevolution of the rotation rate is then obtained by adding theeffects of both tidal torques acting on the planet. From ex-pressions (23) and (60) we get for ε = 0 ◦ and to the secondorder in the eccentricity: ˙ ωτ − eq = ω − (cid:0) e (cid:1) n − ω s (cid:2)(cid:0) − e (cid:1) sign( ω − n ) − e sign(2 ω − n ) + 9 e sign(2 ω − n ) (cid:3) , (107)where ω s = F s H k K a ∆ t a K g ∆ t g ∝ L ⋆ m ⋆ Rm a . (108)When e = 0 , we saw in the case of Venus that finalpositions of the rotation rate at zero obliquity are given by22ig. 19.— Equilibrium positions of the rotation rate for “Super-Earths” as a function of the product am ⋆ for three different values ofthe eccentricity ( e = 0 . , . , . ). Each curve corresponds to a different final state (dotted lines for ω ± and solid lines for ω ± ). For e ≈ (case of Venus), we always count two final states that are symmetrical about n . For small values of am ⋆ , the two equilibriumpossibilities are so close to n that the most likely scenario for the planet is to be captured in the synchronous resonance (case of GJ 581 e).As we increase the eccentricity, we can count at most three final equilibrium rotations, depending on the value of ω s /n (computed fromEq. 114). When e ≈ . , only one equilibrium state exists for am ⋆ < . , resulting from ω s /n < e (1 + e / . This is the presentsituation of µ Arae c and most of the “Super-Earths” listed in Table 4 (Correia et al. 2008). (Eq. 90): | ω − n | = ω s , (109)i.e., there are two final possibilities for the equilibrium ro-tation of the planet, given by ω ± = n ± ω s . When e = 0 ,the above expression (109) is no longer valid and additionalequilibrium positions for the rotation rate may occur. Formoderate values of the eccentricity, from expression (107)we have that the effect of the eccentricity is eventually tosplit each previous equilibrium rotation rate into two newequilibrium values. Thus, four final equilibrium positionsfor the rotation rate are possible (eight if we consider thecase ε = 180 ◦ ), obtained with ˙ ω = 0 (Fig. 18): ω ± , = n ± ω s + e δ ± , , (110)with δ − = 6 n + 12 ω s , δ +1 = 6 n − ω s , (111)and δ − = 6 n + 52 ω s , δ +2 = 6 n − ω s . (112)Because the set of ω ± , values must verify the additionalcondition ω − < n/ < ω − < n < ω +1 < n/ < ω +2 , (113)the four equilibrium rotation states cannot, in general, ex-ist simultaneously, depending on the values of ω s and e . Inparticular, the final states ω − and ω +1 can never coexist with ω − . At most three different equilibrium states are thereforepossible, obtained when ω s /n is close to / , or more pre-cisely, when / − e / < ω s /n < / e / .Conversely, we find that one single final state ω +1 = (1 +6 e ) n +(1 − e / ω s exists when ω s /n < e (1+ e / . The Earth and Venus are the only planets that can beincluded in the category of “Super-Earths” for which theatmosphere and spin are known. Only Venus is tidallyevolved and therefore suitable for applying the above ex-pressions for tidal equilibrium. We can nevertheless in-vestigate the final equilibrium rotation states of the alreadydetected “Super-Earths”. For that purpose, we consideredonly exoplanets with masses smaller than 12 M ⊕ that weclassified as rocky planets with a dense atmosphere, al-though we stress that this mass boundary is quite arbitrarily.Using the empirical mass-luminosity relation L ⋆ ∝ M ⋆ (e.g. Cester et al. 1983) and the mass-radius relationship forterrestrial planets R ∝ m . (Sotin et al. 2007), expres-sion (108) can be written as: ω s /n = κ ( a m ⋆ ) . m − . , (114)where κ is a proportionality coefficient that contains all theconstant parameters, but also the parameters that we are un-able to constrain such as H , k , ∆ t g or ∆ t a . In this con-text, as a first order approximation we consider that for all“Super-Earths” the parameter κ has the same value as forVenus. Assuming that the rotation of Venus is presently sta-bilized in the ω − final state, that is, π/ω − = − days(Carpenter 1970), we compute π/ω s = 116 . days. Re-placing the present rotation in expression (114), we find forVenus that κ = 3 . M . ⊕ M − . ⊙ AU − . . We can thenestimate the ratio ω s /n for all considered “Super-Earths”in order to derive their respective equilibrium rotation rates(Table 4).The number and values of the allowed equilibrium ro-tation states are plotted as a function of aM ⋆ for differenteccentricities in Figure 19. All eccentric planets have a ratio ω s /n that is lower than × − (Table 4), which verifiesthe condition ω s /n < e (1 + e / . As a consequence,only one single final state exists ω +1 /n ≈ (1 + 6 e ) , cor-23 ABLE HARACTERISTICS AND EQUILIBRIUM ROTATIONS OF SOME “S UPER -E ARTHS ” WITH MASSES LOWER THAN M ⊕ Name m ⋆ Age τ eq m sin i a e ω s /n π/n π/ω − π/ω − π/ω +1 π/ω +2 [ M ⊙ ] [Gyr] [Gyr] [ M ⊕ ] [AU] [day] [day] [day] [day] [day]Venus 1.00 4.5 2.3 0.82 0.723 0.007 1.92 224.7 − ∗ − GJ 581 e 0.31 7-11 − − HD 40307 b 0.77 — − GJ 581 c 0.31 7-11 − − GJ 876 d 0.32 9.9 10 − − HD 40307 c 0.77 — − GJ 581 d 0.31 7-11 0.02 7.09 0.22 0.38 0.0011 67.6918 47.8226 HD 181433 b 0.78 — − GJ 176 b 0.5 — − HD 40307 d 0.77 — − HD 7924 b 0.83 — − HD 69830 b 0.86 4-10 − µ Arae c 1.1 6.41 −
55 Cnc e 1.03 5.5 − GJ 674 b 0.35 0.1-1 − − HD 69830 c 0.86 4-10 − ∗ ) Moon tidal effects were not included. τ eq was computed with k = 1 / and ∆ t g = 640 s (Earth’s values). References: [1]Mayor et al. (2009a); [2] Mayor et al. (2009b); [3] Correia et al. (2010); [4] Bouchy et al. (2009); [5] Forveille et al. (2009); [6]Howard et al. (2009); [7] Lovis et al. (2006); [8] Pepe et al. (2007); [9] Fischer et al. (2008); [10] Bonfils et al. (2007). responding to the equilibrium rotation resulting from gravi-tational tides (Eq. 44). The main reason is that the effect ofatmospheric tides is clearly disfavored relative to the effectof gravitational tides on “Super-Earths” discovered orbitingM-dwarf stars: the short orbital periods strengthens the ef-fect of gravitational tides, which are proportional to /a ,while the effect of thermal tides varies as /a . Moreover,the small mass of the central star also strongly affects theluminosity received by the planet and hence the size of theatmospheric bulge driven by thermal contrasts.For the planets with nearly zero eccentricity (GJ 581 e,HD 40307 b, c, d, and GJ 176 b), two equilibrium rotationstates ω ± are possible. However, the two final states ω ± are so close to the mean motion n , that the quadrupole mo-ment of inertia ( B − A ) /C will probably capture the rota-tion of the planet in the synchronous resonance. We thenconclude that “Super-Earths” orbiting close to their hoststars (in particular M-dwarfs), will be dominated by grav-itational tides and present a final equilibrium rotation rategiven by ω e /n ≈ f ( e ) /f ( e ) (Fig. 4), or present spin-orbitresonances like Mercury.
4. FUTURE PROSPECTS
The classical theory of tides initiated by Darwin (1880,1908) is sufficient to understand the main effects of tidalfriction upon planetary evolution. However, the exactmechanism on how tidal energy is dissipated within the in- ternal layers of the planet remains a challenge for planetaryscientists. Kaula (1964) derived a generalization of Dar-win’s work, with consideration of higher order tides andwithout the adoption of any dissipation model. The tidalpotential is described using infinite series in eccentricityand inclination, which is not practical and can only be cor-rectly handled by computers. Ever since many efforts havebeen done in order to either simplify the tidal equations,or to correctly model the tidal dissipation (for a review seeFerraz-Mello et al. 2008; Efroimsky and Williams 2009).Many Solar System phenomena have been successfullyexplained using the existent tidal models, so we expect thatthey are suitable to describe the tidal evolution of exoplan-ets. Nevertheless, many exoplanets are totally differentfrom the Solar System cases, and we cannot exclude to ob-serve some unexpected behaviors. For instance, it is likelythat dissipation within “Hot-Jupiters” is closer to dissipa-tion within stars (e.g. Zahn 1975), while dissipation within“Super-Earths” is closer to dissipation observed for rockyplanets (e.g. Henning et al. 2009). It is then necessary tocontinue improving tidal models in order to get a more re-alistic description for each planetary system. In particular,a correct description of the tidal dissipation and on how itevolves with the tidal frequency is critical for the evolutiontime-scale.The orbital architecture of exoplanetary systems is rela-tively well determined from the present observational tech-niques. However, the spins of the exoplanets are not easy to24easure, as the light curve coming from the planet is alwaysdimmed by the star light. The continuous improvementsthat have been made in photometry and spectrography letus believe that the determination of exoplanets’ spins canbe a true possibility in the near future. In particular, infra-red spectrographs are being developed, which will allow toacquire spectra of the planets if we manage to subtract thestellar contribution (e.g. Barnes et al. 2010).Some additional methods for detecting the rotationand/or the obliquity of exoplanets have also been testedand suggested so far. For instance, indirect sensing ofthe planetary gravitational quadrupole and shape, which islinked to both spin rate and obliquity (e.g. Seager and Hui2002; Ragozzine and Wolf 2009), or transient heating ofone face of the planet, which then spins into and out ofview, as it has been attempted for the system HD 80606(Laughlin et al. 2009). The effect of planetary rotation onthe transit spectrum of a giant exoplanet is another possibil-ity. During ingress and egress, absorption features arisingfrom the planet’s atmosphere are Doppler shifted by of or-der the planet’s rotational velocity ( ∼ − ) relativeto where they would be if the planet were not rotating (e.g.Spiegel et al. 2007). Finally, for planets whose light is spa-tially separated from the star, variations may be discerniblein the light curve obtained by low-precision photometry dueto meteorological variability, composition of the surface, orspots (e.g. Ford et al. 2001).Although the spin states of exoplanets cannot be mea-sured, for exoplanets that are tidally evolved we can stilltry to make predictions for the rotation rates. When the ec-centricity is large, the rotation of many of the observed exo-planets can still be tidally evolved even if the planets are notvery close to their central stars (Fig. 12). For tidally evolved“Hot-Jupiters”, we can conjecture that the rotation periodsare the limit values P orb × f ( e ) /f ( e ) (Fig. 4). It becomesa new challenge for the observers to be able to confirm thesepredictions.Thermal atmospheric tides may very well destabilize thetidal equilibrium from gravitational tides and create addi-tional possible stable limit values, with the possibility ofretrograde rotations, as for planet Venus (Fig.8). Thermaltides should be particularly important for “Super-Earths”,which are expected to have a distinct rocky body sur-rounded by a dense atmosphere. In a paradoxical way, thefinal rotation rate of “Super-Earths” are the most difficultto predict, as the equilibrium configurations depend on thecomposition of the atmospheres. Thermal tides are never-theless more relevant for exoplanets that orbit Sun-like starsat not very close distances, like Venus (Fig.19).We also assumed that the final obliquity of exoplanets iseither ◦ or ◦ , as the two values represent the final out-come of tidal evolution. However, each planetary systemhas its own architecture, and planetary perturbations on thespin can lead to resonant capture in a high oblique Cassinistates or even to chaotic motion. Thus, the final spin evolu-tion of a planet cannot be dissociated from its environment,and a more realistic description of exoplanets rotation can only be achieved with the full knowledge of the system or-bital dynamics. Acknowledgments.
We thank to an anonymous refereefor valuable suggestions, who helped to improve this work.We acknowledge support from the Fundac¸˜ao para a Ciˆenciae a Tecnologia (Portugal) and PNP-CNRS (France).
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