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Spin-orbit coupling for tidally evolving super-Earths

We investigate the spin behavior of close-in rocky planets and the implications for their orbital evolution. Considering that the planet rotation evolves under simultaneous actions of the torque due to the equatorial deformation and the tidal torque, both raised by the central star, we analyze the possibility of temporary captures in spin-orbit resonances. The results of the numerical simulations of the exact equations of motions indicate that, whenever the planet rotation is trapped in a resonant motion, the orbital decay and the eccentricity damping are faster than the ones in which the rotation follows the so-called pseudo-synchronization. Analytical results obtained through the averaged equations of the spin-orbit problem show a good agreement with the numerical simulations. We apply the analysis to the cases of the recently discovered hot super-Earths Kepler-10 b, GJ 3634 b and 55 Cnc e. The simulated dynamical history of these systems indicates the possibility of capture in several spin-orbit resonances; particularly, GJ 3634 b and 55 Cnc e can currently evolve under a non-synchronous resonant motion for suitable values of the parameters. Moreover, 55 Cnc e may avoid a chaotic rotation behavior by evolving towards synchronization through successive temporary resonant trappings.


INTRODUCTION
Tidal theories have gained great attention in the last decade due to the large number of discovered close-in planets which orbit their parent stars with periods of a few days. According to classical theories (e.g. Darwin 1880; Kaula 1964;Mignard 1979), the long-term tidal evolution of these planets results in orbital decay and circularization, on time-scales which depend on the physical properties of the interacting systems (Hut 1981;Dobbs-Dixon, Lin & Mardling 2004;Ferraz-Mello, Rodríguez & Hussmann 2008;Jackson, Barnes & Greenberg 2009;Rodríguez & Ferraz-Mello 2010).
The rotation of short-period planets is also affected by tidal interactions. It is nowadays accepted that the spin rate of planets soon after their formation was much higher than their orbital mean motion (e.g. Agnor, Canup & Levison 1999;Kokubo & Ida 2007). However, the cumulative tidal effects causes the planet to spin down, reaching its stationary value in time-scales much shorter than those needed to circularize the planet orbit. This evolution ultimately results in the synchronous rotation (the spin state in which the orbital and rotation periods coincide), which occurs when the planet orbit is circularized (Hut 1981;Ferraz-Mello et al. 2008). For non-circular orbits, the synchronous rotation only occurs if an additional torque acts to counteract the tidal torque.
The additional gravitational torque may be generated by planets with a rocky-like composition having a permanent equatorial deformation or frozen-in figure (see Goldreich & Peale 1966;Greenberg & Weidenschilling 1984). These torques, in contrast to the tidal ones, are conservative. In the context of the two-body problem, the gravitational interaction of the star with an asymmetric planet drive the planet rotation into different regimes of motion, including oscillations around exact spinorbit resonances (Goldreich & Peale 1966). When dissipative tidal effects are taken into account, the oscillations are damped and the planet rotation is trapped into a resonance. Indeed, the rotational motion of some Solar System bodies can be explained by the combined effects of the tidal and asymmetric torques (Goldreich & Peale 1966;Greenberg & Weidenschilling 1984;Correia & Laskar 2009).
Mercury is an example of a rocky planet with a rotation state currently evolving in the 3/2 spin-orbit resonance (Pettengill & Dyce 1965). Several models of the secular spin evolution of Mercury have been developed, including i) the chaotic evolution considered; planar problem). In addition, we assume that the equatorial asymmetry (a = b) corresponds to a frozen-in body or permanent deformation of the planet. The assumption above can be supported under consideration of a rocky composition of the planet body. This is the case of super-Earth planets, which are the subject of the present investigation. The gravitational potential of the planet at the distance r from its centre is given by (e.g. Beutler 2005, p. 102) where G is the gravitational constant and the angles 1 f and θ are defined in Figure 1. Note thatθ = Ω is the angular rotation velocity of the planet. The parameter C22 describes the equatorial ellipticity of the planet's gravity field and is related to the principal momenta of inertia A, B, C through C22 = (ξ/4)(B − A)/C, where ξ is the structure constant given by ξ = C/(mR 2 ) and C is the polar moment of inertia. We neglect the contribution of terms arising from the oblateness of the rotating body, that is, J2 and higher order terms. In fact, in the planar problem, the gravitational potential corresponding to the J2-term depends only on the distance r. In this case, the potential generates a radial force with zero torque that does not affect the rotation evolution.
At the distance r from the centre of the planet, the gravitational field U22 gives rise to a force on a point mass M which is given by 2 wherer andf are unit vectors with origin in the star (see Fig. 1). The force F22 alters the orbital component of the angular momentum, whereas the reaction force governs the evolution of the spin through the creation of the torque on the planet given by T22 = −r × F22 = −M (∂U22/∂f )k, where r = rr andk =r ×f . Thus, Note that, for CΩ = T22 and C22 = (ξ/4)(B − A)/C, the above equation agrees with the classical expression for the torque (e.g. Danby 1962;Goldreich & Peale 1966). The effects of the dynamical tides are raised by the gravitational action of the star on the inelastically deformed rotating planet. Due to the internal viscosity of the body, there exists a delay ∆t (time lag) between the star perturbation and the maximum deformation. The relationship between ∆t and tidal frequencies (which are linear combinations of Ω and n) depends on the internal rheology, which is poorly known even for Solar System bodies. For sake of simplicity, in this work, we adopt a linear model with constant ∆t (Darwin 1879;Mignard 1979;Hut 1981;Rodríguez et al. 2011a); for a review of other tidal models, the reader is referred to Efroimsky &Williams (2009) andFerraz-Mello (2012).
According to Mignard (1979), the tidal force and torque are given by and where v =ṙ and k2 is the second degree Love number. In this paper, we only consider the tides raised by the star on the planet, neglecting the tides raised on the star by the planet. Indeed, the strength of stellar tide is proportional to the planet mass and can be safely neglected in the case of super-Earths (see Rodríguez & Ferraz-Mello 2010). Note however that, after circularization of the planet orbit, stellar tides may be the only source of orbital decay.
In the reference frame centered in the star (see Figure 1), the exact equations of motion of the planet can be now written asr The perturbing forces F22 and F tide alter the Keplerian orbit of the planet which is defined by the first term of the first equation, while the corresponding torques in the second equation produce variations of the planet rotation. Note that, in the planar case, this last equation can be written in scalar form, because both torques are normal to the orbital plane.

NUMERICAL SIMULATIONS
In this section we apply the analytical model described in the previous section, in order to investigate the motion of three known super-Earths planets, namely Kepler-10 b, 55 Cnc e and GJ 3634 b. For this task, the exact equations of motion (6) were integrated numerically using the RA15 code (Everhart 1985). We stress that no scaling parameter, often used to accelerate the tidal evolution, was used in our simulations (see, for instance, Tittemore & Wisdom (1988), for the validity of the usage of the scaling parameter). Table 1 shows the physical parameters and current orbital elements of the systems considered in this paper.

Setting initial configurations
Numerical integrations require a setting of the initial configurations of the systems. In the case of problems including dissipative processes, the choice is complicated, mainly, due to the inadequacy of the simulations of the past evolution of dissipative systems through integrations backward in time (see details in Michtchenko & Rodríguez 2011). However, as shown in that paper, the final state of a migrating system is independent of its initial position in the phase space, defined by three values of parameters: the mass and semi-major axes ratios and the total angular momentum. As a consequence, to assess the orbit configuration in the past, we have to set solely the starting semi-major axis value. The initial eccentricity should satisfy the conservation of the total angular momentum of the system. The total angular momentum of the system is composed of the orbital component written as m G(M + m) a(1 − e 2 ), and the rotational component given by CΩ = ξmR 2 Ω. Using Kepler's third law, it is easy to show that the ratio of rotational and orbital components is of the order of (R/a) 2 (see Rodríguez et al. 2011a); since R ≪ a, the contribution of rotation to the total angular momentum can be neglected. Therefore, considering that the orbital angular momentum is nearly conserved during tidal migration, we obtain that where the subscripts "ini" and "current" stand for initial and current orbital elements, respectively. In this way, choosing an initial value for the semi-major axis and knowing the current values of a and e, equation (7) allows us to constrain the initial value of the eccentricity (see next section and specially Figure 6). It should be emphasized that the value obtained in such a way must be considered as an approximation of the initial value of the eccentricity, since the planets may belong to systems with two or more planets. The value of ∆t is determined by the relationship between the tidal frequency ν and the quality factor Q. In the case of a linear approximation, we have 1/Q = ν∆t, where ∆t = const. It is known that, in the Fourier spectrum of the tidal potential, the most important frequencies (that is, those involved in terms of order zero and order one in the eccentricity) are ν0 = 2Ω − 2n, ν1 = 2Ω − 3n and ν2 = 2Ω − n . The first one is associated to the term which is independent of the eccentricity, while the two last frequencies correspond to terms which are proportional to e (Darwin, 1879;Ferraz-Mello et al. 2008). Using the frequency ν0 and a modified version of the quality factor written in terms of the Love number k2 as Q ′ = 3Q/2k2, we obtain Actually, the spin rotation of exoplanets is not available from observations, thus we choose its initial value arbitrarily but considering a super-synchronous motion (Ω > n).

Results
The integration of the exact equations of motion usually requires a long computation time. For this reason, we decided to work with a small set of initial spin-orbit configurations and physical parameters. A statistical analysis taking into account many different initial conditions could give us a more accurate global picture of the past spin-orbit evolution of the planets. As pointed out by Correia & Laskar (2009), such statistical study is more adequate in the context of the averaged equations of the spin-orbit problem (see Section 4).

Kepler-10 b
Kepler-10 b is a very close-in planet with a period of ∼ 0.84 days, orbiting an old (≃ 12 Gyr) Sun-like star (Batalha et al. 2011). The physical parameters and the current orbital elements of the planet are summarized in Table 1. The estimated mass and radius values (m = 4.56m⊕, R = 1.416R⊕) suggest a rocky composition of the planet, which can be considered as a super-Earth (Valencia 2011). The transit technique measurements do not allow a reliable estimation of the current orbital eccentricity, which was arbitrarily fixed at zero in this work. This choice is supported by our numerical simulations of the whole system, which includes also the exterior companion, Kepler-10 c. The results have shown that the long-term evolution, due to the coupled effect of tides and mutual interaction, could result in the complete circularization of both planetary orbits (Rodríguez et al., in preparation).
We adopt the initial semi-major axis at aini = 0.02 au, which provides eini = 0.397 through equation (7). In addition, we set Ωini = 2.7 nini and ξ = 0.35. For the tidal parameter, we assume Q ′ = 100 (a value usually adopted for rocky planets) and, solving equation (8), we obtain k2∆t ≃ 1.1 min. In Section 3.1 we have estimated that, for Kepler-10 b on its current orbit, C22 ≃ 10 −3 . It should be emphasized that C22 would be smaller if the frozen-in shape of the planet was attained at larger distances than acurrent. Hence, in order to investigate different possible scenarios, we consider three different values of C22, namely, 10 −3 , 10 −4 and 10 −5 . Figure 2 shows the variation of Ω/n as a function of the eccentricity of Kepler-10 b. The curve with segments line corresponds to the stationary tidal solution (hereafter STS) of the rotation (Ω = 0), obtained when only the tidal torque is considered. The STS is a function which depends only on the planet eccentricity 3 , and is given by (see equations (42) or (43) of Hut (1981); see also Section 4 for more details). The stationary rotation evolves with the tidally damped eccentricity and the system reaches synchronization (Ω = n) only for circular orbits. Note that for small values of e, there follows that Ω ∼ n; for this reason, the STS is also called pseudo-synchronous solution (but this is valid only for e ≪ 1). The solid black curve in Figure 2 corresponds to the result of the numerical simulation performed with C22 = 0 (only tidal forces are considered). In this case, the stationary rotation is attained almost instantaneously (when compared to the time of circularization). Note that both curves are in good agreement, at least for e < 0.4; hence, in this eccentricity interval, we can refer to the solution obtained with C22 = 0 as STS. We also note that the curve for C22 = 0 reaches the STS for Ω/n close to two; however, this is incidental and unrelated with the 2/1 trapping. Figure. 3 shows the time variation of Ω/n. For the three considered values of C22 > 0, the rotation rapidly encounters the value Ω/n = 2 and thus the planet rotation is captured in the 2/1 spin-orbit resonance (hereafter SOR). On one hand, for C22 = 10 −3 and C22 = 10 −4 , the rotation escapes from the 2/1 trapping approximately at 80 Kyr, when the rotation drops to the value Ω/n = 1, implying capture into 1/1 SOR and synchronous motion. However, it is worth noting that, for C22 = 10 −3 , the synchronism is attained in a form of jump, whereas for C22 = 10 −4 the rotation evolves without capture for a very short time before the 1/1 SOR (see small panel in Fig. 3). We have also plotted the result for C22 = 0, showing that synchronization is attained around 400 Kyr. It is interesting to note that the introduction of the equatorial asymmetry of the planet reduces the time to reach synchronization; for instance, for C22 = 10 −3 , it is reduced by a factor of 5.
On the other hand, when C22 = 10 −5 , the 2/1 SOR is destabilized at around 48 Kyr giving rise to the trapping in the 3/2 SOR (Ω/n = 3/2). Thus, the permanence of the rotation in the 2/1 SOR is temporary and its duration clearly depends on the C22 value. The rotation evolves in a Mercury-like state for some time, when a new instability occurs, after which an evolution free of trapping starts. Note that during the period of non-capture, the rotation returns to the stationary value, as can be seen in the dotted curve of Fig. 2    We note that, when the rotation evolves under capture in resonance (that is, for C22 > 0), the rates of variations of the elements are larger than the one corresponding to the pure tidal case (C22 = 0), with the difference increasing as the eccentricity is damped (when the comparison is done for a given SOR). Moreover, for the same value of C22, the rate of orbital decay and circularization seems to depend on the specific SOR in which the planet rotation is trapped in. For example, in the case of C22 = 10 −5 , the evolutions of a and e show two different regimes as the rotation evolves first in the 2/1 and then in 3/2 trappings (see Section 4).
We note that the time necessary to get an orbit with e = 0.05 is much greater for C22 < 10 −3 . In fact, when the rotation evolves without capture, the time-scale for circularization increases, according to Figures 4 and 5.
It is worth noting that, in the case of C22 > 0, the synchronous motion can be attained even for e = 0. For instance, in the case of C22 = 10 −4 , the system is captured in 1/1 SOR at roughly 120 Kyr, when e ≃ 0.079. This fact confirms the classical result that, for eccentric orbits, the synchronous motion is only possible if there exists an additional torque which Eq.(7) current initial Figure 6. Evolution of the orbit of Kepler-10 b in the plane (a, e). The solution of equation (7) is shown by the dashed line and is compared to the result of the numerical simulation for the case C 22 = 10 −3 , giving a good agreement (the curves for the other two C 22 values also agree with the solution of equation (7), but are not shown here for sake of simplicity). The starting position of the planet is indicated by the star symbol.
counteracts the tidal torque (see Goldreich & Peale 1966;Ferraz-Mello et al. 2008 and references therein). We also note that after circularization, the 1/1 SOR is the only possible resonant motion 4 . Figure 6 shows the evolution of the system in the plane of orbital elements a and e. The solid line shows the result of numerical simulation in the case of C22 = 10 −3 . The dashed line corresponds to equation (7), which is independent on the value of C22. Note as both curves are in very good agreement, in such a way confirming our assumption about the conservation of the orbital angular momentum (Section 3.1).

GJ 3634 b
GJ 3634 b is a super-Earth planet with mass m = 7.0m⊕ orbiting a M2.5 dwarf each 2.65 days with an eccentricity of e = 0.08 (Bonfils et al. 2011). Since there is no transit detection for this planet, its radius is unknown. However, Valencia, Sasselov & O'Connell (2007) have shown that there is a maximum radius that a rocky terrestrial planet may achieve for a given mass. A larger radius would imply a large amount of water and even the possibility of a massive volatile envelope. Indeed, for a planet with mass 7.5m⊕ the maximum terrestrial radius would be 11,600 km (≃ 1.82R⊕), which result in a mean density of ρ ≃ 1.25ρ⊕. Hence, we assume for GJ 3634 b that ρ = ρ⊕ to obtain R ≃ 1.9R⊕. The adopted initial orbital elements are aini = 0.033 au and eini = 0.369, while Ωini = 3.7 nini, ξ = 0.35 and Q ′ = 100 (k2∆t = 2.1 min). At the current value of the semi-major axis the estimated value of C22 is C22 ≃ 1.6 × 10 −4 (Section 3.1); in this work, we perform the analysis using two values of C22: 10 −4 and 10 −5 . Figure 7 shows the variation of Ω/n as a function of e (left panel) and the time variation of a (right panel). For C22 = 10 −4 , the rotation is rapidly trapped in the 2/1 SOR, remaining in such resonant configuration during the most time of the tidal eccentricity damping. Only at low eccentricities, close to 0.032, the system leaves the 2/1 SOR dropping directly into the synchronous motion.
At variance, for C22 = 10 −5 , the rotation quickly reaches the stationary (non-resonant) state given by the STS curve (C22 = 0) in Figure 7 (left). The system evolves along the STS curve up to close approach to the 3/2 SOR. Then, the system is captured and remains in this resonance until dropping in the synchronized state, at e ≃ 0.012.
In both cases, the final configuration of the planet's orbit is characterized by circularization and synchronization, as in the case of Kepler-10 b. However, the current eccentricity of the GJ 3634 b is different from zero (its location at 0.08 is shown by the dotted vertical line in the left panel of Figure 7). Note that the current eccentricity is reached when the rotation still evolves in the 2/1 or 3/2 SOR (depending on C22). This fact suggests that GJ 3634 b could be presently involved in one of such resonant trappings, in particular, it could be found in a Mercury-like rotational state (Ω/n ≃ 3/2). However, it should be kept in mind, that this conclusion depends on the choice of specific values of C22 and Q ′ . Indeed, the results of an additional simulation performed with C22 = 10 −6 , have shown that the trapping of the system in the 3/2 SOR is destabilized at e ≃ 0.11. Since ecurrent = 0.08 < 0.11, the planet rotation would be currently evolving without capture (along the STS) for such a small C22 value.
The semi-major axis evolution (Figure 7, right panel) shows that the tidal decay is faster when the rotation of the planet is trapped in the 2/1 and 3/2 spin-orbit resonant motions. The fact that the current eccentricity of the planet orbit is different Depending on C 22 , the planet rotation may be currently evolving in either 2/1 or 3/2 SOR. Note that the time-scale of orbital decay is drastically reduced provided the planet can support a permanent deformation resulting in a nonzero C 22 value. The vertical and horizontal dotted lines indicates the current values of e and a, respectively from zero suggests that, in the absence of other perturbations, the orbital decay 5 of the planet is presently in progress and, consequently, the final semi-major axis will be smaller than the current one.

55 Cnc e
55 Cnc e is a hot super-Earth planet, with the mass m = 8.63m⊕ and the radius R = 2.17R⊕, orbiting a Sun-like star each 0.74 days with e = 0.057 (Winn et al. 2011;Gillon et al. 2012). The averaged density of the planet is comparable to that of Earth, suggesting a rock-iron composition supplemented by a significant mass of water, gas, or other light elements 6 (Winn et al. 2011).
For numerical simulations, we adopt aini = 0.0186 au, eini = 0.4, Ωini = 2.7 nini and ξ = 0.35. The tidal history of the system indicates that the magnitude of Q ′ must be higher than the commonly adopted values for rocky planets (Ferraz-Mello 2012). In this work, we use Q ′ = 5.5 × 10 3 ; this value allows us to obtain k2∆t ≃ 1.1 s. We perform numerical simulations for two different values of C22: 10 −3 and 10 −4 . For the purpose of further comparison with the results reported in Callegari & Rodríguez (2012), we also consider the values C22 = 2.45 × 10 −3 and C22 = 2.45 × 10 −4 . Figure 8 shows the evolution of rotation as a function of the planet's eccentricity. The captures inside the 5/2, 2/1, 3/2, 4/3 and 1/1 resonances can be observed. For C22 = 2.45 × 10 −3 , the planet rotation evolves under capture in the 3/2 SOR almost up to the orbital circularization, with Ω/n roughly varying between 1.46 and 1.54.
In a recent work, Callegari & Rodríguez (2012) have shown that, for e = 0.057 and (B − A)/C = 0.028 (or C22 = 2.45 × 10 −3 ), the phase space of the rotation (θ/n, θ) of 55 Cnc e, between the main spin-orbit resonances, is dominated by a large region of chaotic motion. This region increases as larger values of the eccentricity are considered.
Capture in spin-orbit resonance can work as a mechanism to prevent a chaotic rotation, provided that the dissipation is large enough in order to ensure the corresponding trapping. According to our simulations (see Figure 8), destabilizations of different resonances occur as the eccentricity is tidally damped (see also next section), giving rise to new temporary trappings with a subsequent evolution to synchronism. In this way, a chaotic behavior of the rotation can be avoided by a continuous evolution under resonant motion. It is important to note that, at least for the cases shown in Figure 8, there is no interval of non-resonant motion between two successive trappings, as in the case of Kepler-10 b (see Figure 3 for C22 = 10 −4 and 10 −5 ).
An additional simulation done with C22 = 2.45×10 −5 (not shown in figures), has resulted in a 2/1 trapping until e ≃ 0.04, indicating that the planet rotation may be currently evolving in resonant motion, for such a small value of C22. Figure 9 shows the time variation of semi-major axis. It is worth noting that the orbital decay results in a more rapid evolution towards acurrent when the planet rotation is trapped in a SOR. Also, the rate of variation depends on the specific resonance. From Figure 9 we see that the evolution when C22 ∼ 10 −4 , (capture in 2/1) is almost two times faster than the one for C22 = 2.45 × 10 −3 (3/2 capture). We will return to this discussion in the next section.

AVERAGED EQUATIONS
In order to better understand the results previously obtained through numerical simulations of the exact equations of motion, in this section, we develop an analytical approach in the context of the averaged equations of the spin-orbit problem.
The equation of motion for the spin in equation (6) cannot be solved analytically. However, because we are interested in the evolution inside spin-orbit resonances, we will consider planetary spin rates having values such that Ω ≃ pn, where p is a half-integer. Our investigation is restricted to the cases p = 1, p = 3/2 and p = 2 but can be easily extended to any other SOR.
We introduce the "resonant" angle δ = θ − p l, where l is the mean anomaly of the planet orbit and θ was defined in Section 2. Note that the angle δ gives the orientation of the long axis whenever the planet passes through the pericenter of its orbit (i.e., δ = θ when l = 0, see Figure 1). It is easy to show that, inside the spin-orbit resonance given by the parameter p, the resonant angle is a slowly varying variable when compared to the orbital period of the planet. Indeed, we can write thatθ =δ + pn, whereθ = Ω ≃ pn and, consequently,δ ≃ 0. Now, replacing θ by δ + pl in equation (3) and averaging over the orbital period, we obtain the classical result for the averaged torque on the planet produced by the permanent equatorial deformation (see Goldreich 1966;Goldreich & Peale 1966): where H(p, e) are power series of e with the leading term of order e 2|p−1| (see Appendix A for a literal expansion up to seventh order). The averaged tidal torque on the planet is given by where K tide = 3k2∆tM R 5 and T (p, e) is a suitable function, whose expansion in powers of e is shown in Appendix A (Hut 1981). At resonance, the equilibrium solution of the rotation is given by the condition and, resolving for C22 sin 2δ, we obtain If we know the maximum value that δ can reach without disturbing the trapping in the p-SOR, we may obtain the minimum value of C22 necessary for capture in the same resonant motion.

1/1 Spin-Orbit Resonance
Replacing p = 1 in equation (13) We note that, for O(e 4 ), the above expression is the same obtained in Ferraz-Mello et al. (2008), for synchronous asymmetric planets 7 .

Semi-major axis evolution
The averaged variation of the orbital elements can be obtained using the Euler-Gauss equations (see Beutler 2005, p. 230), which are written in terms of the radial and transverse components of the perturbing force. The result for the semi-major axis is Replacing C22 sin 2δ by the corresponding expression given in equation (14), we obtain <ȧ22 >= 3k2∆t M m R a 5 n 2 a 12e 2 + 363 4 e 4 + 2911 8 In the context of the linear tidal model, the averaged variation of the semi-major axis due to tides is given by equation The total averaged variation of the semi-major axis is then given as At the same order of approximation, the above equation coincides with the result of previous works (see Ferraz-Mello et al. (2008) and references therein). When only the tidal torque on the planet is taken into account, the non-resonant STS is given by equation (9). In this case, the synchronism is possible only for circular orbits. If instead of p = 1, we replace p = pstat = Ωstat/n in equation (17), we obtain n 2 a 7e 2 + 54e 4 + 1133 4 e 6 + O(e 8 ).
We define r p a as the ratio between averaged semi-major axis variation in the p-SOR (p = 1, 3/2, 2, ...) and in the tidal stationary solution of the rotation (p = pstat), that is Comparing equations (20) and (21), we note that r p=1 a > 1, indicating that the rate of semi-major axis decay is stronger when the rotation is locked in the 1/1 SOR, compared to the case when it evolves following the STS. This result is in agreement with Wisdom (2008), who has shown that the rate of energy dissipation in a body with synchronous rotation is larger than in a asymptotic non-synchronous rotation.

Eccentricity
We repeat the procedure described in the previous section to obtain the averaged variation of the planet eccentricity. The resulting expressions are: Similarly to the semi-major axis variation case, r p=1 e > 1, where r p e is the analogous for the eccentricity of equation (22).
(28) Figure 10 shows a plot of r p a for the three analysed resonances as a function of the eccentricity. On one hand, we note that r p=3/2 a and r p=2 a increase for small eccentricity, indicating that the major effect on the orbital evolution due to the resonance captures occurs as e decreases, in agreement with the results of the numerical simulations (see Figures 4, 7 and 9). On the other hand, for large e, ra is small (mainly in the case p = 3/2) as also shown in Figures 4 and 7, where there is a small difference between the evolutions with and without C22 (i.e, ra ≃ 1).
When the synchronous motion is analysed, we note in Figure 10 that r p=1 a is an increasing function of e, indicating that the orbital decay when the rotation is trapped in the 1/1 SOR is large for large e. However, as simulations show, synchronism is attained for very low values of e, making the orbital decay almost negligible.
The results above are also valid for r p e . It is interesting to note that r p a and r p e are only functions of the eccentricity and do not depend on any physical parameter.
It is worthwhile mentioning that T22 averages to zero unless the condition Ω ≃ pn is assumed (whit p integer), that is, the averaged orbital evolution due to the triaxial torque is expected only for resonant cases.  Figure 10. The variation of r p a as a function of orbital eccentricity, for p = 2, 3/2 and 1. In the cases of p > 1 the ratio of rates of variation of a in the SOR and in the STS increases for small e, showing that the orbital decay is enhanced (when compared with the evolution in the STS) whenever the planet rotation evolves under resonance with the orbital motion.

Critical C22
As it has been noted, equation (13) can be used to obtain the minimum value of C22 necessary for the stability of the p-SOR. Calling C * 22 the one value given by the right hand of equation (13), the stability condition for a resonant spin state requires that C22 > C * 22 . The above condition is frequently found in literature as strength criterion of the resonance (see Murray & Dermott 1999, p. 205). Figure 11 shows the functional dependence of the absolute value of C * 22 with the eccentricity, obtained for Kepler-10 b. The dashed horizontal lines represent three values of C22 used in the numerical simulations shown in Section 3.2.1, namely, 10 −3 , 10 −4 and 10 −5 .
The case p = 2, shown by the blue curve, indicates that C * 22 = 10 −5 for e ≃ 0.3, C * 22 = 10 −4 for e ≃ 0.1 and C * 22 = 10 −3 for e ≃ 0.03. A good agreement is obtained doing the comparison with the results of the numerical simulations for Kepler-10 b, shown in Figure 12. Note as the 2/1 SOR destabilizes when e ≃ 0.28, e ≃ 0.1 and e ≃ 0.03.
The same can be observed for the 3/2 SOR, in which the rotation escapes from the trapping when e ≃ 0.11 if C22 = 10 −5 , in concordance 9 with the result of Figure 11 (orange curve).
Since for roughly e 0.1 the condition C22 > C * 22 is satisfied more easily for p = 1 as compared to p > 1, there is no problem in understanding the occurrence of synchronous motion for very small eccentricity, as can be also verified in Figure  12.
It is worth noting that instead to compute a critical C22 value, the condition for stationary solutions < T tide > + < T22 >= CΩ = 0 also enables us to obtain a critical value of e for a given fixed C22. The resonant motion would destabilize whenever the eccentricity becomes smaller than the critical value for each SOR. This approach is more convenient in those cases for which the observations allow an estimation of C22 (see Correia & Laskar (2004) for the study of the rotation of Mercury).

DISCUSSION AND CONCLUSIONS
The numerical exploration of the planet rotation which is subject to the conservative triaxial gravitational torque of the star shows different behaviors, including the oscillation around the equilibrium points of spin-orbit resonances (see Callegari & Rodríguez 2012 and references therein). When dissipative effects produced by tidal torques are included, the rotation of close-in bodies can be captured in a resonant motion (Goldreich 1966;Goldreich & Peale 1966;Celletti & Chierchia 2008). The capture in a specific resonance depends on the eccentricity e and on parameters related to tides and equatorial ellipticity (Q and C22, respectively). Hence, since e is tidally damped during the inward migration, each specific capture should become unstable and the rotation can achieve another resonant state, which, at the same time, should result in a temporary trapping. When the orbit is circularized, the final evolution results in the synchronization between the orbital and rotational periods (i.e, the 1/1 spin-orbit resonance).
In this paper, we analysed the coupled spin-orbit evolution of Kepler-10 b, GJ 3634 b and 55 Cnc e short-period super-Earth planets. The numerical simulation of the exact equations of motion have shown that, depending on C22, the rotation  (13) and calculated with parameters of the planet Kepler -10 b. The calculation of C * 22 can be used as a guide to investigate the stability of a given SOR as a function of eccentricity (see text for discussion). Note that in the case of p = 1 we have C * 22 = 0 for e = 0 (see equation (14)), indicating that the synchronism is the only possible resonant motion for circular orbits (see Goldreich & Peale 1966;Celletti & Chierchia 2008;Callegari & Rodríguez 2012 Figure 11 is done, we obtain a good agreement for the values of e for which each SOR destabilizes (see text for details). displays a number of resonance trappings for which Ω ≃ pn, including the cases p = 5/2, 2, 3/2, 4/3 and 1. Moreover, the results indicate that GJ 3634 b can be an example of an exoplanet with a rotation state similar to Mercury's rotation (p = 3/2), provided the body figure can support a permanent equatorial asymmetry as small as C22 = 10 −5 . 55 Cnc e can currently evolve under 5/2 or 3/2 trappings if C22 ∼ 10 −3 and under 2/1 if C22 ∼ 10 −4 or even if C22 is as small as 2.45×10 −5 . The rotation of 55 Cnc e may have a chaotic behavior for C22 ∼ 10 −3 (Callegari & Rodríguez 2012). However, we have shown that a scenario is possible in which the rotation evolves to synchronism through successive trappings in SOR. Indeed, the eccentricity damping due to tides provides the destabilization mechanism required to escape from one given capture and subsequently evolve to lower order resonances (see Correia & Laskar 2004).
We have shown that different spin-states corresponding to various resonance captures can occur in extrasolar systems. Although the 1/1 spin-orbit coupling is the final state of evolution, it is not reached in every case as present configuration. For current non-circular orbits, trappings in the 3/2 SOR or other resonance configurations can be sustained over long times. The latter would depend on the orbital eccentricity, the planet's C22 value and its dissipation factor. Given that these parameters are not well-determined for terrestrial extrasolar planets, we note that the 1/1 coupling may not be the current rotation state of some planets. This would have consequences on the stellar insulation and therefore on the evolution of the atmosphere and surface of the planet.
When only the tidal torque is taken into account (C22 = 0), the rotation evolves to the stationary tidal solution (STS). We have shown that under assumption of C22 > 0, the evolution along spin-orbit resonance (SOR) leads to higher rates of orbital decay and circularization than the corresponding ones when the rotation evolves in the STS. Moreover, for a given C22 value, the variation rates of the elements depend on the specific trapping in which the rotation is trapped in, and increases with p (see Figure 4).
We have also calculated the averaged rates of variations of semi-major axis and eccentricity including the gravitational and tidal torques. Moreover, through the analytical calculation of the critical value of C22 it is possible to predict the values of the eccentricity for which a given SOR would become unstable. The comparison with the direct numerical simulations have shown a good concordance for the analysed planets.
The importance of the C22 value on the rotational history of the planet is illustrated in the example of Kepler-10 b. It is worth noting that the planet rotation of Kepler-10 b can evolve under the influence of the 2/1 SOR for longer time-scales (i.e., small e) provided the planet figure can support a permanent equatorial deformation characterized by C22 ∼ 10 −3 . For smaller C22, the 2/1 resonant state would become unstable for relatively large eccentricity (Figures 11 and 12).
We stress that the results of the numerical simulations provide different possible spin-orbit evolutions depending on the considered initial values of elements and rotation. The Q value and the specific tidal model would also play an important role in determining the rotation history of the planet, since the temporary trapping in SOR depends on the rate of eccentricity damping, which is Q-dependent. In fact, in a recent paper, Makarov (2012) has applied the model by Efroimsky (2012) and successfully explained the current resonant state of Mercury within the considered model.
Our results concerning the association of the critical C22 value with the capture probability are not new in the context of averaged equations of the spin-orbit problem (see for instance Correia & Laskar 2004). However, in this work, we are able to compare the results of the averaged equations with those resulting from the numerical simulations of the exact equations of motion.
A model of planet interior including the effect of viscous friction at the core-mantle boundary would provide quantitatively different results (see Peale 2005, andCorreia &Laskar 2009, for the case of Mercury). According to previous works the chances of resonant capture increase if the latter effect is taken into account (Goldreich & Peale 1967;Peale & Boss 1977). However, in order to highlight the contribution of C22 to the orbital evolution, in this paper we adopt a simple model neglecting a possible layered structure of the planet.
In the present study we have assumed constant C22 values. However, depending on the adjustment to hydrostatic equilibrium C22 may be itself time-dependent. The time-scale for adjustment to the completely relaxed state depends on material parameters, e.g., rigidity and viscosity which are highly temperature dependent. Furthermore, it will depend on the spin-orbit coupling the planet is trapped in. For the application to terrestrial planets given here the constant, i.e. frozen, C22 configuration is justified because adjustment of solid rocky planets to equilibrium usually occurs on geological time-scales. Therefore, the states described here do not necessarily represent hydrostatic configurations.
In this work we have considered a model of one interacting planet, despite the fact that Kepler-10 b and 55 Cnc e belongs to multiple-planet systems with exterior (more massive) companions. Previous works have shown that the mutual secular interaction with the companion may affect the eccentricity evolution of the tidally evolved planet, increasing the circularization time-scale (Mardling 2007;Rodríguez et al. 2011a;Rodríguez et al. 2011b). Moreover, the planetary perturbations can produce large excursions of the innermost planet eccentricity, providing a mechanism for increasing or decreasing the capture probability of the spin-orbit resonances (Correia & Laskar 2004). However, a numerical simulation of 55 Cnc system including the exterior companion 55 Cnc b has shown little differences with the single planet case. Indeed, the evolution of the inner orbit eccentricity is almost not affected by the mutual interaction between the planets and thus the rotation evolution of 55 Cnc e can be safely analysed within the one planet model. An investigation about the rotation of asymmetric close-in super-Earths including an exterior companion will be addressed in a future investigation. The exact expression for the function T (p, e) is given by (Hut 1981) T (p, e) = (1 − e 2 ) −6 f2(e) − (1 − e 2 ) 3/2 f5(e) p , where f2(e) = 1 + 15 2 e 2 + 45 8 e 4 + 5 16 e 6 (31) and f5(e) = 1 + 3e 2 + 3 8 e 4 .