A&A 630, A102 (2019) Astronomy https://doi.org/10.1051/0004-6361/201936336 & © ESO 2019 Astrophysics

Spin-orbit coupling for close-in planets Alexandre C. M. Correia1,2 and Jean-Baptiste Delisle2,3

1 CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal e-mail: [email protected] 2 ASD, IMCCE, Observatoire de Paris, PSL Université, 77 Av. Denfert-Rochereau, 75014 Paris, France 3 Observatoire de l’Université de Genève, 51 chemin des Maillettes, 1290 Sauverny, Switzerland

Received 17 July 2019 / Accepted 20 August 2019

ABSTRACT

We study the spin evolution of close-in planets in multi-body systems and present a very general formulation of the spin-orbit problem. This includes a simple way to probe the spin dynamics from the orbital perturbations, a new method for computing forced librations and tidal deformation, and general expressions for the tidal torque and capture probabilities in resonance. We show that planet–planet perturbations can drive the spin of Earth-size planets into asynchronous or chaotic states, even for nearly circular orbits. We apply our results to Mercury and to the KOI-1599 system of two super-Earths in a 3/2 mean motion resonance. Key words. celestial mechanics – planets and satellites: general – chaos

1. Introduction resonant rotation with the orbital libration frequency (Correia & Robutel 2013; Leleu et al. 2016; Delisle et al. 2017). The inner planets of the solar system, like the majority of the Despite the proximity of solar system bodies, the determi- main satellites, currently present a rotation state that is differ- nation of their rotational periods has only been achieved in ent from what is believed to have been the initial state (e.g., the second half of the 20th century. Mercury and Venus using Goldreich & Soter 1966). Just after their formation, they are sup- radar ranging (Goldstein 1964; Carpenter 1964; Pettengill & posed to have rotated much faster (e.g., Kokubo & Ida 2007), but Dyce 1965), Hyperion after the Voyager 2 flyby (Smith et al. subsequently slowed down owing to tidal interactions with the 1982), and the minor satellites of the Pluto with images from the central more massive body (e.g., MacDonald 1964). Hubble Space Telescope (Showalter & Hamilton 2015). We thus The last stage for tidal evolution is synchronous rotation with do not expect that it will be easy to observe the rotation of the orbital mean motion, since it corresponds to the minimum recently discovered exoplanets. Nevertheless, many of these of energy (Hut 1980; Adams & Bloch 2015). This equilibrium, planets are close to their host star, and we can assume that their also known as the 1/1 spin-orbit resonance, was first observed spins have already undergone enough dissipation and evolved for the Moon. However, other possibilities exist, such as the into a final equilibrium. Therefore, we can try to anticipate all 3/2 spin-orbit resonance of Mercury (Pettengill & Dyce 1965), the possible scenarios for the rotation of these planets. In particu- which is possible because the eccentricity is different from zero lar, we aim to determine when asynchronous rotation is possible, (Colombo 1965; Goldreich & Peale 1966; Correia & Laskar which is important for habitability studies. 2004; Noyelles et al. 2014). Another interesting example is the In this paper we generalize previous studies to any kind of chaotic rotation of Hyperion (Wisdom et al. 1984; Harbison et al. orbital perturbations. In Sect.2, we write the equation for the 2011), which also results from nonzero eccentricity. rotation of a planet that is perturbed by other bodies, and show When the eccentricity is nearly zero, asynchronous spin- that the orbital forcing can be decoupled from the rotation. In orbit resonances are still possible, provided that a companion Sect.3, we determine the possible configurations for the rotation is present in the system. The rotation of Venus is also tidally of a rigid body by analyzing the orbital forcing. In Sect.4, we evolved and not synchronous (Goldstein 1964; Carpenter 1964); consider that the planet can deform under the action of tides, and it is close to a 5/1 spin-orbit resonance with the Earth, present a new method for determining the equilibrium shape and − 1 which suggests that the rotation can be trapped at this ratio the forced oscillations in resonance. In Sect.5, using a Maxwell (Goldreich & Peale 1966, 1967). Indeed, one possibility for model, we obtain the tidal torque as a function of the orbital asynchronous rotation is a spin-orbit resonance with a con- forcing and the corresponding capture probabilities in resonance. junction frequency. The minor satellites of the Pluto–Charon In Sect.6, we perform complete numerical simulations including system may also have ended up trapped in a resonance of tidal dissipation for the KOI-1599 system. Finally, we discuss our this kind (Correia et al. 2015) or even present chaotic rotation results in Sect.7. (Showalter & Hamilton 2015). Another possibility, which is very important for planets involved in mean motion resonances, is 2. Orbital forcing

1 The expected libration width in resonance does not seem sufficient We consider a system composed by a central star of mass M, to cover this scenario. The current explanation is that Venus rotation a companion planet of mass m, and additional companions in is an equilibrium between gravitational and thermal atmospheric tides coplanar orbits. The planet is considered an oblate ellipsoid with (Gold & Soter 1969; Dobrovolskis 1980; Correia & Laskar 2001). mean radius R and gravity field coefficients given by C22 and Article published by EDP Sciences A102, page 1 of 12 A&A 630, A102 (2019)

S 22. We furthermore assume that the spin axis of the planet is For the shape of the planet, Z, we can split it in two main con- orthogonal to the orbital plane (zero obliquity). The torque of the tributions: one rigid or permanent, Z0, due to inhomogeneous star on the planet changes the rotation angle, θ, whose variations mass distribution inside the planet, and another that accounts are given by (e.g., Correia et al. 2014): for the deformation, Zt, due to external tidal forcing, such that Z = Z0 + Zt. In the solar system, we observe that Z0 is a function 6GMmR2 h i of the planet mass. We have Z0 0.1 for bodies with diame- θ¨ = C22 sin 2(θ f ) + S 22 cos 2(θ f ) , (1) − Cr3 − − ters smaller than 100 km (e.g.,| Correia| ∼ 2018), but larger bodies 5 evolve into spherical shapes ( Z0 0); we have Z0 10− for where G is the gravitational constant, C is the moment of inertia | | → 6 | | ∼ with respect to the rotation axis, r is the distance and f is the Mercury (Genova et al. 2013) and Z0 10− for the Earth and Venus (Yoder 1995). On the other hand,| | ∼ Z (Eq. (16)), that true longitude of the center of mass of the planet with respect to | t| ∼ A 3 is, it depends on the mass and distance of the star. For Mercury the star. We neglect terms in (R/r) (quadrupolar approximation) 7 we have Zt 10− Z0 , so we can assume Z Z0. However, and the torques of the companions on the planet’s spin. | | ∼  | | ' 4 The above expression can be rewritten in a more compact for the Galilean satellites of Jupiter, we have Zt 10− Z0 , so we can assume Z Z (Correia & Rodríguez| |2013 ∼ ).  | | and usable form with complex numbers as ' t  2iθ θ¨ = Z x e− , (2) B = 3. Rigid bodies with As in most previous studies on spin-orbit coupling, let us assume for now that Z = Z0, either because Zt Z0 (case of Mercury) Z = C22 iS 22, (3) | |  | | − or because Zt is frozen from a time the planet was more fluid a3 (case of the Moon). Then, Eq. (2) becomes2 x = e2i f , (4)   r X 2  i(k νt 2θ) 6GMmR θ¨ = Z xk e · −  . (9) = 18 n2, (5) B =  0  B Ca3 ≈ k where a is a scale constant with units of distance, that is usually Each term in this expression individually behaves like a made equal to a mean semimajor axis, n is the mean motion, pendulum, with islands of rotational libration centered at and C/mR2 1/3, which is the usual value for rocky planets (Yoder 1995).≈ This way, we are able to isolate in Eq. (1) all the k ν θ˙ = νk = · , (10) important quantities that change the rotation, namely, the shape h i 2 (or deformation) of the planet, Z, the orbital forcing, x, and the 2iθ and libration width rotation angle, e− . This factorization allows us to treat each contribution separately. p p p βk = 2 Z0 xk 6n Z0 xk . (11) The orbital and spin evolutions are coupled, such that the B| | ≈ | | | | total angular momentum is conserved. Therefore, a change in the In general, the rotation can be trapped in any of these rotation angle (Eq. (1)) also modifies the position r (see Eqs. (15) islands, provided they are far apart. However, when the libra- and (16) in Correia et al. 2014). However, the corrections in the tion widths of some individual resonant islands overlap, the orbital motion are on the order of (R/r)2 Z 1 (the orbital | |  rotation can be chaotic (Chirikov 1979; Morbidelli 2002), i.e., angular momentum is much larger than the rotational angu- the rotation exhibits random variations in short periods of time. lar momentum), so their contribution can usually be neglected. Overlap between two resonances with νk1 < νk2 then occurs for For planets in multi-body systems, the important contributions νk1 + βk1 νk2 βk2 , that is, when to the orbital forcing thus come from the mutual gravitational ≈ − perturbations. We let β¯ = βk + βk νk νk = ∆ν. (12) 1 2 ≈ 2 − 1 X = r ei f = (r cos f, r sin f ) (6) This simple condition allows us to differentiate easily between the different dynamical regimes. Indeed, when Z0 is be the position vector of the planet, whose spin we want to study. known for a given kind of body, we can infer its dynamics simply The evolution of X can be obtained by integrating a system of N by analyzing the orbital forcing (Eq. (8)). For β¯ ∆ν the rota- point-mass bodies (the star, the planet and the remaining com-  tion can be trapped in any isolated island with θ˙ = νk, for β¯ ∆ν panions). The orbital forcing on the spin (Eq. (4)) can then be the rotation is chaotic, while for β¯ ∆ν the rotation can be∼ sta- directly computed from X as  ble again because the νk are engulfed in a single libration island a3X2 with only a small chaotic zone at the separatrix. x = . (7) X 5 | | 3.1. Single planet case Assuming quasi-periodic orbital motion, we can expand the previous expression as a Fourier series In the classical case of a single planet in an elliptical orbit, the X orbital forcing (Eq. (4)) is given by ik νt x = xk e · , (8) X 3,2 2i$ iknt k x = − (e) e e , (13) Xk where ν is the vector of fundamental frequencies, and the ampli- k P tudes xk decay with k , such that xk is convergent. The 2 | | || || k | | Many authors adopt a frame where Z0 is a real number (Z0 = C22,0), orbital forcing on the spin, x, can then be characterized by the which simplifies expression (1). This is correct for the rigid case, but it main terms in the above series. is no longer valid when we consider the tidal deformation of the planet.

A102, page 2 of 12 A. C. M. Correia and J.-B. Delisle: Spin-orbit coupling for close-in planets

1 Table 1. Parameters for the KOI-1599 system used in this study.

Parameter (Unit) Inner Outer 0.8 . . Z -2 m (M ) 8 3 4 3 | 0| = 2.9 x 10 Hyperion R (R ⊕) 1.9 1.9 0.6 a (au)⊕ 0.11230 0.14727 | k

x . . | -5 e 0 0067 0 0177 |Z | = 1.5 x 10 Mercury 0 λ (deg) 217.4 251.5 0.4 $ (deg) 40.6 220.1

0.2 Notes. The mass of the star is 1.02 M . This orbital solution corresponds to the stable fit GEA I solution obtained by Panichi et al.(2019).

0 0 0.5 1 1.5 2 2.5 mutual perturbations are strong and need to be taken into account νk /n in the spin-orbit coupling analysis. Fig. 1. Fast Fourier transform of the orbital forcing x (Eq. (7)) for the One possibility is to expand the disturbing function in its two-body problem with eccentricity e = 0.165. The horizontal lines main perturbing terms (see Chap. 6, Murray & Dermott 1999), 5 give the amplitudes βk for Z0 = 1.5 10− , which corresponds to and use it to obtain the dominating contributions to x (Eq. (8)). | | × 2 Mercury’s value (Genova et al. 2013), and Z0 = 2.9 10− , which Closely packed systems usually have small eccentricities, so a corresponds to Hyperion’s value (Thomas et al.| 2007| ). × common approach is to truncate the series for some power in the eccentricities (e.g., Goldreich & Peale 1966, 1967; Correia & Robutel 2013; Correia et al. 2015; Leleu et al. 2016; Delisle et al. l,m where (e) are the Hansen coefficients (Hughes 1981) and $ 2017). Although modern series manipulators allow us to perform Xk is the longitude of the pericenter. Thus, resonant islands occur this calculation rapidly, the number of terms to retain is large 3,2 for νk = kn/2, with amplitude xk = k− (e). The two largest (even for a two-planet system), and their amplitudes depend on | | X 3,2 the masses and semimajor axes. Moreover, for resonant systems terms correspond to k = 2 and k = 3, for which 2− (e) 1 and 3,2 X 2 ≈ the eccentricities can be high and the libration amplitude is also − (e) 7e/2, respectively (we neglect terms in e ). Then X3 ≈ important. Therefore, the dominating terms vary from system to p  p  system and it is difficult to predict the terms to be retained. β/¯ ∆ν 12 Z 1 + 7e/2 . (14) ≈ | 0| A powerful alternative to analytical studies is to obtain the series expansion for x numerically. This method is exact (with- 5 For Mercury ( Z0 10− , e = 0.206), we get β/¯ ∆ν out approximations), fast, and valid for an unlimited number of | | ≈ ≈ 0.07 1, so we can expect stable rotation. Indeed, the planet is planets in the system. We simply need to run a simulation of  observed in a stable spin-orbit resonance with k = 3 (Colombo the full system and then perform a frequency analysis of the 1965; Goldreich & Peale 1966). On the other hand, for Saturn’s orbital motion. The only requirement is that we integrate the moon Hyperion ( Z0 0.03, e = 0.123), we get β/¯ ∆ν 3.5 system for a time long enough to extract the frequencies whose | | ≈ ≈ ∼ 1. According to condition (12), resonance overlap occurs for amplitudes may be dominating. However, very small frequencies Hyperion, which shows chaotic rotation (Harbison et al. 2011). (corresponding to very long periods) have no perceptible effect The single planet case can be treated analytically. However, on the spin, since they are engulfed by the synchronous reso- we can also study it numerically, in order to understand more nance (case where β¯ ∆ν). Thus, in general a short integration complex situations. We can integrate the orbital motion of the of the system (104 times the longest orbital period) is enough to two-body problem with an eccentricity e = 0.165 (mid-value of characterize the spin dynamics. the orbits of Mercury and Hyperion), which provide us a time As an example, we apply the numerical study to the KOI- series for X. We then compute a fast Fourier transform of the 1599 system, whose planets were detected using the transit quantity x (Eq. (7)). The frequencies νk/n = k/2 immediately method (Steffen & Hwang 2015) and validated with transit time show up as well as the corresponding amplitudes xk . Using variations (Panichi et al. 2019). This system is a good candi- | | expression (11) we obtain the rotational libration width corre- date for a rich spin dynamics because it shows some transit time sponding to the resonance islands θ˙ = νk for a given value of Z0 . variations (Delisle et al. 2017). The best stable fits yield a two | | In Fig.1 we horizontally draw the width of the libration islands super-Earth system in nearly circular orbits and involved in a 3/2 centered at each νk for the Z0 values of Mercury and Hyperion. mean motion resonance (Table1). We integrated the KOI-1599 | | This plot clearly shows why Mercury can be trapped in isolated system for 1000 years, and then performed a frequency analysis spin-orbit resonances, while Hyperion shows chaotic motion. of the forcing function x (Laskar 1993). In Table2, we show the results for the outer orbit because ∆ν/n is larger, i.e., the individ- 3.2. Multi-planet systems ual resonant islands are more distant. We observe that the classic perturbing terms at half-integers of the mean motion (Sect. 3.1) In general, planets are not alone in their orbits. For widely spaced are still present, but with very low amplitude, since the eccen- orbits, as in the solar system, planet–planet mutual interactions tricity is always very small (e < 0.025). The three dominating are relatively weak, and only modify the orbital parameters on terms correspond to harmonics of the mean motion, n, and the secular timescales. As a result, the single planet case from the orbital libration frequency, υ, with similar amplitudes. previous section is a good approximation. However, a large num- We need to estimate the rotational libration width βk for ber of exoplanets has been observed in compact configurations each frequency in Table2 to determine the behavior of the and/or showing resonant interactions. For these systems, the spin. Since we know the orbital forcing amplitudes, the only

A102, page 3 of 12 A&A 630, A102 (2019)

Table 2. Quasi-periodic decomposition of the orbital forcing function x Z -6 | 0| = 1.5 x 10 for the outer planet in the KOI-1599 system (Eq. (8)). 0.01 2

k ν · νk/n 1 xk ϕk (◦) 0.005 1.5 n υ g − | |

2 0 0 0.000000 0.93142 173.21 - 1 n

/ 0 1 2 1 0 0.003316 0.25878 79.54 0 γ 2 1 0 0.003316 0.24821 86.87 2 −2 0 −0.006632 0.04461 0.54 3 −1 1− 0.498128 0.04138 150.57 -0.005 0.5 3− 0 −1 0.501445 0.03730 123.10 2 2− 0 0.006632 0.02197 − 14.13 − -0.01 0 3 2 1 0.494812 0.01212 64.24 -90 -60 -30 0 30 60 90 − − γ 3 1 1 0.504761 0.01070 143.24 0 (deg) 1 0− 1 0.501445 0.00753 70.49 − − Z -7 2 3 0 0.009949 0.00676 85.79 | 0| = 1.0 x 10 − − − 0.01 2 Notes. We provide the dominating terms of x by decreasing ampli- 3 1 tude ( xk > 5 10− ). We have νk = 2 k ν (Eq. (10)), and ϕk = arg xk. ν = (n|, υ,| g) is× the fundamental frequencies· vector of the orbital motion, 0.005 1.5 1 1 where n = 112.305 rad yr− is the mean motion, υ = 0.744857 rad yr− 1 the libration frequency, and g = 0.324472 rad yr− the precession rate. - 1

− n / 0 1 0 γ

1 -0.005 0.5

0.8 -0.01 0 Z -6 -90 -60 -30 0 30 60 90 | 0| = 1.5 x 10 Earth-size γ 0.6 0 (deg) | k Fig. 3. Frequency map analysis of KOI-1599 outer planet rotation angle x | Z -7 6 7 | 0| = 1.0 x 10 Super-Earth assuming a rigid body with Z0 = 1.5 10− (top) or Z0 = 1.0 10− 0.4 (bottom). Each dot corresponds| | to a grid× with 181 201| | different× ini- tial conditions of γ(0) and γ˙(0), respectively, with×γ(t) = θ(t) λ(0). The color index gives the derivative ∂η/∂γ˙(0), where η is the main− fre- 0.2 quency of γ(t). The blue dots correspond to resonant motion, green dots to nonresonant regular motion, and red dots to chaotic motion.

0 -0.00662 -0.00331 0 0.00331 0.00662 integrator. In Fig.3 we show a frequency map analysis of the νk /n - 1 rotation angle (Laskar 1993; Leleu et al. 2016) for the two Z | 0| Fig. 2. Fast Fourier transform of the orbital forcing x (Eq. (7)) for the values in the neighborhood of the synchronous resonance. We outer planet in the KOI-1599 system. The horizontal lines give the conclude that our predictions for the spin of the outer planet 6 7 amplitudes βk for Z0 = 1.5 10− and Z0 = 1.0 10− . based on the Chirikov(1979) general criterion are correct: for | | × | | × 7 Z0 10− the planet can present stable asynchronous rotation, | | ∼ 6 while for Z0 10− it likely presents chaotic rotation. uncertainty is related to the permanent deformation of the planet, | | ∼ Z Z = . 6 0 (Eq. (11)). We can adopt the Earth’s value, 0 1 5 10− 4. Tidal deformation and forced oscillations because KOI-1599 planets are likely rocky, as their| | densities× are similar to that of the Earth. However, their surface gravity is a Planets are not completely rigid bodies. They deform under tidal bit stronger, so maybe the inhomogeneities in the mass distribu- interactions with the parent star because the mass distribution tion are smaller, such that Z 10 7. These two values can be inside the planet adjusts to the tidal potential (Darwin 1879, | 0| ∼ − seen as maximal and minimal estimations for Z0 , respectively. 1880). Viscoelastic rheologies have been shown to reproduce the In Fig.2 we horizontally draw the libration width| | of the islands main features of tidal effects. The planet can respond as an elas- centered at each νk for both Z0 values over the Fourier spectra of tic solid or as a viscous fluid, depending on the frequency of the | | 7 the orbital forcing x. We observe that, for Z0 10− , each island perturbation (for a review see Henning et al. 2009). is isolated, so the rotation can be trapped| in| any ∼ of them, which One of the simplest viscoelastic models is to consider that the includes the synchronous resonance, along with the super- and planet behaves like a Maxwell material, which is represented by 6 sub-synchronous resonances. On the other hand, for Z0 10− , a purely viscous damper and a purely elastic spring connected in the libration widths overlap, so we can expect chaotic| rotation.| ∼ series (e.g., Turcotte & Schubert 2002). More complex models, In order to study in more detail the spin dynamics, we have such as the Andrade model (Andrade 1910), can be thought of as to run direct numerical simulations of the rotation angle of the the Maxwell model equipped with extra terms describing hered- outer planet. We then integrated Eq. (1) together with the orbital itary reaction of strain to stress (Efroimsky 2012). Therefore, the solution of the KOI-1599 system given by a classical N body main features of tidal response are similar to the Maxwell model. − A102, page 4 of 12 A. C. M. Correia and J.-B. Delisle: Spin-orbit coupling for close-in planets

For simplicity, we adopt the Maxwell viscoelastic rheology. with The viscous response of the planet to the tidal excitation is mod- X τ zk = Z0 δr,k + br k x¯k+lyl, (24) eled by the parameter , which corresponds to the relaxation A − time of the planet, while the elastic response in modeled by the l parameter τe. The deformation Z follows (Correia et al. 2014) and

Z + τZ˙ = Z0 + Ze + τeZ˙e, (15) 1 + i(k ν)τe bk = · . (25) 1 + i(k ν)τ with ·  3 The bk function depends on the tidal model, in this case, the kf M R Z = x¯ e2iθ, where = , (16) Maxwell model. Different tidal models would imply different e A A 4 m a expressions for bk, but for all models we get b0 = 1. kf is the fluid second Love number for potential, and x¯ is the com- In the case of very weak or no forced oscillations, expres- plex conjugate of x. The function Ze is the equilibrium shape, i.e., sions (17) and (18) simplify as the maximal deformation that the planet can attain owing to the r ν tidal perturbation. θ = θ0 + · t, (26) 2 We denote the average rotation rate of the planet by ω = θ˙ . 2iθ 2iθ0 ir νt For simplicity, we assume that θ has a small forced amplitudeh i e = e e · , (27) of libration due to orbital forcing and a zero free amplitude of 2iθ0 that is, y0 = e , and yk,0 = 0. Replacing in expression (24), we libration, such that get for the deformation of the planet X ik νt θ = ω + θ , 2iθ0 t ke · (17) zk = Z0 δr,k + br k e x¯k. (28) k A − j  P ik νt Finally, to get the permanent tidal deformation, we can X i k 2θke · X 2iθ 2iωt i(2ω+l ν)t average over one rotational period e = e = yle · , (18) j! j 0 l 2iθ0 Zt = Z Z = e x¯r, (29) ≥ h i h i − 0 A where all θ are small, and the series is absolutely convergent P k or ( θk converges). Each coefficient yl can be expressed as a k | | power series of the coefficients θk, by identifying terms on both Zt = xr . (30) sides of Eq. (18). |h i| A | | From expressions (8), (16), and (18), we obtain For instance, in the classical case of a single planet in an   elliptical orbit (Sect. 3.1), we have (Eq. (13)) X X    i(2ω k ν)t 3,2 2i$ Ze =  x¯k+lyl e − · , (19) x¯ = − (e) − , (31) A   k k e k l X thus, for a planet trapped in the νk/n = k/2 resonance and replacing this in expression (15), we get 3,2 = − . t/τ Zt k (e) (32) Z = Z0 + Zt + Ce− , (20) |h i| AX For the outer planet in the KOI-1599 system (Sect. 3.2), we with have for the synchronous resonance (Table2)   X X  1 + i(2ω k ν)τ 6   e i(2ω k ν)t . . − , Zt =  x¯k+lyl − · e − · . (21) x[2,0] 0 93 Zt 2 5 10 (33) A   1 + i(2ω k ν)τ | | ≈ ⇒ |h i| ≈ × k l − · while for the super- and sub-synchronous resonances t/τ In the following, we neglect the term Ce− because it vanishes 7 x[2, 1] 0.25 Zt 7.0 10− . (34) after some time, and we only consider the solution Z = Z0 + Zt. | ± | ≈ ⇒ |h i| ≈ × We note that Z (Eq. (29)) does not depend on the tidal h ti 4.1. No resonant rotation model function bk (Eq. (25)). Indeed, Zt is independent of the tidal model, as it corresponds to the maximalh i deformation that We assume that ω , νk = k ν/2 for all k (no resonant rotation). the planet can achieve while trapped in a given resonance. Then, over one rotational period· we have (Eq. (21)) Z = 0, (22) 4.3. Resonant rotation with forced oscillations h ti that is, there is no permanent tidal deformation. In case of resonant motion (ω = νr), replacing the deformation of the planet, Z (Eq. (23)), in expression (2), we get     4.2. Resonant rotation without forced oscillations X X      ik νt θ¨ =   xk+l+my¯lzm e ·  , (35) B =     The situation changes if the rotation is locked in some spin-orbit k l,m resonance, that is, we assume that ω = νr = r ν/2 (Eq. (10)). We can then rewrite expression (20) as · and from expression (17) X X i(r k) νt 2 ik νt Z = zke − · , (23) θ¨ = (k ν) θke · . (36) − · k k A102, page 5 of 12 A&A 630, A102 (2019)

Then, by identifying terms in both series, we find terms. Applying to the case of Mercury, we have r = 3, ν = n, and Z0 . Thus, expression (42) becomes 1 | |  A θk = (αk α¯ −k) , (37) i θ 2i − 18 e 2 0 αk Z¯0 x¯3 k. (43) ≈ k2 − where The only αk , 0 is for x3 k = x2, that is, for k = 1. Then, the X − αk = B x¯l+m−kylz¯m. (38) rotation angle becomes (Eq. (39)) (k ν)2 · l,m 3 θ nt + θ0 + 18 Z0 x2 sin(nt + φ1). (44) We also note that θ k = θ¯k. Therefore, with φk = arg αk, we ≈ 2 | | − can rewrite expression (17) as This simple approximation gives a forced libration period X equal to the orbital period, and a forced libration amplitude θ = νr t + θ0 + αk sin(k ν t + φk). (39) α = | | · 1 18 Z0 5600, in perfect agreement with a full numerical k,0 |simulation| ≈ | for| Mercury (see Fig. 2 in Peale et al. 2007).

We recall that the coefficients yl are expressed as power series of the coefficients θk (Eq. (18)), and that the coefficients 4.3.2. Multi-planet systems z y k are linear combinations of the coefficients l (Eq. (24)). We consider the outer planet in the KOI-1599 system, for which Equations (18), (24), and (37) thus define a closed system of the dominating terms of the orbital forcing are x = 0.93, equations, which can be solved for the amplitudes of the forced [2,0] x[2,1] = 0.26, and x[2, 1] = 0.25 (Table2), and we neglect the oscillations θk and for the deformation of the planet zk. − remaining terms. In this case, we have Z0 , so replacing We can estimate the order of magnitude of the expected expression (24) in (42) gives | |  A forced oscillations. Since xk 1, yk 1 and zk = | | ≤ | | ≤ | | ≤ Z max( Z0 , ), we deduce from expression (37) 18 n2 ei2θ0 X X | | A α − y , k A 2 bm r x¯m k xm+l ¯l (45) !2 ≈ (k ν) − n · m l θ . k BZ 2 18 (40) | | ≤ (k ν) ≈ Z 2νk which depends on the tidal model through bm r (Eq. (25)). For · − simplicity, we further assume that τ = 0 and τνk 1 (high For an Earth-size planet and a perturbing frequency on the e  5 frequency regime, see Sect.5), such that bm r δr,m. Then order of the orbital frequency (νk n), we get θk . 10 , that − ≈ ∼ | | − is, the amplitude of the forced oscillations remains very small. 18 n2 ei2θ0 X αk A x¯r−k xr+l y¯l. (46) We then linearize expression (18) and find ≈ (k ν)2 · l i2θ0 i2θ0 i2θ0 y = e , and yl 2i θl e = (αl α¯ −l) e . (41) 0 ,0 ≈ − Let us consider, for example, that the planet is captured in the The coefficients zk thus become linear in θk, so we only need r = [2, 1] spin-orbit resonance, with frequency ν[2,1] = n + υ/2. The only two αk , 0 are for x¯[2,1] k = x¯[2,0] and x¯[2, 1], that is, for to solve a system of linear equations to get the amplitudes of − − forced libration, θk, and the corresponding corrections to the k = [0, 1] and [0, 2], respectively, with associated frequencies υ deformation of the planet, zk. For planets involved in mean- and 2υ, where υ is the orbital libration frequency motion resonances, we may have νk/n 0.01 (Delisle et al.  2 X ∼ i2θ0 n 2017), so the amplitude of forced oscillations is expected to be αυ 18 e x¯ , x , +l y¯l, (47) ≈ A υ [2 0] [2 1] stronger ( θk . 0.1). In this case, higher order terms should be l | | included and we would need to solve a system of polynomial  2 X i2θ0 n equations (see Delisle et al. 2017, Appendix D). α2υ 18 e x¯[2, 1] x[2,1]+l y¯l. (48) ≈ A υ − For the sake of simplicity and readability, we restrict our 2 l study to the first order case in the following, as it already provides Since α /α 0.067, we can also neglect the contribution a good approximation in most cases. Then, from expressions (38) | 2υ υ| ≈ and (41) we have of α2υ. Thus, the rotation angle (Eq. (39)) simply becomes 1 θ θ (n + υ) t + θ + α sin(υt + φ ), (49) 18 n2 ei2 0 X ≈ 2 0 | υ| υ αk x¯m−kz¯m, (42) ≈ (k ν)2 and the rotation rate · m h i θ˙ n + υ 1 + α cos(υt + φ ) . (50) where z¯m is given by expression (24). In order to be captured in ≈ 2 | υ| υ a given resonance (ω = νr), we have that xr , 0 (Sect.3). Since we always have k , 0 (Eq. (39)), it is needed that at least another From expression (41) we additionally have xm,r , 0 to observe forced librations. Indeed, if there is only one X x[2,1]+l y¯l = x[2,1]y¯[0,0] + x[2,0]y¯[0, 1] + x[2, 1]y¯[0, 2] orbital forcing term, as in the case of a single planet in a circular − − − l orbit, all αk = 0, and hence there are no forced librations. 2iθ0
e− x , x , αυ . (51) ≈ [2 1] − [2 0] 4.3.1. Single planet case Inserting in expression (47) and solving for αυ we finally get We first consider a single planet in an elliptical orbit with small 2 18 (n/υ) x¯[2,0] x[2,1] eccentricity. The two dominating terms of the orbital forcing are αυ A . (52) 2 2 x 1 and x 7e/2 (Sect. 3.1), so we neglect the remaining ≈ 1 + 18 (n/υ) x[2,0] 2 ≈ 3 ≈ A | | A102, page 6 of 12 A. C. M. Correia and J.-B. Delisle: Spin-orbit coupling for close-in planets

1.5 In addition, assuming very small or no forced oscillations, that 4 2iθ0 τn = 10 yr is, θ θ = θ0 + ω t (Eq. (17)), we have y0 = e and yk,0 = 0, ) 1 ≈ h i -3 so the average net torque can be simplified as 0.5 X 2 (2ω k ν)τ 0 (ω) = xk − · , (55)

)/ n (x 10 2 2

r T −K | | + ω k ν τ k 1 (2 ) -0.5 − · - ν ( θ -1 where   2 5   ) 6 τe 3GM R τe

-7 = = . 1 kf 6 1 (56) 4 K AB − τ 2Ca − τ | (x 10

t 2 The initial rotation rate of the planets in unknown. A small

| Z 0 number of large impacts at the end of the formation process 0 20 40 60 80 100 can significantly change the spin rate or direction (Dones & time (Myr) Tremaine 1993; Kokubo & Ida 2007). However, since plan- Fig. 4. Forced libration of the rotation rate for the super-synchronous ets contract from an initial larger protoplanet, in general it is resonance (νr = n + υ/2) of the outer planet in the KOI-1599 system. expected that they rotate fast. Indeed, based on the solar sys- 4 We show the evolution of (θ˙ νr)/n adopting τe = 0 and τn = 10 . tem planets, MacDonald(1964) proposed an empirical relation − ω m4/5/R2, which leads to about 12 h for Earth-size planets. 0 ∝ Therefore, in general we expect ω0 k ν, and since τe < τ, the With = 2.7 10 6 (Eq. (16)), and (n/υ) 150 (Table2), we  · A × − ≈ initial rotation rate can only decrease (Eq. (55)). estimate αυ 0.14 rad. From expression (50) we also estimate As the planet arrives in a regime where ω k ν the situa- | | ≈ 4 ∆θ/˙ n = (υ/n) αυ 9 10− . In Fig.4 we show the forced libra- τ ∼ · | | ≈ × tion changes. Depending on and the amplitudes xk , the torque tions of the rotation rate for the r = [2, 1] spin-orbit resonance, may become positive, and hence create stable equilibria| | for the which is obtained performing a numerical simulation of the KOI- rotation rate. In Fig.5 we draw the net torque as a function of τ 4 2 1599 system with τn = 10 (that is, τυ 10 ). We observe that for two different orbital forcing functions. The left-hand column the libration frequency of the spin is indeed∼ υ, and that the forced 3,2 represents the case of Mercury, for which xk = k− (e) with libration amplitude also matches the estimated value. e = 0.206, while the right-hand column is for| the| outerX planet in We can now estimate the correction that the forced libration the KOI-1599 system, for which xk are given in Table2. introduces in the tidal deformation (Eq. (24)) as follows: We observe that, for small τ| values,| only one equilibrium
point near the synchronous rotation is possible. However, for Zt x¯[2,1]y[0,0] + x¯[2,0]y[0, 1] h i ≈ A − large τ values, several equilibria are possible, and they coincide 2iθ0
= e x¯ , x¯ , α¯ υ exactly with the resonant islands identified in Sect.3. This result A [2 1] − [2 0] ! was already known for the classical case of a single planet in an 18 (n/υ)2 x 2 2iθ0 [2,0] = e x¯[2,1] 1 A | | . (53) eccentric orbit (see Correia et al. 2014), but it is remarkable that 2 2 A − 1 + 18 (n/υ) x[2,0] it prevails for any kind of orbital forcing. This property of the A | | tidal torque enhances the chances of observing the rotation of We thus obtain for the outer planet in the KOI-1599 system, 7 close-in planets trapped in some asynchronous resonance. Zt 3.6 10− , which is about half of the value without con- sidering|h i| ≈ the× forced librations (Eq. (34)); this value corresponds to what is observed in the numerical simulations (Fig.4). We 5.2. Capture probabilities conclude that the forced libration contributes to decreasing the The total torque on the rotation (Eq. (2)) can be written as tidal deformation (see also Delisle et al. 2017, Appendix D).  2iθ  2iθ θ¨ = Z x e− + Zt x e− , (57) B = 0 B = 5. Tidal dissipation and capture probabilities since Z = Z0 + Zt. The average of the second term is the tidal The tidal deformation of the planet is not instantaneous; there torque (ω) (Eq. (55)). The average of the first term is different T is a delay between the perturbation and maximal deformation from zero when the rotation rate is close to some resonant island, that leads to a nonzero net torque on the spin of the planet. As a ω νr, with νr = r ν/2 (Eq. (9)): ≈ · consequence, tidal effects slowly modify the average rotation rate D  2iθE  2i(γ φ) Z x e− = Z xr e− − , (58) of the planet and may drive it into the different configurations B = 0 B = 0 described in Sect.3. 1 where γ = θ νr t + φ, and φ = arg(Z x ) is a constant phase. − 2 0 r 5.1. Tidal dissipation Thus, the average of the total torque (Eq. (57)) becomes β2 Replacing the tidal deformation of the planet Zt (Eq. (21)) in the r γ¨ = sin 2γ + (νr + γ˙), (59) general expression of the torque (Eq. (2)), and averaging over the − 2 T fundamental frequencies of the system we get where β2 = 2 Z x (Eq. (11)). If Z = 0, capture could only D  E r 0 r 0 2iθ be possible atB| the equilibrium| of the tidal torque, that is, when = Zt x e− T B = = 0. However, in a more general situation, the tidal torque   T X 1 + i(2ω k ν)τ  can be counterbalanced by the conservative torque resulting from  e  =  xk+l x¯k+mymy¯l − ·  . (54) βr. Therefore, even when , 0, each time the planet crosses a AB =  1 + i(2ω k ν)τ  T k,l,m − · resonant island, there is some chance of being captured.

A102, page 7 of 12 A&A 630, A102 (2019)

Mercury KOI-1599 outer

τn = 10-1 τυ = 10-1

0 0

τn = 100 τυ = 100

0 0

τn = 101 τυ = 101

0 0

τn = 102 τυ = 102

0 0

0 0.5 1 1.5 2 2.5 3 -0.006 -0.004 -0.002 0 0.002 0.004 0.006 ω/n ω/n-1 3,2 Fig. 5. Normalized net torque / (Eq. (55)) for different τ values and two orbital forcing functions: Mercury, for which xk = −k (e) and e = 0.206 (left column); the outerT planetK in the KOI-1599 system, for which the x are given in Table2( right column). | | X | k|

Goldreich & Peale(1966) computed a first estimation of the resonance). The “energy” variation of the planet after a cycle capture probability Pcap, and subsequently more detailed studies around the resonance is then given by the function proved their expression to be essentially correct (for a review, Z γ2 Z γ3 see Henrard 1993). We follow the same approach described in ∆h(γ ) = 2 dγ + 2 dγ, 1 T T Sect. 5.3 of Correia & Laskar(2010). Expression (59) can be γ1 γ2 rewritten as where γ1 is the γ value when the planet crosses the separatrix dh dγ between the circulation and libration zones. The values γ2 and = 2 , (60) dt T dt γ3 are the first two following γ values corresponding to γ˙ = 0. In order to be captured, after a cycle inside the resonance, the planet where must remain within the libration zone, that is, the “energy” vari- 2 β ation after a cycle must be negative ∆h(γ1) < 0. Assuming that h = γ˙ 2 r cos 2γ (61) − 2 γ1 is uniformly distributed between π/2 and π/2, capture inside the resonance occurs for “energy” variations− within is an integral of motion in the conservative problem (Sect.3), which can be related to the total energy. The separatrix equation : ∆h( π/2) < ∆h(γ1) < 0 , (62) 2 2 {E − } between dynamical regimes is h = βr /2, where h > βr /2 gives from a total of possibilities the trajectories in the circulation zone (outside the resonance) < β2/ T : ∆h( π/2) < ∆h(γ1) < ∆h(π/2) , (63) and h r 2 the trajectories in the libration zone (captured in {E − } A102, page 8 of 12 A. C. M. Correia and J.-B. Delisle: Spin-orbit coupling for close-in planets

100 100

80 80

60 60 (%) (%) cap cap P P 40 40

20 20

0 0 -4 -2 0 2 4 6 10-3 10-2 10-1 100 101 102 103 10 10 10 10 10 10 τ n τn Fig. 6. Capture probabilities in the presently observed 3/2 spin-orbit Fig. 7. Capture probabilities for the super-synchronous resonance νr = 5 7 resonance for Mercury (e = 0.206 and Z = 1.25 10− ). The solid n+ν/2 for the outer planet in the KOI-1599 system with Z0 = 10− . The 0 | | line gives the theoretical approximation given| | by expression× (66), while solid line gives the theoretical approximation given by expression (66) the dots give the results of numerical simulations with Eqs. (1) and (15). with all terms from Table2, while the dashed line is obtained with only We run 180 initial conditions with θ0 differing by one degree. the first three dominating terms. The dots give the results of numerical simulations with Eqs. (1) and (15). We run 180 initial conditions with θ0 differing by one degree. that is, R γ π d 1 ∆h( 2 ) P = E = − . (64) cap R π π remain in the 3/2 spin-orbit equilibria even with Z = 0 (Correia dγ1 ∆h( 2 ) ∆h( 2 ) 0 ET − − et al. 2014). 2 In Fig.7, we show the theoretical capture probabilities Usually the tidal torque is very weak, that is, (˙γ) βr , 2 |T |  (Eq. (66)) for the super-synchronous resonance νr = n + ν/2 for which gives γ3 π/2, γ2 π/2, and h βr /2. Then, from expression (61) we' − have ' ≈ the outer planet in the KOI-1599 system (r = [2, 1], Table2). We adopt Z = 10 7, since for larger values of Z the libra- | 0| − 0 γ˙ = sign(˙γ) βr cos γ, (65) tion widths of the individual resonances overlap, and thus it is no longer possible to capture the rotation in individual reso- and replacing it in expression (64) we finally get nances (see Fig.2). As for the case of Mercury, we observe Z π that for τn 1 the probability stabilizes at a constant value T 2 (54%), while for τn 1 the probability is 100%. However, Pcap = 1 − , with T = (νr βr cos γ) dγ. (66)  − T+ ± π T ± there are two transitions of regime. The main transition occurs − 2 around τn 102 τυ 1 because the dominating terms of This expression is valid for any kind of tidal torque. For the the orbital∼ forcing⇔ are those∼ involving the orbital libration fre- Maxwell rheology that we adopt in this paper, we use expression quency, υ (Table2). If we only consider the main three terms of (55) for , which leads to T the orbital forcing, this transition, for which the capture proba- X " + δ+ ! + δ !# bility drops to zero, would be the only observed (green dashed 2 1 1 k 1 1 −k T = xk ln ln , (67) line). The other transition occurs around τn 1 and results from ± −K | | ∆+ 1 δ+ − ∆ 1 δ ∼ k k − k −k − −k terms that involve the precession rate of the orbit, g. These terms have smaller amplitudes than in the case of Mercury because p 2 2 with ∆ = η (λk i) , δ = ( (η λk) i)/∆ , η = 2τβr, e 0.02 (Table1), but they contribute to increase the capture ±k − ± ±k ± − − ±k ± ≈ and λk = τ(r k) ν. The in η corresponds to T . Since both η probabilities in the super-synchronous resonance. − · ± ± and λk depend on τ and Z0 (through βr), the chances of capture In Fig.7, we additionally show the results from numer- in each resonant island change| | with these two parameters. ical simulations (Eqs. (1) and (15)). Contrary to the case of In Fig.6, we show the capture probabilities in the presently Mercury (Fig.6), in this case the agreement between the the- observed 3/2 spin-orbit resonance for Mercury (e = 0.206 and oretical and numerical capture probabilities is not so good. For 5 2 Z0 = 1.25 10− ). We see there is a very good agreement τn 10 , this is because the theoretical probabilities (Eq. (66)) between| | the× theoretical approximation (Eq. (66)) and the results were estimated using the approximation (59), which assumes of numerical simulations (Eqs. (1) and (15)). For τn < 0.1, the that the effect of the nearby resonances can be averaged to probability is constant and equal to about 8.5%, the same value zero. However, although isolated resonant islands exist for Z0 = 7 | | obtained with a linear tidal model (Correia & Laskar 2004). For 10− , there is already a small chaotic zone around these islands τn 1, the probability slightly decreases and then increases very (Fig.3, bottom). Therefore, in the numerical simulations the fast.∼ For τn > 10, the probability is always 100%. The transition rotation is chaotic for some time before it is stabilized in one occurs because for τn 1 the tidal torque changes from a regime of these islands (see Sect.6) and the probabilities are altered. proportional to the forcing∼ frequency to a regime inversely pro- For τn 102, in addition to the previous effect, there is a  portional (Eq. (55)). As a result, for τn > 1, multiple equilibria significant increase in the permanent tidal deformation Zt > 7 h i for the tidal torque are possible (Fig.5). One of this equilibrium 10− (Eqs. (33) and (34)) that extends the size of the chaotic points is placed exactly at ω/n = 3/2, so the rotation rate of the zone (Fig.3, top). As a result, the rotation may remain perma- planet is naturally driven to the corresponding resonant island nently chaotic or evolve into some totally unexpected spin-orbit and capture is unavoidable. Indeed, for τn > 10, the planet can resonance (see next section for more details).

A102, page 9 of 12 A&A 630, A102 (2019)

Table 3. Number of captures in each spin-orbit resonances (in percentage) for the KOI-1599 outer planet using different τ values.

4 3 2 1 0 1 2 3 4 5 6 τn 10− 10− 10− 10− 10 10 10 10 10 10 10 n + υ 5.1 9.4 3.3 n + υ/2 64.0 47.5 58.3 74.0− 10.6 − − − −59.2 − − − n 30.3 39.2 36.7 26.0 89.4 100− 100 − 100 − − − n υ/2 0.6 3.9 1.7 40.8− − − n − υ − − − − − −86.7 − Chaotic− − − − − − − − − −100 − 13.3 − − − − − − − − −

6. Complete evolution is completely outside the resonant area, which is bounded by θ/˙ n . 1.01 (Fig.3). For any τ value, the rotation rate of the planet The spin evolution of a planet is governed by Eq. (1). For a pre- decreases owing to tidal effects until it approaches the region liminary analysis, this equation can be decoupled in the shape where multiple spin-orbit resonances are present. The shape of of the planet Z (Eq. (3)) and in the orbital forcing function x the tidal torque depends on the τ value (Fig.5), so at this stage (Eq. (4)). The orbital forcing determines the equilibrium states different scenarios can occur. In addition, capture in resonance for the spin, and for a given shape we can predict the possi- is a stochastic process. Therefore, for each τ value we have run ble dynamical regimes. However, when we take into account 180 simulations with θ0 differing by one degree. In Table3 we tidal deformation, the shape itself depends on the orbital forc- list the final distribution of the spin in the different resonances ing (Eq. (20)). As a result, the exact behavior of the spin depends for each τ value. on the tidal model that we adopt. For τn 102, the spin is in the “low frequency regime” From the analysis of the orbital forcing, we can anticipate (τυ 1), which is usually known as the viscous or linear regime which systems can present a rich spin dynamics, such as the (e.g., Singer 1968; Mignard 1979). Because the relaxation time KOI-1599 system. However, a deep understanding of all possible is small, the deformation of the planet is very close to the equi- scenarios can only be achieved by performing numerical simu- librium deformation, Z Z + Z (Eq. (15)). As a consequence, ≈ 0 e lations of the complete equations of motion, namely the orbital Z Z0, that is, the behavior of the spin follows that of a rigid equations of an N-body system, which give the solution for x(t), bodyh i ≈ (Sect.3). We can thus predict the dynamics from the fre- together with Eq. (1) for the spin and Eq. (15) for the deforma- quency map analysis shown in Fig.3 (bottom). We see that tion (tidal model). At the end of the previous section, we already the rotation can be trapped in multiple spin-orbit resonances. did that for estimating the numerical probabilities of capture in a Close to the synchronous resonance, there is a small chaotic zone specific spin-orbit resonance (Figs.6 and7). where the spin can wander for some time before it finds a path In this section, we study the complete behavior of the outer into a stable resonant island. In Fig.8 we show three examples planet in the KOI-1599 system (Table1). This planet has an of capture in different spin-orbit resonances. From Table3 we estimated mass of 4.3 M , that is, it can be classified as a conclude that the most likely resonance is the super-synchronous super-Earth. An upper bound⊕ of ten Earth masses is commonly resonance with ω = n + υ/2. In the linear regime, the value of accepted to warrant that the planet is mostly rocky (e.g., Valencia τ only impacts the damping timescale, so the statistics do not et al. 2007). Therefore, the internal structure and rheology of this change much with this parameter. planet is likely similar to the terrestrial planets in the solar sys- For τn 102, the spin is in a “frequency transition regime” 7 ∼ tem. We adopt for the residual deformation Z = 1.0 10− . For (τυ 1). In this case, the deformation of the planet is still 0 × ∼ the tidal deformation we use a Maxwell model (Eq. (15)), with Z Z0, but the tidal torque has an important modification. In 2 h i ≈ kf = 0.9, τe = 0, and C/mR = 1/3. the low frequency regime it decreases linearly, allowing capture For rocky planets, such as the Earth and Mars, we have in all spin-orbit resonances. In the transition regime, however, a dissipation factor Q = 10 (Dickey et al. 1994) and Q = 80 the torque drives the rotation away from all spin-orbit equilib- (Lainey et al. 2007), respectively. We then compute for the Earth ria except for θ/˙ n 1 < 0.002, where there is an inversion τ = 1.6 day, and for Mars τ = 14.7 day. However, in the case of (Fig.5). As a result,| − only| the synchronous resonance can be the Earth, the present Q factor is dominated by the oceans, the reached (Table3). The impact of this inversion in the tidal Earth’s solid body Q is− estimated to be 280 (Ray et al. 2001), torque is also pretty clear in Fig.7, the capture probability in the which increases the relaxation time by more than one order of super-synchronous resonance sharply decreases for τn 102. magnitude (τ = 46 day). Although these values provide a good For τn 102, the spin is in the “high frequency∼ regime” estimate for the average present dissipation ratios, they appear (τυ 1). In this regime, the relaxation time is large and the  to be inconsistent with the observed deformation of the plan- planet is thus able to retain some tidal deformation Zt , 0 ets. Indeed, in the case of the Earth, the surface post-glacial (Sect.4). For the outer planet in the KOI-1599 system weh i have 4 6 rebound from the last glaciation about 10 yr ago is still going = 2.7 10− Z0, so for the main spin-orbit resonances the on, suggesting that the Earth’s mantle relaxation time is some- tidalA deformation×  dominates over the permanent deformation. As thing like τ = 4400 yr (Turcotte & Schubert 2002). Since τ can a result, the phase space of the spin is modified with respect to 7 range from a few minutes up to thousands of years, we adopt the phase space drawn for Z0 = 10− , it resembles more to that 6 6 log τ 4, which is equivalent to 4 log τn 6. for Z 10− (Fig.3, top). We therefore expect chaotic motion − ≤ 10 yr ≤ − ≤ 10 ≤ 0 ∼ In our numerical simulations we use the initial conditions for the spin in this regime. However, in our simulations we verify from Table1. The initial rotation period is set at 19.5 day, which that, although chaotic rotation is indeed a possibility for some corresponds to θ/˙ n 1.05. Since the libration frequency of the τ values, stable rotation is also possible in some asynchronous outer planet is υ/n ≈ 0.0066 (Table2), the initial rotation rate spin-orbit resonances (Table3). One reason for this is that in ≈

A102, page 10 of 12 A. C. M. Correia and J.-B. Delisle: Spin-orbit coupling for close-in planets

τn = 10-4 τn = 106

1.04 1.04 n n 1.02 1.02 θ / θ /

1.00 1.00 ) -6

1.00 1.04

Z | (x 10 0.00 | 0 2 4 6 8 10 time (Myr) n 1.02 Fig. 9. Spin evolution of the outer planet in the KOI-1599 system (top) θ / and the corresponding deformation Z (bottom) for τn = 106 (high fre- quency regime). The blue lines give| the| position of the main spin-orbit 1.00 resonances n, n υ/2, and n υ. ± ±

starts with an initial rotation rate faster than the synchronous rotation, it finally ends up with a rotation slower than that. 1.04 7. Discussion

n 1.02 In this paper we have revisited the spin-orbit coupling of close-

θ / in planets. These bodies are assumed to be tidally evolved, and thus can be trapped in some equilibrium states. We derived 1.00 our results using a very general approach, that is suitable to apply to multi-planet systems in various orbital configurations (e.g., very eccentric orbits or involved in orbital resonances). We 0 2 4 6 8 10 introduced a simple way to probe the spin dynamics from the orbital perturbations (Eq. (12)), a new method for computing the time (Myr) forced librations (Eq. (35)) and the permanent tidal deformation Fig. 8. Some examples of the spin evolution of the outer planet in the (Eq. (53)), and a general expression for the tidal torque (Eq. (55)) τ = 4 KOI-1599 system for n 10− (low frequency regime). The blue lines and for the capture probabilities in resonance (Eq. (67)). give the position of the main spin-orbit resonances n, n υ/2, and n υ. ± ± We considered that the spin of the planet is only subject to tidal torques from the parent star (Eq. (1)). Tidal torques between this regime the tidal torque facilitates the capture in all these planets are also present, but their contribution is very small resonances (Fig.5). The other reason is because the tidal defor- for the main spin-orbit resonances (see Table II in Goldreich & mation varies from resonance to resonance (Eq. (30)). For the Peale 1966), and accounts for less than a few percent of the total 6 synchronous resonance Zt 10− , so even if the rotation is tidal dissipation (Hay & Matsuyama 2019). In addition, the posi- somehow captured there,|h it becomesi| ∼ chaotic after some time. For tions of the spin-orbit resonances do not change because they 7 the super-synchronous resonance Zt 3.6 10− (Eq. (53)), depend on the fundamental frequencies of the system; the addi- so it is not impossible to remain trapped|h i| ≈ there,× provided that the tional torques only modify the libration width of the resonant spin finds a way in. islands. However, if we consider a system with two or more stars, In Fig.9 we show an example for τn = 106, which clarifies our model should be corrected to include the torques from all the behavior of the spin in the high frequency regime. In this bodies on the spin (Correia et al. 2015; Leleu 2017). figure we show the evolution of the rotation rate (top) together In our model we assumed for simplicity that the spin axis of with the evolution of the shape (bottom). Before the spin reaches the planet is normal to the orbit, since this is the final outcome of 7 the resonant area, we have Z = Z0 = 10− , so in principle the tidal evolution (Hut 1980). For multi-planet systems, the obliq- rotation can be trapped in ah stablei spin-orbit resonance (Fig.3, uity does not evolve exactly to zero degrees, it is rather captured bottom). As soon as the spin enters the resonant area, the defor- in an equilibrium configuration known as Cassini state (Colombo 6 mation increases and can experience values Z > 10− . Then, 1966; Peale 1969). Although high obliquity states exist, they are the rotation becomes chaotic (Fig.3, top). However,| | just outside usually unstable under strong tidal interactions. Therefore, the the chaotic zone there exists the sub-synchronous resonance with only stable equilibrium state has in general very small obliquity ω = n υ. Somehow, after some time in the chaotic region, the (less than 1◦), for which our model applies. We note, however, spin finds− a path into this stable region. Once there, the defor- that this may not be the case for planetary systems showing large 7 mation of the planet decreases again to a value Z 10− . Thus, mutual inclinations (Correia 2015). the sub-synchronous resonance is no longer connected| | ∼ with the We also assumed coplanar orbits. This restriction was used to chaotic region and the spin can remain there forever (Fig.3, simplify the orbital forcing function x (Eq.(6)), since this func- bottom). In this example, it is remarkable that although the spin tion has only two dimensions in the planar case. If the system

A102, page 11 of 12 A&A 630, A102 (2019) is not coplanar we can still use our model because the forcing Correia, A. C. M., & Robutel, P. 2013, ApJ, 779, 20 function captures the effect of the inclination. It is nevertheless Correia, A. C. M., & Rodríguez, A. 2013, ApJ, 767, 128 convenient to align the inertial plane where we compute the forc- Correia, A. C. M., Udry, S., Mayor, M., et al. 2008, A&A, 479, 271 Correia, A. C. M., Boué, G., Laskar, J., & Rodríguez, A. 2014, A&A, 571, A50 ing function with the mean orbital plane of the planet whose Correia, A. C. M., Leleu, A., Rambaux, N., & Robutel, P. 2015, A&A, 580, L14 rotation we want to study. This alignment is also important to Darwin, G. H. 1879, Philos. Trans. R. Soc. London, 170, 1 ensure that the obliquity remains close to zero. Darwin, G. H. 1880, Philos. Trans. R. Soc. London, 171, 713 In our model we only considered gravitational tidal torques. Delisle, J.-B., Correia, A. C. M., Leleu, A., & Robutel, P. 2017, A&A, 605, A37 Dickey, J. O., Bender, P. L., Faller, J. E., et al. 1994, Science, 265, 482 For Earth-size or larger planets, we can expect that a dense atmo- Dobrovolskis, A. R. 1980, Icarus, 41, 18 sphere is present. As a result, thermal atmospheric tides may give Dones, L., & Tremaine, S. 1993, Icarus, 103, 67 rise to additional equilibrium states (Correia & Laskar 2001). Efroimsky, M. 2012, Celest. Mech. Dyn. Astron., 112, 283 However, it has been shown that gravitational tides dominate for Genova, A., Iess, L., & Marabucci, M. 2013, Planet. Space Sci., 81, 55 close-in planets with a . 0.3 au (Correia et al. 2008; Leconte Gold, T., & Soter, S. 1969, Icarus, 11, 356 Goldreich, P., & Peale, S. 1966, AJ, 71, 425 et al. 2015). Thus, as long as we stay in this regime, the presence Goldreich, P., & Peale, S. 1967, AJ, 72, 662 of an atmosphere should not modify the global picture drawn in Goldreich, P., & Soter, S. 1966, Icarus, 5, 375 this paper much. Goldstein, R. M. 1964, AJ, 69, 12 We have shown that the spin dynamics of Earth-size plan- Harbison, R. A., Thomas, P. C., & Nicholson, P. C. 2011, Celest. Mech. Dyn. Astron., 110, 1 ets in compact systems can be very rich. In particular, we have Hay, H. C. F. C., & Matsuyama, I. 2019, ApJ, 875, 22 shown that synchronous rotation is not the only possibility; the Henning, W. G., O’Connell, R. J., & Sasselov, D. D. 2009, ApJ, 707, 1000 rotation is allowed to evolve into some unexpected spin-orbit res- Henrard, J. 1993, in Dynamics Reported (New York: Springer Verlag), 117 onances or even present chaotic behavior. In the high frequency Hughes, S. 1981, Celest. Mech., 25, 101 regime, we have shown that the tidal torque has equilibrium Hut, P. 1980, A&A, 92, 167 Kokubo, E., & Ida, S. 2007, ApJ, 671, 2082 points at exactly the same position as the spin-orbit resonances, Lainey, V., Dehant, V., & Pätzold, M. 2007, A&A, 465, 1075 which enhances the chances of capture in these equilibria. Future Laskar, J. 1993, Physica D, 67, 257 studies can take into account all the neglected contributions in Leconte, J., Wu, H., Menou, K., & Murray, N. 2015, Science, 347, 632 our model, which may increase the diversity of spin states for Leleu, A. 2017, ArXiv e-prints [arXiv:1701.05585] Leleu, A., Robutel, P., & Correia, A. C. M. 2016, Celest. Mech. Dyn. Astron., these planets even more. 125, 223 MacDonald, G. J. F. 1964, Revs. Geophys., 2, 467 Acknowledgements. We thank G. Boué, A. Leleu, and P. Robutel for discussions. Mignard, F. 1979, Moon Planets, 20, 301 A.C. acknowledges support by CFisUC strategic project (UID/FIS/04564/2019), Morbidelli, A. 2002, Modern Celestial Mechanics: Aspects of Solar System ENGAGE SKA (POCI-01-0145-FEDER-022217), and PHOBOS (POCI-01- Dynamics (London: Taylor & Francis) 0145-FEDER-029932), funded by COMPETE 2020 and FCT, Portugal. J.B.D. Murray, C. D., & Dermott, S. 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