The Astrophysical Journal Letters, 744:L23 (5pp), 2012 January 10 doi:10.1088/2041-8205/744/2/L23 C 2012. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

PUMPING THE ECCENTRICITY OF BY TIDAL EFFECT Alexandre C. M. Correia1,2, Gwenael¨ Boue´2,3, and Jacques Laskar2 1 Department of Physics, I3N, University of Aveiro, Campus de Santiago, 3810-193 Aveiro, Portugal; [email protected] 2 Astronomie et Systemes` Dynamiques, IMCCE-CNRS UMR8028, 77 Av. Denfert-Rochereau, 75014 Paris, France 3 Centro de Astrof´ısica, Universidade do Porto, Rua das Estrelas, 4150-762 Porto, Portugal Received 2011 October 20; accepted 2011 November 21; published 2011 December 19

ABSTRACT Planets close to their host are believed to undergo significant tidal interactions, leading to a progressive damping of the . Here we show that when the orbit of the planet is excited by an outer companion, tidal effects combined with gravitational interactions may give rise to a secular increasing drift on the eccentricity. As long as this secular drift counterbalances the damping effect, the eccentricity can increase to high values. This mechanism may explain why some of the moderate close-in exoplanets are observed with substantial eccentricity values. Key words: celestial mechanics – planetary systems – planets and satellites: general Online-only material: color figure

1. INTRODUCTION where G is the gravitational constant, R is the radius of the planet, and k2 is the second Love number for potential. Close-in exoplanets, as for Mercury, Venus, and the major- Since we are interested in the secular behavior of the system, ity of the natural satellites in the solar system, are supposed we average the motion equations over the mean anomalies of to undergo significant tidal interactions, resulting in their spins both orbits. The averaged potential, quadrupole level for the and orbits being slowly modified. The ultimate stage for tidal spin (e.g., Correia & Laskar 2010a), octopole level for the orbits evolution is the synchronization of the spin and the circular- (e.g., Lee & Peale 2003; Laskar & Boue´ 2010), and with general ization of the orbit (e.g., Correia 2009).Thespinevolvesin relativity corrections (e.g., Touma et al. 2009), is given by a shorter timescale than the orbit, so long-term studies on the  −  − tidal evolution of exoplanets usually assume that their rotation =− − 2 1/2 − − 2 3/2 U C0 1 e1  C1 1  e1  is synchronously locked, and therefore limit the evolution to the 3 2 3 2 1+ 2 e1 e1e2 1+ 4 e1 orbits. However, these two kinds of evolution cannot be disso- − C2   + C3   cos , (2) − 2 3/2 − 2 5/2 ciated because the total angular momentum must be conserved. 1 e2 1 e2 Synchronous rotation can only occur when the eccentricity is very close to zero. Otherwise, the rotation rate tends to be locked with with the orbital speed at the periapsis, because tidal effects are 2 2 2 stronger when the two bodies are closer to each other. In ad- 3β1G (m0 + m1) Gm0m1J2R C0 = ,C1 = , (3) dition, in the presence of a companion body, the eccentricity a2c2 2a3 undergoes oscillations (e.g., Mardling 2007), and the rotation 1 1 rate of the planet shows variations that follow the eccentricity Gβ m a2 15Gβ m a3 (m − m ) (Correia & Laskar 2004). As a consequence, some unexpected = 1 2 1 = 1 2 1 0 1 C2 3 ,C3 4 , (4) behaviors can be observed, such as a secular increase of the ec- 4a2 16a2 m0 + m1 centricity. In this Letter, we provide a simple averaged model for the orbital and spin evolution of an with a companion where ai is the semimajor axis (that can also be expressed using = − (Section 2) and apply it to the HD 117618 planetary system the mean motion ni), ei is the eccentricity, and  1 2 is (Section 3). We then give an explanation for the eccentricity the difference between the longitudes of the periastron, i .We = = pumping (Section 4) and derive some conclusions (Section 5). also have β1 m0m1/(m0 + m1) and β2 (m0 + m1)m2/(m0 + m1 + m2). The contributions to the orbits are easily obtained using the 2. THE MODEL Lagrange planetary equations (e.g., Murray & Dermott 1999):   We consider here a system consisting of a central of − 2 − 2 m0, an inner planet of mass m1, and an outer companion of 1 ei ∂U 1 ei ∂U e˙ = , ˙ =− . (5) mass m2. We use Jacobi canonical coordinates, with r1 being the i 2 i 2 βi ni ai ei ∂i βi ni ai ei ∂ei position of m1 relative to m0, and r2 the position of m2 relative to the center of mass of m1 and m0. We further assume that Thus,  |r1||r2|, that the system is coplanar, and that the obliquity   2 − 2 of the planet is zero. The inner planet is considered an oblate e2 1+3/4e1 1 e1 ellipsoid with gravity field coefficient given by J , rotating about e˙1 =−ν31   sin , (6) 2 − 2 5/2 the axis of maximal inertia, with rotation rate ω, such that (e.g., 1 e2 Lambeck 1988)    e 1+3 4e2 ω2R3 ˙ = 1 1 = e2 ν32   sin , (7) J2 k2 , (1) − 2 2 3Gm1 1 e2

1 The Astrophysical Journal Letters, 744:L23 (5pp), 2012 January 10 Correia, Boue,´ & Laskar and where   ν ν x2 3 2 ˙ =  0   1  3k2 m0β1 R K m1 R  + 2 K = n1 ,K= , (19) 1 − e2 1 − e2 ξQ m2 a 1/ξ β a 1 1 1 1 1 1   − 2 3 2 −1 2 4 2 9/2 1 e1 1+ e Q = n Δt, and f (e) = (1 + 3e +3e /8)/(1 − e ) , − 2 1 1 1 + ν21   ν22 = 2 4 6 − 2 6 = − 2 3/2 (1 − e2)2 f2(e) (1 + 15e /2+45e /8+5e /16)/(1 e ) , f3(e) 1 e2 2 2 4 6 8 − 2 15/2    (1 + 31e /2 + 255e /8 + 185e /16 + 25e /64)/(1 e ) , = 2 4 − 2 5 = 2 e 1 − e2 1+ 9 e2 f4(e) (1 + 3e /2+e /8)/(1 e ) , f5(e) (1 + 15e /4+ 2 1 4 1 4 6 − 2 13/2 − ν31   cos  15e /8+5e /64)/(1 e ) . − 2 5/2 e1 1 e2 We neglect the effect of tides over the longitude of the    periastron as well as the flattening of the central star. Their effect e 1+ 3 e2 1+4e2 1 4 1 2  + ν32   cos , (8) is only to add a small supplementary frequency to 1, similar to − 2 3 e2 1 e2 the contributions from general relativity (for a complete model see Correia et al. 2011). where x = ω/n1, and the constant frequencies Under the effect of tides alone, the equilibrium rotation rate,   obtained when ω˙ = 0, is attained for (Equation (16)) n a 2 ν = 3n 1 1 , (9) 0 1 ω f (e )   c = 2 1 = 2 O 4  1+6e1 + e1 . (20) 5 n1 f1(e1) = k2 m0 + m1 R ν1 n1 , (10) 2 m1 a1 Usually K K, so tidal effects modify the rotation rate much  3 faster than the orbit. It is thus tempting to replace the equilibrium 3 m2 a1 rotation in expressions (17) and (18). With this simplification, ν21 = n1 , (11) 4 m0 + m1 a2 one always obtains negative contributions for a˙1 and e˙1 (Correia  2 2009), 3 m0m1 a1 a˙ ν = n , (12) 1 =− 2 22 2 2 7K f6(e1)e1, (21) 4 (m0 + m1) a2 a1  m m − m a 4 7   = 15 2 0 1 1 ˙ =− − 2 ν31 n1 , (13) e1 K f6(e1) 1 e1 e1, (22) 16 m0 + m1 m0 + m1 a2 2  = 2 4 6 8 15 m m m − m a 3 with f6(e) (1 + 45e /14 + 8e + 685e /224 + 255e /448 + ν = n 0 1 0 1 1 . (14) 25e10/1792)(1−e2)−15/2/(1+3e2 +3e4/8). Thus, the semimajor 32 2 2 16 (m0 + m1) m0 + m1 a2 axis and the eccentricity can only decrease until the orbit of the Note that the variations in e1 and e2 (Equations (6) and (7)) planet becomes circular (Figure 1(a)). However, planet–planet are related by the conservation of the total angular momentum interactions can produce eccentricity oscillations with a period 2 (after dividing by m1n2a2 ): shorter than, or comparable to, the damping timescale of     the spin. In that case, expression (20) is not satisfied and ω R 2 β n a 2 β multi-planetary systems may show non-intuitive eccentricity ξ 1 1 1 − e2 2 − e2 = Cte, + 1 1 + 1 2 (15) evolutions, such as eccentricity pumping of the inner orbit (e1 n2 a2 m1 n2 a2 m1 increases while e2 decreases). where ξ is a structure coefficient. The conservative system (Equation (2)) can thus be reduced to one degree of freedom. 3. APPLICATION TO EXOPLANETS In our model, we additionally consider tidal dissipation raised As an illustration of the eccentricity pumping, we apply by the central star on the inner planet. The dissipation of the our model to the HD 117618 system. This Sun-like star mechanical energy of tides in the planet’s interior is responsible (m ≈ M) has been reported to host a single Saturn-like planet for a time delay Δt between the initial perturbation and the 0 on a eccentric orbit (Butler et al. 2006). The residuals of the maximal deformation. As the rheology of planets is badly − best-fit solution to the observational data are 5.5 m s 1,so known, the exact dependence of Δt on the tidal frequency is we assume that any additional companion with a Doppler shift unknown. Several models exist (for a review see Correia et al. semi-amplitude smaller than this value is presently undetected, 2003; Efroimsky & Williams 2009), but for simplicity we adopt that is, any planet with m < 0.2 M and a > 1.4AU. here a model with constant Δt, which can be made linear (Singer 2 J 2 In our simulations we adopt for the observed planet the same 1968; Mignard 1979). The contributions to the equations of geophysical parameters as for Saturn, R = R , k = 1/2, motion are given by (e.g., Correia 2009) Sat 2 ξ = 1/5, and a dissipation time lag Δt = 200 s (which is ˙ equivalent to Q ≈ 3 × 104). Since the semimajor axis of the ω =− − K (f1(e1)x f2(e1)) , (16) planet undergoes tidal dissipation, its value was certainly larger n 1 when the system formed. We then adopt a = 0.25 AU as the ˙ 1 a1 = − initial value for all simulations. The initial eccentricity is chosen 2K (f2(e1)x f3(e1)), (17) ≈ = a1 such that e1 0.4 when a1 0.175, the present observed  values (Table 1). We further assume initial  = 180◦, and 11 = e˙ = 9K f (e )x − f (e ) e , (18) 2π/ω 50 , which quickly evolves near the equilibrium 1 18 4 1 5 1 1 rotation (Equation (20)).

2 The Astrophysical Journal Letters, 744:L23 (5pp), 2012 January 10 Correia, Boue,´ & Laskar

0.8 0.8 (a) (b) 0.7 0.7

0.6 0.6

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0 0 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000

0.6 0.6 (c) (d) 0.5 0.5

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0 0 0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000

time (Myr) time (Myr)

Figure 1. Long-term evolution of the HD 117618 system in different situations. We show the semimajor axis a1 in AU (blue), and the eccentricities e1 (red) and e2 (green). (a) Without a companion; (b) with an m2 = 0.2 MJ companion at a2 = 1.8 AU, but without dissipation on the orbit; (c) same as (b), but with a full model; (d) with an m2 = 0.15 MJ companion at a2 = 1.4 AU, with a full model. The dashed line gives the slope of the linear approximation (Equation (37)). (A color version of this figure is available in the online journal.)

Table 1 We now add a companion to the system with m2 = 0.2 MJ , Single Planetary Systems with 0.1 0.3 1 1 a2 = 1.8 AU, and e2 = 0.4, and set e1 = 0.3. At first, we only consider dissipation in the spin (Equation (16)) and neglect its Star a1 e1 m1 m0 Age τ (Name) (AU) (MJ)(M) (Gyr) (Gyr) effect on the orbit (Equations (17) and (18)) in order to highlight the eccentricity pumping (Figure 1(b)). We then clearly observe HD 108147 0.102 0.53 0.26 1.19 2.0 0.01 CoRoT-10 0.105 0.53 2.75 0.89 3.0 0.24 this effect, with the eccentricity of the inner planet rising up to HD 33283 0.145 0.48 0.33 1.24 3.2 0.34 0.7. We also observe that the eccentricity of the outer planet is HD 17156 0.163 0.68 3.19 1.28 3.4 0.44 simultaneously damped because of the conservation of the total HIP 57050 0.164 0.31 0.30 0.34 ... 39.4 angular momentum (Equation (15)). HD 117618 0.176 0.42 0.18 1.05 3.9 2.06 Orbital and spin evolution cannot be dissociated, so we HD 45652 0.228 0.38 0.47 0.83 ... 93.3 then integrate the full system (Figure 1(c)). We observe that HD 90156 0.250 0.31 0.06 0.84 4.4 35.8 the initial behavior of the system is identical to the situation HD 37605 0.260 0.74 2.84 0.80 10.7 10.6 without dissipation on the orbit (Figure 1(b)). However, as the HD 3651 0.284 0.63 0.20 0.79 5.1 15.5 eccentricity increases, the inner planet comes closer to the star at periastron, and tidal effects on the orbit become stronger. As Notes. Data are taken from http://exoplanet.eu/. 2m a8(1−e2)8 a consequence, the semimajor axis decreases and the damping = 1 1 1 Δ = 2 τ Δ 2 5 with t 10 s (Correia & Laskar 2010b). effect on the eccentricity (Equation (18)) overrides the pumping t21k2Gm0R drift. The system ultimately evolves into a circular orbit. The present configuration is attained around 4 Gyr of evolution, In the absence of a companion, the eccentricity and the which is compatible with the present estimated age of the star. semimajor axis are damped following an exponential decay The pumping effect is then responsible for a delay in the final (Correia & Laskar 2010b), and the present configuration is evolution of planetary systems and may explain the high values attained after 1 Gyr (Figure 1(a)). At present the observed observed for some of them (Table 1). eccentricity would be around 0.1, and we still needed to explain Finally, we repeat the integration of the full system, but with the high initial value near 0.7. a smaller-mass companion m2 = 0.15 MJ at a2 = 1.4AU.

3 The Astrophysical Journal Letters, 744:L23 (5pp), 2012 January 10 Correia, Boue,´ & Laskar = The companion eccentricity is still e2 0.4, but the inner =− − νe K(f1(e10)x0 f2(e10)), (31) planet now begins with e1 = 0.15 (Figure 1(d)). The initial = 3 5 − 2 11/2 = evolution is still similar to the previous simulation (Figure 1(c)), where f1(e) 15(e +3e /2+e /8)/(1 e ) and f2(e) except that the eccentricity oscillations of both planets are higher 3(9e +65e3/2 + 125e5/8+5e7/8)/(1 − e2)7. We neglected the because the orbits are closer. Since the companion mass is octopole terms since ν3i ν2i , the contributions from δe2, and smaller, its eccentricity also decreases more than before and assumed that ei0 = 0. reaches zero around 3 Gyr. At this stage, the angle  stops ◦ At first order, the precession of the periastron is constant, circulating and begins librating around 180 . The two orbits are ˙ g, and the eccentricity is simply given from expression then tightly coupled and evolve together, showing an identical (23)as behavior to close-in planets with an of a few days δe1 = Δe cos(gt + 0), (32) (e.g., Mardling 2007; Laskar et al. 2011). As a consequence, Δ = the evolution timescale is much longer, allowing the inner where e A/g. That is, the eccentricity e1 presents periodic Δ planet to maintain high eccentricity for longer periods of time variations around an equilibrium value e10, with amplitude e (Figure 1(d)). The present eccentricity is only observed after and frequency g. Since gx δx,geδe1 g, the above solution for 6Gyr. the eccentricity can be adopted as the zeroth-order solution of the system of Equations (23)–(25). With this approximation, the 4. ECCENTRICITY PUMPING equation of motion (25)ofδx becomes that of a driven harmonic oscillator whose steady state solution is In order to understand the unexpected behavior of the eccen- tricity during the initial stages of the evolution, we can perform δx = Δx cos(gt + 0 − φ), (33) some simplifications in the equations of motion without loss of Δ = Δ 2 2 = 2 2 generality (Section 2). We can neglect tidal effects on orbital with x νe e/ νx + g and sin φ g/ νx + g .Therota- quantities (Equations (17) and (18)), which is justified since tion rate thus presents an oscillation identical to the eccentricity K K (Equation (19)). The only contribution of tides is then (Equation (32)), but with smaller amplitude and delayed by on the rotation rate (Equation (16)). The semimajor axis and the an angle φ (see Correia 2011). Using the above expression in mean motion are thus constant, and the eccentricity only varies Equation (24) and integrating gives for the periastron: due to the gravitational perturbations (Equation (6)). In addition, gx ge we linearize the set of equations of motion in the vicinity of the  = gt+0 + Δx sin(gt+0 −φ)+ Δe sin(gt+0). (34) averaged values of x, e1, and e2.Letx = x0 + δx, where x0 is the g g solution of Equation (20), e = e + δe , and e = e + δe .In 1 10 1 2 20 2 Finally, substituting into Equation (23) and using the approxi- the following, δe2 is expressed as a function of δx and δe1 us- Δ Δ ing the conservation of the angular momentum (Equation (15)). mation gx x,ge e g gives Then, the set of equations of motion (6), (8), and (16) reduces geA to δe˙1 ≈−A sin(gt + 0) − Δe sin(gt + 0) cos(gt + 0) δe˙ =−A sin , (23) g 1 g A − x Δx sin(gt +  − φ) cos(gt +  ), (35) g 0 0 ˙ = g + gx δx + geδe1, (24) or, combining the two products of periodic functions, δx˙ =−νx δx + νeδe1, (25) with  gx A   δe˙1 =−A sin(gt + 0) − Δx sin(2gt +20 − φ) 2 − 2 2g e20 1+3/4e10 1 e10 geA gx A A = ν31   , (26) − Δ Δ − 2 5/2 e sin(2gt +20)+ x sin φ. (36) 1 e20 2g 2g The two middle terms in the above equation can be neglected 2 ν0 ν1x0 since they are periodic and have a very small amplitude g =   +   Δ Δ 1 − e2 − 2 2 (gx x,ge e g). However, the last term in sin φ is constant 10 1 e10    and it adds a small drift to the eccentricity, − 2 2 1 e10 1+3e 2   −  10  2 + ν21 3/2 ν22 2 , (27) νegx A − 2 − 2 e1 =  t. (37) 1 e20 1 e20 2 2 2g νx + g 2x0 The drift is maximized for g ∼ ν , and it vanishes for gx = ν1   , (28) x − 2 2 weak dissipation (Δx → 0), and also for strong dissipation 1 e10 (φ → 0). Note that the phase lag φ between the eccentricity

2 (Equation (32)) and the rotation variations (Equation (33)) is 2e10 4x0 e10 3e10 essential to get a drift on the eccentricity. That is why the ge = ν0   + ν1   − ν22   − 2 2 − 2 3 − 2 2 eccentricity pumping was never observed in previous studies 1 e10 1 e10 1 e20 that did not take into account the spin evolution. −  e10 ν21   , (29) The major difference when we consider the full − 2 − 2 3/2 1 e10 1 e20 non-linearized problem is that the drift (Equation (37)) cannot grow indefinitely. Indeed, when the eccentricity reaches high νx = Kf1(e10), (30) values, the drift vanishes (Figure 1(b)). Moreover, tidal effects

4 The Astrophysical Journal Letters, 744:L23 (5pp), 2012 January 10 Correia, Boue,´ & Laskar are also enhanced for high eccentricities and counterbalance the forcing (Equation (32)) and the precession angle oscillations drift (Equation (18)). As a consequence, the drift on the eccen- (Equation (34)). The result is that the drift effect on the tricity is never permanent, although it can last for the age of the eccentricity can be even more pronounced than the one presented system (Figures 1(c) and (d)). here. In order to observe the pumping effect, the eccentricity should not be damped, while the damping timescale of the spin of We acknowledge support from the PNP-CNRS, the FCT the planet should be of the order of the period of eccentricity (grant PTDC/CTE-AST/098528/2008), and the European Re- oscillations. This is valid for gaseous planets roughly within search Council. 0.1

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