Tidal decay and orbital circularization in close-in two-planet systems
Adrián Rodríguez, Sylvio Ferraz-Mello, Tatiana A. Michtchenko, Cristian Beaugé, Octavio Miloni
aa r X i v : . [ a s t r o - ph . E P ] A p r Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 30 October 2018 (MN L A TEX style file v2.2)
Tidal decay and orbital circularization in close-in two-planetsystems
A. Rodr´ıguez, ⋆ S. Ferraz-Mello, T. A. Michtchenko, C. Beaug´e and O. Miloni Insituto de Astronomia, Geof´ısica e Ciˆencias Atmosf´ericas, IAG-USP, Rua do Mat˜ao 1226, 05508-900 S˜ao Paulo, Brazil Observat´orio Astron´omico, Universidad Nacional de C´ordoba, Laprida 854, (X5000BGR) C´ordoba, Argentina Facultad de Ciencias Astron´omicas y Geof´ısicas, Universidad Nacional de La Plata, Paseo del Bosque S/N B1900 FWA, La Plata, Argentina
Released 2011 Xxxxx XX
ABSTRACT
The motion of two planets around a Sun-like star under the combined effects of mutualinteraction and tidal dissipation is investigated. The secular behaviour of the system is ana-lyzed using two different approaches. First, we solve the exact equations of motion throughthe numerical simulation of the system evolution. In addition to the orbital decay and circu-larization, we show that the final configuration of the system is affected by the shrink of theinner orbit. Our second approach consist in the analysis of the stationary solutions of meanequations of motion based on a Hamiltonian formalism.We consider the case of a hot super-Earth planet with a more massive outer companion.As a real example, the CoRoT-7 system is analyzed solving the exact and mean equationsof motion. The star-planet tidal interaction produces orbital decay and circularization of theorbit of CoRoT-7b. In addition, the long-term tidal evolution is such that the eccentricity ofCoRoT-7c is also circularized and a pair of final circular orbits is obtained. A curve in thespace of eccentricities can be constructed through the computation of stationary solutionsof mean equations including dissipation. The application to CoRoT-7 system shows that thestationary curve agrees with the result of numerical simulations of exact equations.A similar investigation performed in a super-Earth-Jupiter two-planet system shows thatthe doubly circular state is accelerated when there is a significant orbital migration of the innerplanet, in comparison with previous results were migration is neglected.
Key words: celestial mechanics – planetary systems.
It is well known that close-in planets undergo tidal interactionswith their host stars. The tidal effect produces orbital decay,circularization and spin-orbit synchronization of a planet orbit-ing a slow-rotating star (Dobbs-Dixon, Lin & Mardling 2004;Ferraz-Mello, Rodr´ıguez & Hussmann 2008; Jackson, Greenberg& Barnes 2008). When the tidally affected planet has an eccentriccompanion, its evolution is more complicated. As we shall show inthis paper for two different systems, new features appear due to theinterplay of tidal effects and mutual interactions between planets.Several works have already investigated the evolution of two-planet systems accounting for tidal dissipation (Wu & Goldreich2002; Mardling & Lin 2004; Mardling 2007; Zhou & Lin 2008,Mardling 2010, Greenberg & Van Laerhoven 2011). Particularly inMardling (2007), a secular model based on the Legendre expan-sion of the disturbing forces up to fourth order in semi-major axesratio, a /a , was introduced. She has shown that the outer com- ⋆ E-mail: [email protected] panion excites the eccentricity of the inner planet, accelerating itsmigration toward the star due to tidal dissipation. Moreover, theevolution of the system under tidal effect follows several stages,depending on the circulation or oscillation of the angle betweenthe lines of apses. The excitation of the inner planet eccentricityreaches a quasi-equilibrium value and, in a long time-scale (whichdepends on physical parameters and initial orbital configuration),both eccentricities are damped to zero. If the companion planet cansustain a non-zero eccentricity of the inner planet for times of theorder of the age of the system, the mechanism could explain someobserved oversized planets, since the body can inflate in response totidal heat. The companion planet hypothesis would thus explain thelarge size of some planets, e.g. HD 209458b, HAT-P1b and WASP-1b (see Mardling 2007 for details).Although Mardling’s model has no restrictions on the valuesof the planetary mass ratio, its applications have given priority tosystems with more massive inner planets . In this case, in a first Except in the case of HAT-P-13 non-coplanar system (see Mardling2010)c (cid:13)
A. Rodr´ıguez et al. approximation, we may assume that the planetary semi-major axesdo not vary during the tidal evolution. This approximation may besufficient for systems with a less massive outer planet, in whichcase the excitation of the inner planet eccentricity is not strongenough to induce its rapid orbital decay due to the tidal friction.As a consequence, the orbital decay of the inner planet occurs intime-scales much longer than the ages of the systems analyzed inMardling (2007). In this scenario, analytical results have shown agood agreement with results of numerical simulations, performedthrough numerical integrations of the Lagrange’s planetary equa-tions.In the present work, we use a different approach, with no re-strictions on the semi-major axes variation, which allows us to in-vestigate the tidal evolution of two-planet systems with an arbi-trary mass ratio. We perform numerical simulations, using New-ton’s equations of the coplanar motion of two planets and includ-ing the forces due to tidal friction according to Mignard (1979). Weshow that the angular momentum conservation constrain the vari-ation of the planetary orbital elements, even in the case of rapidorbital decay.We perform an adaptation of the conservative analytical modelof secular motion to the systems with slow loss of the energy, suchas tidally affected pair of planets. Assuming that the planets are farenough from any mean-motion resonances, only secular terms aremaintained in the disturbing function. Previous works have shownthat for a given value of the total angular momentum, all possi-ble motions of the secular system are constant-amplitude oscilla-tions around one of the two equilibria of the secular Hamiltonian,known as Mode I and Mode II (Michtchenko & Ferraz-Mello 2001,Michtchenko & Malhotra 2004). We also know that the location ofthe stationary solutions in the phase space is uniquely defined bytwo parameters: the ratios of planet masses and semi-major axes.When we consider the non-conservative secular system, inwhich the rate of dissipation is sufficiently slow (more precisely,it is slower than the proper frequency of the system), the parame-ter a /a varies slightly due to the slow orbital decay of the innerplanet. For all possible values of a /a and the total angular mo-mentum, we may calculate the locations of the equilibria in thephase space. A curve of stationary solutions can be constructed andthen compared with the result of the numerical simulation of thesystem (see Michtchenko & Rodr´ıguez 2011 for a detailed discus-sion).In this paper, we restrict our investigations to systems with ahot super-Earth and a more massive outer companion. By super-Earth, we mean planets with an upper limit of m ⊕ , where m ⊕ is the mass of the Earth (Fortney, Marley & Barnes 2007; Valencia,Sasselov & O’Connell 2007). Up to date, twenty-four known super-Earth have been discovered with masses ranging from . m ⊕ to . m ⊕ and the majority of these planets evolve in multi-planetsystems . Almost half of the known super-Earth have orbital peri-ods smaller than 5 days; so small periods indicate that the planetshave experienced an orbital decay due to the tidal interaction withthe star.We apply our approach to the CoRoT-7 extrasolar system,which is composed by one super-Earth and a more massive outercompanion (Queloz et al. 2009; L´eger at al. 2009; Ferraz-Melloet al. 2011). Previous investigations involving tidal evolution ofCoRoT-7 planets include the analysis of tidal heating of CoRoT-7b and the consequences for its orbital migration. It was shown http://exoplanet.eu/ that the past history of the planet may have included a tidal heatinglarge enough to be compared with the corresponding tidal heatingof the Jovian satellite Io (Barnes et al. 2010). The coupled interac-tion between tides and secular evolution in the CoRoT-7 system isinvestigated in this paper showing that a final orbital configurationwith two circular orbits is obtained in a few Myr.This paper is organized as follow: In Section 2 we briefly de-scribe the tidal evolution for a single-planet system. In Section 3we analyze the two-planet secular behaviour through the numer-ical integrations of the exact equations of motion including thetidal force (in the coplanar case). We consider two applications:the real system CoRoT-7 and an hypothetical system containing ahot super-Earth with a Jupiter companion. The angular momentumconservation of the system is discussed in Section 4, providing asimple explanation for the damping of the outer planet eccentricityand the orbital decay of the inner planet. In Section 5, we inves-tigate the secular dynamics of CoRoT-7 planets through the studyof stationary solutions of the mean equations using the Hamilto-nian formalism. The role of stellar tides is analyzed in Section 6.Finally, Section 7 is devoted to provide a general discussion andconclusions. We first consider the tidal interaction between one planet and itshost star. We call m , R , the mass and stellar radius, whereas m , R are the corresponding planetary values. In the case of zero or-bital inclination, the mean variations of orbital elements producedby the joint effect of both planetary and stellar tides are given by < ˙ a > = − na − ˆ s [(1 + 23 e ) + 7 e D ] (1)and < ˙ e > = − nea − ˆ s [9 + 7 D ] , (2)where D ≡ ˆ p/ s and ˆ s ≡ k Q m m R , ˆ p ≡ k Q m m R , (3)are the strengths of the stellar and planetary tides, respectively(Dobbs-Dixon et al. 2004; Ferraz-Mello et al. 2008). In above equa-tions, a, e and n are semi-major axis, eccentricity and mean orbitalmotion, respectively. k i and Q i are the Love number and dissi-pation function of the deformed body, for i = 0 , . The dissipa-tion function is associated to the efficiency for which the energyis dissipated in the interior of the tidally deformed body. The re-sults are valid up to second order in eccentricity, assuming a quasi-synchronous planetary rotation and a slow-rotating star (see Section3.2.3). To obtain the variations produced by stellar tide, a linearmodel with constant time lag was assumed to relate phase lags andtidal frequencies (see Section 3.1).From equations (1) and (2) we obtain d a/a = F ( e, D ) d e/e ,where F ( e, D ) ≡ D + 23) e ]7 D + 9 . (4)After integration, we obtain a = a ini exp " (7 D + 23)( e − e ini ) + 2 log ( e/e ini )7 D + 9 , (5) c (cid:13) , 000–000 uper-Earth tidal evolution m m rr m f −f g−g Figure 1.
Schematic illustration of tidal interaction in the system. where the subscript ini indicates initial values. It should be empha-sized that, when stellar tide is neglected ( D → ∞ ) we have a = a ini exp( e − e ini ) , (6)indicating that the final position of the planet is completely deter-mined by initial values of the elements.It is important to note from equations (1) and (2) that the cu-mulative effect of tides produces orbital decay and circularizationon time-scales which varies according to k i /Q i - values (see Jack-son et al. 2008; Rodr´ıguez & Ferraz-Mello 2010). Moreover, thestellar tide is the only source of orbital decay after total circular-ization is achieved. However, we see from equation (3) that thestrength of the stellar tide is proportional to the planetary mass m .Hence, tides on the star due to large planets are more efficient thantides produced by small planets, in order to induce a planetary mi-gration on time-scales comparable to the age of the system. In this section we study the dynamical evolution of a system com-posed by two planets orbiting a central star. The purpose is to in-vestigate how the presence of an outer companion affects the tidalevolution of the inner planet.
We will consider systems in which the inner planet is a super-Earth.As discussed in the previous section, in this case, the contributiondue to stellar tides is negligible due to the small planet mass. Weassume that both planets are deformed under the tides raised bythe central star. In addition to tidal force, we consider that the in-ner planet is also affected by the correction of Newtonian potentialof the star due to general relativity. The reference frame chosenis centered in the star and the motion of the planets occurs in thereference plane (i.e coplanar motion).According to our assumptions, the orbits evolve under thecombined effects of mutual interaction and tides raised by the star.The equations of motion of the planets are written as ¨ r = − G ( m + m ) r r + Gm r − r | r − r | − r r ! (7) + ( m + m ) m m ( f + f rel ) + g m , ¨ r = − G ( m + m ) r r + Gm r − r | r − r | − r r ! +( m + m ) m m g + ( f + f rel ) m , (8) Table 1.
The adopted orbital elements and physical data of the CoRoT-7system (L´eger et al. 2009; Queloz et al. 2009, Ferraz-Mello et al. 2011). Thevalue of R was computed assuming an Earth mean density for CoRoT-7c.Body m i R i a i current (au) e i current Q ′ i . m ⊙ . R ⊙ - - -1 . m ⊕ . R ⊕ . m ⊕ . R ⊕ where f rel is the general relativity contribution acting on the innerplanet, which is approximately given by f rel = Gm m c r h(cid:16) Gm r − v (cid:17) r + 4( r · v ) v i (9)where v = ˙ r and c is the speed of light (see Beutler 2005). f and g are the tidal forces acting on the masses m and m , respectively.We use the expression for tidal forces given by Mignard (1979): f = − k ∆ t Gm R r [2 r ( r · v ) + r ( r × Ω + v )] , (10) g = − k ∆ t Gm R r [2 r ( r · v ) + r ( r × Ω + v )] , (11)where Ω i is the rotation angular velocity of the i -planet, for i =1 , . It is worth noting that the Mignard’s force is given by a closedformula and, therefore, is valid for any value of eccentricity . ∆ t i is the time lag and can be interpreted as a delay in the deformationof the tidally affected body due to its internal viscosity.The time lag is related to the dissipation function ( Q ) of thedeformed planet, which is a quantity used in the study of tidal evo-lution of extra-solar planets. However, in many classical theories,the tidal potential appears written as a function of phase lags ( ε j ) oftidal waves, introduced in each periodic term in order to considerthe effect of viscosity (Darwin 1880; Kaula 1964; Ferraz-Mello etal. 2008). In fact, the expansion of equations (10)-(11) leads to thesame forces used in the Darwin’s theory when the phase lags of thetide components are assumed proportional to their frequencies (theso-called linear model): ε j = ν j ∆ t , with the same ∆ t for all j -frequencies. For small lags, Q can be associated to the phase lag ofthe tidal wave through ε = 1 /Q (see Efroimsky and Lainey 2007,Efroimsky and Williams 2009 for a complete discussion). For plan-ets in stationary or synchronous rotation, the tidal evolution is welldescribed by the phase lag corresponding to the frequency n (an-nual tide, see Ferraz-Mello et al. 2008). Thus, in the linear modelwe have /Q ≃ n ∆ t or, using a modified definition which absorbsthe Love number, Q ′ ≡ Q/ k , we obtain k ∆ t = 3 / (2 Q ′ n ) (seeLeconte et al. 2010). We numerically investigate the recently discovered CoRoT-7 plan-etary system, which is composed by two short-period planets or-biting a central star. The star, CoRoT-7, is a G9V with mass m = 0 . m ⊙ , radius R = 0 . R ⊙ and age of 1.2 - 2.3 Gyr(Bruntt et al. 2010). The inner planet, CoRoT-7b, and outer planet, Note however that, for a more accurate description, a large number ofharmonics in the expansion of the tidal potential should be considered whenthe star-planet distance is small enough (see Taylor and Margot 2010)c (cid:13) , 000–000
A. Rodr´ıguez et al. a [ au ] time [Myr] 0.0459 0.04595 0.046 0.04605 0.0461 0.04615 0.0462 0 1 2 3 4 5 6 7 a [ au ] time [Myr] Figure 2.
Time variation of semi-major axes for the planets of CoRoT-7system. The excitation of e enables the orbital decay of CoRoT-7b dueto tides on the planet raised by the star. CoRoT-7c also migrates due totides but with a slower rate, resulting in a divergent migration between theplanets. CoRoT-7c, are assumed to have the masses given in Ferraz-Melloet al. (2011), m = 8 . m ⊕ and m = 13 . m ⊕ , respectively.The radius of CoRoT-7c is not known. We adopted values of Q ′ and mean density consistent with a rocky terrestrial planet. Theirorbital periods are P orb = 0 . and P orb = 3 . days (L´egeret al. 2009; Queloz et al. 2009). CoRoT-7b is the third exoplanetwith shortest orbital period discovered up to the present date.The adopted physical and orbital parameters of the system arelisted in Table 1. As we will shown, in the current configurationof the system, both orbits must be circular. We emphasize that themasses of the CoRoT-7 planets satisfy the relation m /m < .Choosing Q ′ = Q ′ = 100 (a typical value for terrestrial planets)and using current values of semi-major axes, we obtain k ∆ t = 3 min and k ∆ t = 12 min. A scaling factor is often used in thenumerical simulations in order to accelerate the process of tidalevolution. In the simulations reported in this paper, we did not usea scaling factor. We also assume that the inclinations of both orbitalplanes with respect to the reference plane are zero.The next step to perform the simulation is the choice of the ini-tial configuration of the CoRoT-7 system. (It should be emphasizedthat, since we are studying a dissipative process, we cannot use asimple backward simulation for that sake). The choice is arbitrary,but it can be constrained by some considerations. We assume thatthe planets have experienced an orbital decay due to tidal effects,thus its past location must be more distant from the central star thanthe current one. Finally, as will be discuss in Section 4, the angularmomentum of the system can be considered as invariable duringthe evolution of the system under the interplay of the gravitationaland tidal forces. Gathering this information, we assume the follow-ing initial values for the orbital elements: e ini = 0 , e ini = 0 . , a ini = 0 . au and a ini = 0 . au.Figs. 2 and 3 show the result of a numerical integration ofequations (7) and (8) using the RA15 code (Everhart 1985). It is e cc en t r i c i t i e s time [Myr]e e e ∆ ω time [Myr] Figure 3.
Time variation of eccentricities and ∆ ̟ (in degrees) for the plan-ets of the CoRoT-7 system. The eccentricity of the initial inner circular orbitis excited by the outer companion. Tides on the inner planet circularize itsorbit in approximately 7 Myr. The eccentricity of the outer orbit is alsodamped because of the total angular momentum conservation (see Section4). At the end of the simulation, a state of double circularization is obtained(see text for discussion). shown the orbital decay and circularization of the orbits due totidal effects. The inner planet eccentricity initially undergoes largevariations, but, after a short time of damped oscillations, it reachesa quasi-equilibrium value (Mardling 2007, see Sec. 3.2.1). Afterrelaxation, e decreases smoothly to reach near-circularization inabout 7 Myr. The eccentricity of the outer planet is also dampedand a pair of circular orbits is obtained as final configuration of thesystem.The excitation of e activates the orbital decay of the innerplanet, which requires a non circular orbit according to equation(1) (with ˆ s = 0 , because we are not considering the contribution ofstellar tides in our simulations). The migration of CoRoT-7b towardthe star occurs until the circularization of its orbit. Meanwhile, a also decreases (and is only weakly affected when the 4/1 mean-motion resonance is crossed). However, when equations (10)-(11)are expanded up to third order in eccentricities (see Ferraz-Mello etal. 2008), we obtain that that the ratio of forces on CoRoT-7b andCoRoT-7c is approximately 16, indicating a more intense migrationof the inner planet.The bottom panel of Fig. 3 shows the time variation of the sec-ular angle ∆ ̟ ≡ ̟ − ̟ , where ̟ means longitude of pericen-ter. Initially, ∆ ̟ oscillates around ∆ ̟ = 0 ◦ with large amplitude;this stage of the evolution corresponds to the damped oscillationsof the planetary eccentricities. Reaching the relaxed state, in whichthe system oscillates with amplitudes close to zero, the oscillationamplitude of the secular angle also tends to zero.It is worth mentioning that, due to the divergent migration be-tween the planets, the capture in mean motion resonance is not pos-sible. c (cid:13) , 000–000 uper-Earth tidal evolution e m /m γ =0 γ≠ Figure 4.
Quasi-equilibrium e eq given by equation (12), as a function ofthe mass ratio. The vertical dashed line corresponds to the mass ratio ofthe CoRoT-7 system. The curves show the cases with and without generalrelativity. Fig. 3 shows that, approximately at t = 0 . Myr, the amplitudesof oscillation of the eccentricities are damped and the variation be-comes smooth. This situation is called quasi-equilibrium. Mardling(2007) derived an expression to compute the quasi-equilibriumvalue of the inner planet eccentricity, e eq , using the Legendre ex-pansion of the disturbing function, up to fourth order in a /a . Itis given by e eq = (5 / a /a ) e ε − | − p a /a ( m /m ) ε − + γε | , (12)where ε ≡ p − e and γ ≡ n a /c ) ( m /m )( a /a ) .Fig. 4 shows the variation of e eq given by equation (12), forCoRoT-7, as a function of m /m . The relevant value of e whichshould be used is the one when e eq is reached. In addition, the con-dition (12) was obtained under assumption that semi-major axes areconstants. We show the cases with ( γ = 0 ) and without ( γ = 0 ) thecontribution of general relativity. The inclusion of γ = 0 reducesthe value of e eq . Hence, since the mean rate of orbital decay is pro-portional to e , the omission of the relativistic potential can resultin an underestimation of the circularization and orbital decay time-scales. Mardling & Lin (2004) also showed that the inclusion of therelativistic potential is very important when the planet approachesthe star. Our numerical simulation has shown that the eccentricity of theouter planet is strongly affected during the orbital decay (see Fig.2, top panel). However, the source of e - damping is not the directaction of tides on the outer planet since, as we have already shown,the magnitude of the tidal force on CoRoT-7c is much smaller thanthe corresponding force on CoRoT-7b. Mardling (2007) also studied the long-term variation of theouter planet eccentricity and provided an expression to calculatethe e - damping: One can take an idea about which is the contribution of tides raised inthe outer planet to the circularization process of its orbit. For that sake, wenote that the ratio between the circularization time-scales associated withthe direct tidal interaction with the star and the one given by equation (13)is of the order of ( a /a ) ≫ . e Time [Myr]simulationequation (13)
Figure 5.
Time variation of e in the CoRoT-7 system. The comparison isdone between the integration of equation (13) and the result of numericalsimulation (see text for more discussion). ˙ e = − λτ circ e F ( e ) , (13)where τ circ is the circularization time-scale of the inner orbit in thesingle-planet case (i.e. e / ˙ e , where ˙ e is given by equation (2)); λ ≡ (25 / m /m )( a /a ) / and F ( e ) ≡ ε (1 − α ε − + γ ε ) , with α ≡ ( m /m )( a /a ) / .The comparison between the integration of the equation (13)and the result of the simulation of exact equations of motion (7)and (8) is shown in Fig. 5. The difference between both resultsmay arise from the fact that for obtaining equation (13) is explicitlyassumed that semi-major axes are constant (see Mardling 2007) .In Section 3.3, we will show that the disagreement increases in asystem for which m /m ≪ .We note that, according to equation (13), the rate of e -damping decreases with m /m , enabling a non-circular outer or-bit be maintained over long times. In addition, e eq increases with e as equation (12) indicates. Hence, since the mean rate of orbitaldecay is proportional to e , a significant inner orbital shrinkage canoccur for large m /m (see next section). The tidal force produces a torque that directly affects the rotationevolution of the deformed bodies, in such a way that C ˙ Ω = r × f and C ˙ Ω = r × g . Thus, C i ˙ Ω i = − k i ∆ t i Gm R i r i [ − r i Ω i + r i × v i ] , (14)where C i is the polar moment of inertia of the i -planet.To obtain the time evolution of the angular velocity Ω i , theabove equations ( i = 1 , ) were integrated simultaneously withthe equations of motion (7) - (8). The result of the integration isshown in Fig. 6. The rotation periods, defined as P i rot = 2 π/ Ω i ,reach stationary values, which, up to second order in e i , are givenby P i rot = P i orb / (1 + 6 e i ) (Hut 1981; also see Ferraz-Mello etal. 2008; Correia, Levrard & Laskar 2008). In the same figure, wealso plot the orbital periods defined as P i orb = 2 π/n i , which de-crease as a consequence of the orbital decay. The synchronization We stress that equation (13) gives an estimate for the damping time-scaleof e (the full expression is given in equation (57) of Mardling 2007). How-ever, it is shown in that work that the approximate solution (13) underesti-mates the time-scale on which e evolves. Hence, the difference betweenanalytical and numerical results would be even large if the full expressionwere used.c (cid:13) , 000–000 A. Rodr´ıguez et al.
18 23 28 0.01 0.1 1 10 P e r i od s [ h s ] CoRoT-7b rotationorbital 60 80 100 0.01 0.1 1 10 P e r i od s [ h s ] time [Myr] CoRoT-7c Figure 6.
Time variation of the rotation and orbital periods of CoRoT-7planets. Synchronization is reached as the doubly circular state approaches.
Table 2.
Orbital elements and physical data corresponding to an hypotheti-cal Sun-super-Earth-Jupiter system. The value of R was computed assum-ing a terrestrial mean density.Body m i R i a i (au) e i Q ′ i m ⊙ R ⊙ - - -1 m ⊕ / R ⊕ m J - 0.1 0.1 - between the orbital motions and the rotation of the planets occurswhen Ω i = n i or P i rot = P i orb . In Fig. 6 we can observe thatthis condition is only reached after the circularization of the planetorbits. In this section, we investigate the tidal evolution of a system inwhich the inner planet is much smaller than the outer one. We chosethe fictitious system composed by a super-Earth and a Jupiter-likeouter companion, with masses m = 5 m ⊕ and m = 1 m J ,respectively ( m J is the mass of Jupiter). Note that m /m ≃ . ≪ . The initial configuration and physical parameters ofthis hypothetical system are listed in Table 2, in which the valuesof semi-major axes correspond to orbital periods of 2.92 and 11.6days. The large orbital period of the outer planet, in addition to thelarge Q -value for Jupiter-like planets, enable us to neglect the tideson the outer companion. Moreover, as shown in Fig. 7 for the con-sidered system, the contribution of the relativistic potential is alsonegligible. Here, we also assume zero inclinations. In addition, thecalculation of k ∆ t = 3 / (2 Q ′ n ) with Q ′ = 100 results in k ∆ t = 10 min. Fig. 8 shows the time evolution of the inner planet semi-major axis(solid curve in the top panel) and two eccentricities (bottom panel)over the first 50 Myr. Due to the large mass of the outer planet, e is almost constant over the time interval shown in the figure (seeequation (13)). On the contrary, e initially suffers large oscillationswhich are quickly damped to the quasi-equilibrium value. On thesame graph, we plot the variation of e in the case of a single-planet system, when the interactions with one outer planet do notexist. Comparing the two cases, we note that the presence of the e m /m γ =0 γ≠ Figure 7.
Quasi-equilibrium e eq in the second application, given by equa-tion (12), as a function of the mass ratio. The contribution of general rela-tivity does not affect the value of e eq and thus can be neglected. a [ au ] time [Myr]two-planet casesingle-planet case 0 0.02 0.04 0.06 0.08 0.1 0.12 0 10 20 30 40 50 e cc en t r i c i t i e s time [Myr] e e e (single-planet case) Figure 8.
Time variations of orbital elements in the case m /m ≪ (c.f Table 2). The evolution in the case in which the super-Earth is the onlyplanet is also shown. The inner orbit reaches a quasi equilibrium value in 25Myr, retarding the orbital circularization and enhancing the orbital decay. outer companion delays the circularization of the inner planet orbit(see Fig. 9). As a consequence, the presence of the second planetlead to a faster decay of the inner planet orbit. (In the single-planetcase, the semi-major axis a decreases only slightly and reaches afinal constant value (0.0396 au) after total circularization) . Fig. 9 shows the time variation of the planet semi-major axes (toppanel) and the eccentricities (bottom panel) over 300 Myr. After e eq is reached, the inner planet eccentricity decreases smoothly tovery small values. The outer eccentricity follows the evolution ofthe inner one and, at the end of simulation, about ≃ . Gyr, apair of circular orbits is obtained. When the time approximates to This value can be obtained using equation (6) with e = 0 , e ini = 0 . and a ini = 0 . au. c (cid:13) , 000–000 uper-Earth tidal evolution s e m i - m a j o r a x i s [ au ] time [Gyr]a Roche e cc en t r i c i t i e s time [Gyr]e e Figure 9.
Long-term evolution of semi-major axes and eccentricities in aSun-super-Earth-Jupiter system. Due to orbital decay, the final position ofthe inner planet is close to the Roche limit. The value of e is damped tozero and the outer orbit is also circularized. . Gyr, e is strongly excited due to the passage of the systemthrough the 5/1 mean-motion resonance but, immediately after that,is damped to its previous value.The orbital decay of the inner planet occurs until the to-tal circularization of its orbit (see top panel in Fig. 9). Note thatthe final position of the super-Earth is close to the Roche limit a Roche = ( R / . m /m ) / (Faber, Rasio & Willems 2005),which gives a Roche = 0 . au. The outer planet semi-major axisdoes not varies in the considered time interval.In Fig. 10 we compare the damping of e resulting from nu-merical simulation (dashed line) and that obtained through the in-tegration of equation (13) of Mardling’s model (solid line). The re-sults are very different, indicating that equation (13) is not a goodapproximation to compute the decreasing of e in the case of themassive outer companion. The discrepancy can be understood tak-ing into account that, for this particular system, there is a signifi-cant orbital decay due to tides on the inner planet, while the equa-tion (13) was obtained assuming λ and α as constants, i.e. constantsemi-major axes (Mardling 2007). Thus, it is shown that the processof outer planet circularization is accelerated in the case of signifi-cant orbital decay of the inner planet, if compared with the resultof equation (13). Many dynamical features of the planet evolution shown in the pre-vious sections, can be explained considering the conservation of theangular momentum of the system. The total angular momentum ofthe three-body system in the astrocentric reference is the vectorialsum of the orbital momenta of the planets, L orb i ( i = 1 , ), and therotational momenta of the planets and star, L rot i ( i = 0 , , ).If r i is the astrocentrical position of the i -planet, the orbitalcomponent of the angular momentum is given by e Time [Gyr]simulationequation (13)
Figure 10.
Time variation of e in a Sun-super-Earth-Jupiter system. Thecomparison is done between the integration of equation (13) and the resultof numerical simulation (see text for more discussion). L orb = X i =1 m i r i × ˙ r i − M X i =1 m i r i × X i =1 m i ˙ r i , (15)where M ≡ P i =0 m i . Note that the second term of the equation(15) appears due to the use of the astrocentric reference frame. Thisterm is of second order in masses and, assuming m i ≪ m , canbe neglected. Moreover, r i × ˙ r i = p G ( m + m i ) a i (1 − e i ) ˆ k ,where ˆ k is a unitary vector normal to orbital planes. Hence L orb ≃ X i =1 m ′ i p a i (1 − e i ) ˆ k , (16)where m ′ i ≡ m i √ Gm .The rotation component of the angular momentum is given by L rot i = X i =0 C i Ω i , where C i are the moment of inertia with respect to the axes of ro-tation and Ω i are the angular velocities of rotation, for i = 0 , , .Because we account only for the planetary tides on the planets, Ω and Ω vary with time, while Ω is constant.We can show that for planets, L rot i ≪ L orb i ; thus, the contri-bution of the rotational component to the total angular momen-tum can be neglected. Indeed, using the third Kepler’s law, wehave L rot i /L orb i = ξ i ( R i /a i ) (Ω i /n i ) , where we have introduced C i = ξ i m i R i , with < ξ i / . In the case of the quasi-synchronous planet discussed in Section 3.2.3, when Ω i ≃ n i , thisratio is small, of the order of ( R i /a i ) .In contrast, for the star, the quantity L rot /L orb i = ξ ( m /m i )( R /a i ) (Ω /n i ) is close to unity, when we considerclose-in planets and a slow-rotating star, for which the condition Ω ≪ n i is satisfied. Hence, the total angular momentum of thesystem can be approximately written as L ≃ ( m ′ p a (1 − e ) + m ′ p a (1 − e ) + C Ω )ˆ k . (17)Is should be noted that some quantity of angular momentumis removed due to the orbital decay of the inner planet. Thus, ac-cording to above equation, for small e , the eccentricity of the outerplanet would be damped in order to preserve the angular momen-tum conservatation. As a consequence of the total angular momentum conservation, wecan obtain the minimal possible value for a /a during the orbital c (cid:13) , 000–000 A. Rodr´ıguez et al. a m i n [ au ] m /m e =0.1e =0.1a =0.04 aua =0.10 au 0.0392 0.0396 0.01 0.1 1 a m i n [ au ] m /m Figure 11.
Plot of equation (19) showing the dependence of the minimumvalue of the inner planet semi-major axis with m /m . The initial elementsof Table 2 were used to compute κ and κ . The case of a more massiveinner planet is shown in the bottom panel. decay, knowing that this minimum value should be achieved onceorbital circularization is reached. Imposing e = e = 0 in equa-tion (17) we obtain a min a = ( L ′ − (cid:16) m m (cid:17) , (18)where L ′ ≡ ( L − C Ω ) /m ′ √ a and m ′ /m ′ ≃ m /m . Forthose cases in which tides on the outer planet can be neglected, a and thus L ′ are constants of the problem. Applying equation (18)to the super-Earth-Jupiter system we obtain a min /a ≃ . ,reproducing the corresponding result of the numerical simulation.Hence, it is worth noting that the final position of the inner planetcan be obtained from the analysis of the angular momentum con-servation, as given by equation (18).Equation (18) allows us to investigate the variation of a min /a for several mass ratios. However, we note that there is also an im-plicit dependence on m /m through L ′ , which is not shown inequation (18). Hence, in order to express the full dependence wedefine two constants, namely, κ ≡ p ( a ini /a ini )(1 − e ini ) and κ ≡ p − e ini . Using the definition of L ′ and replacing it intoequation (18), we obtain a min = a " ( κ − m m + κ . (19)Fig. 11 shows the variation of a min with the mass ratio m /m according to equation (19), where we have used initial val-ues of the orbital elements listed in Table 2. We see that, for fixedinitial values of the elements, the inner planet will come closer tothe star for high-mass companions. Therefore, as discussed in Sec.3.2.2, the orbital decay of the inner planet is enhanced.For sake of completeness, we show in the bottom panel of Fig.11 the result for m /m < , which is the case of a more massiveinner planet. Note that the orbital decay is very weak, supportingthe assumptions done in Mardling (2007) of constant semi-majoraxes in the case of slow-mass outer companions. In this section we will show that evolutionary routes of the migrat-ing system are traced by stationary solutions of the conservativesecular problem. For this task, we explore the secular dynamics ofthe planar two-planet system, described in detail in Michtchenko &Ferraz-Mello (2001) and Michtchenko & Malhotra (2004).In the context of hamiltonian formalism, the stationary val-ues are defined by the secular Hamiltonian function, in which thefast-period terms, involving arguments with combinations of meanlongitudes, are removed by first-order averaging. In this case, thesecular problem of coplanar two-planet systems is reduced to onedegree of freedom. Regarding to the angle ∆ ̟ , the stationary so-lutions are restricted to ∆ ̟ = 0 ◦ and ∆ ̟ = 180 ◦ , which corre-spond to Mode I and Mode II solutions, respectively. In addition,the stationary values of eccentricities for these modes are e ∗ iI and e ∗ iII , for i = 1 , . Periodic solutions of the secular system are cir-culations or oscillations around Modes I or II and the stationarysolution can be represented by a point in the plane of eccentricities.The position of this pont in the phase space depends on the pa-rameters of the secular problem, namely, angular momentum andsemi-major axes ratio.To understand how the stationary solutions are affected by theintroduction of dissipative forces on the system, we will assumethat the dissipation is very slow. This means that the dissipation rateis much smaller than the characteristic time of the secular system,which is the proper period of the secular angle ∆ ̟ . Under thisassumption, the motion of the pair of planets is nearly conservativeand, for very short time intervals it can be described by the classicalsecular theory.For the sake of simplicity, let us consider that periodic solu-tions are oscillations around Mode I. When the tidal effects aretaken into account, the semi-major axis of the inner planet de-creases slowly. Since the semi-major axes ratio is changed, the po-sition of the equilibrium ( e ∗ I , e ∗ I ) in the phase space will be alsochanged, in order to preserve the total angular momentum of thesystem. In such a way, the system oscillates around a centre whoseposition in the phase space changes slowly due to dissipation.Assuming continuous values of the inner semi-major axis, weobtain the equilibria ( e ∗ I , e ∗ I ) , for all possible values of a . Acurve in the space ( e , e ), which is composed by successive equi-librium points, is a locus of stationary solutions in the space ofeccentricities and we refer to it as LSE. By definition, all solutionin LSE curve have the same value of angular momentum.Fig. 12 illustrates the LSE curve for Mode I in the case of theCoRoT-7 planets , comparing with the curve resulting from numer-ical simulation of the system. For sake of simplicity, we do not in-clude the relativistic potential into the analysis of mean equations.The initial conditions for the numerical integration were chosen as ∆ ̟ = 0 , e = 0 and e = 0 . . Initially, the eccentricities os-cillate rapidly with decreasing amplitudes around an equilibriumwhich slides slowly along the LSE curve. When e ≃ . and e ≃ . , the amplitude of oscillation becomes close to zero andthe solution is almost coincident with the LSE curve, until the sys-tem reaches the total orbital circularization.Detailed discussion about the secular dynamics of non-conservative two-planet systems can be found in Michtchenko &Rodr´ıguez (2011). We have seen in Section 3.2 that the angle ∆ ̟ oscillates with almostzero amplitude around ∆ ̟ = 0 ◦ , indicating motion around Mode I.c (cid:13) , 000–000 uper-Earth tidal evolution e e simulationLSE curve Figure 12.
Comparison between computation of LSE curve and the resultof numerical simulation for CoRoT-7 system (without the contribution ofthe relativistic potential). After the angle ∆ ̟ is close to ◦ (Mode I), bothcurves are almost coincident. In this section, we evaluate the time required for CoRoT-7 planetsreach the Roche limit with the star. As seen in Section 2, in the caseof small-mass planets, the contribution of tides raised by the planeton the star can be neglected if compared with the contribution oftides raised by the star on the planet. However, we know that afterorbital circularization, the stellar tide is the only source of orbitaldecay. Thus, due to the current orbital configuration of the CoRoT-7system, with two circular orbits, only the stellar tides could producethe orbital decay.We define the mean lifetime of CoRoT-7b as the time forwhich the Roche limit is reached, starting with a = a current =0 . au. According to equation (1) with ˆ p = 0 , e = 0 and in thelimit m ≪ m , the mean rate of semi-major axis variation is < ˙ a > = − √ Gm a / k Q m m R . (20)Equation (20) can be integrated as Z a Roche a current da a / = −K τ a , (21)where K ≡ √ Gm ( k /Q )( m /m ) R , whereas τ a is themean lifetime of CoRoT-7b and a Roche = 0 . au ( ≃ . R ).Solving for τ a , we obtain τ a = 239 ( a / current − a / Roche ) √ Gm R Q k m m . (22)Fig. 13 shows the variation of τ a as function of k /Q ac-cording to equation (22). It is easy to see that the mean lifetime ofCoRoT-7b is small for high values of k /Q . On the other hand, τ a increases when the energy within the star is not efficiently dis-sipated (small k /Q ). For a typical value of k /Q = 10 − ,we obtain τ a = 0 . Gyr (see also Jackson, Barnes & Green-berg 2009). If the same analysis is done for CoRoT-7c, we obtain a Roche = 0 . au ( ≃ . R ) and τ a = 367 Gyr, indicating thatCoRoT-7c should not be destroyed by tides within the lifetime ofthe star.
In this work we have investigated the tidal evolution of super-Earthplanets with exterior companions. Previous works have already dis-cussed that problem, giving special attention to the case of a more li f e t i m e [ G y r ] k /Q CoRoT-7b CoRoT-7c
Figure 13.
The mean lifetime τ a of CoRoT-7 planets due to stellar tidesin function of the dissipation on the star. For k /Q = 10 − , τ a = 0 . Gyr and τ a = 367 Gyr. massive inner planet. In Mardling (2007), explicit equations wereobtained giving the variation of eccentricities in the context of con-stant semi-major axes (see equations (12) and (13)). Here, we cen-tre the attention to the case of a more massive outer companion.Through an approach in which the exact equations of motionsare numerically solved, we have shown that the super-Earth orbitaldecay is accompanied by orbital circularization of both planets. Inaddition, we show that the rate of e -damping is small for largevalues of m /m , enabling to sustain a non-circular outer orbitfor longer times. Thus, as e eq increases with e , a significant innerorbital decay can occur for large m /m , since its mean rate isproportional to e .As a real application of tidal evolution of a super-Earth accom-panied by a more massive outer planet, we investigated the CoRoT-7 two-planet system. We have shown that, starting from an initialcircular orbit, e is excited enabling the activation of the orbitaldecay of CoRoT-7b (which requires a non-zero value of e ). Theequilibrium value of e is quickly damped by tides on the planetand a strong decreasing of a is observed in the short-term tidalevolution.In addition, the long-term tidal evolution has shown that thecurrent orbital configuration of the system is achieved, includingthe double-circularization state of CoRoT-7 planets. After circular-ization, the migration toward the star continues due to the action ofstellar tides. We have shown that CoRoT-7b can reach the Rochelimit in approximately 1 Gyr for k /Q = 10 − (see also Jacksonet al. 2009).The study of stationary solutions of mean equations is usefulto better understand the secular dynamics in the space e i cos ∆ ̟ - e i sin ∆ ̟ . The periodic solutions around Mode I (aligned pericen-ters) and Mode II (anti-aligned pericenters) may be easily identifiedin the conservative case. In addition, the analysis can be extendedto the case in which dissipation is included. It is possible to con-struct a curve in the space of eccentricities composed by successivestationary solutions, which are obtained one by one through thesame technique used in the conservative case. We have shown thatthe result of numerical simulation of exact equations for CoRoT-7planets is in good agreement with the curve of stationary solutionsof mean equations of motion.Future studies involving the dynamical evolution of a two-planet pair including tidal dissipation will help to investigate thepast history of the CoRoT-7 system (i.e formation and subsequentevolution). This may bring more information about how the currentorbital configuration of the system was achieved.As an example in which m /m ≪ , we have also investi- c (cid:13) , 000–000 A. Rodr´ıguez et al. gated the tidal evolution of a Sun-super-Earth-Jupiter-like system.The results of numerical simulation have shown that e is not welldescribed by integration of the simple model represented by equa-tion (13), which describes the time variation of e in the long-termevolution. We have shown that the doubly circular state is acceler-ated when there is a significant orbital migration of the inner planet,showing that the final configuration of the system is affected by theshrink of the inner orbit.A simple analysis based on the angular momentum conserva-tion shows that it is possible to predict the final value of the innerplanet semi-major axis, provided the semi-major axis of the outerplanet is fixed. ACKNOWLEDGMENTS
The authors acknowledge the support of this project by CNPq,FAPESP (2009/16900-5) (Brazil), and the CAPES/SECYT jointprogram. The authors also gratefully acknowledge the support ofthe Computation Centre of the University of S˜ao Paulo (LCCA-USP). We also thank the anonymous referee for his/her stimulatingrevision.
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