The Effect of Close-In Giant Planets' Evolution on Tidal-Induced Migration of Exomoons

Home , Regular moon

arXiv:1707.02906v2 [astro-ph.EP] 28 Nov 2019 MNRAS oisi h lntr system planetary the in bodies 2013 al. et Gong e.g. (see systems planetary in phenomenon 2010 Armitage common a is © 1995 2 form original in 8; July 2017 Received 8. July 2017 Accepted ⋆ with interaction tidal and tational discov- planets now. giant to of up hundreds ered the among detected been e.g. to (see intended efforts multiple spite ( exoplanet first the of discovery the Since INTRODUCTION 1 h ffc fcoei in lnt’eouinon exomoons evolution of planets’ migration giant tidal-induced close-in of effect The uvv irto n/rtefrainpoessced in ( moons open succeeds putative still process remains their conditions, formation those if the and/or i.e. migration moons, survive have planets migrated al 0N.5-1 Medell´ın, Colombia 52-21, Colo No. Antioquia, 70 de Universidad Calle FCEN, F´ısica (FACom) - de Group Instituto Astrophysics (SEAP) and Group Physics Physics Computational Planetary and Earth Solar, .A Alvarado-Montes A. J. E-mail: 07TeAuthors The 2017 feoon uvv lntr migration planetary survive exomoons If thsbe la htmgaino in planets giant of migration that clear been has it ) 000 ipn ta.2013 al. et Kipping [email protected] , 1 – n eeecsteen.Teqeto fthose if question The therein). references and 9 ; 21)Pern eebr21 oplduigMRSL MNRAS using Compiled 2019 December 2 Preprint (2017) elr&Pdiz2015 Pudritz & Heller ; elre l 2014 al. et Heller ABSTRACT aelts hsclevolution physical satellites: c discove words: to are Key contribute models. exomoons may structure If results internal our planet. and planets, the evolution giant with outwa close-in collision migrate around a only ture in may or exomoon occurs disruption evolution, rotation tidal orbital Thus and fate. to distribution final comparable its mass or also internal but radius, migration signifi exomoon evolution planetary exo of planetary of that time-scale fate found the have final only We the giants. study gas to close-in paper ing models, this result structure in evolution, In interior taken orbital and process. have tidal-induced lution this them exomoon in of of play None models could formation exom numerical planet. evolution of host the planetary their fate for that of the role mechanism effects determine tida a tidal to the provide developed moon to sy been could and lunar have or of planet models planets disruption star, Several the with the for moons of responsible of be interplay could the processes of These virtue in outwards n h tr ilmodify will star, the and discover ans&OBin2002 O’Brien & Barnes hi host their ⋆ yohtcleoon rudcoei in lnt a irt in migrate may planets giant close-in around exomoons Hypothetical og .ZlaaadMroSucerquia Mario and Zuluaga I. Jorge , ). h rtexomoon first the Additionally, ao Queloz & Mayor hi gravi- their , lnt other planet, oehas none ) lnt n aelts yaia vlto n tblt lnt and planets – stability and evolution dynamical satellites: and planets de- mbia 1 pi 6 April 017 ; ( e hl h lntrmisunchanged. hap- remains migration planet and the interactions while pen exomoon common: in ture 2013 may exomoons migrating orbit, initial and 2010 planet the fate final outwards their and/or inwards migrate 2013 al. et close- Gong around ( exomoons planets of giant dynamics in the studied have ( thors orbits original their nltclyitatbeapcso h rbe ( problem the of and aspects complex intractable account into analytically Others taking beyond. simulations and system numerical solar use formalisms the in tested analytical been well-known have that apply migration, omoon new soio2002 Esposito oto h hoeia models theoretical the of Most ; ) (dwarf) eeetdfo h lntr ytm rbcm a become or system, planetary the from ejected be , aaie l 2012 al. et Sasaki , edsutdo bieae nieRceradius Roche inside obliterated or disrupted be lnt( ‘ploonet’). (a planet ; ol ediverse. be could ) ans&OBin2002 O’Brien & Barnes hro ta.2009 al. et Charnoz ytmtclyfinding systematically ans&OBin2002 O’Brien & Barnes .Alo them of All ). eedn ntermass their on Depending rudteplanet the around however, , on rudevolv- around moons ; sdt describe to used ntan planetary onstraint tm,tecollision the stems, oswe subject when oons oacuttekey the account to fpaeayevo- planetary of s htexomoons that rme Kenyon & Bromley ; iesae lower time-scales not affects cantly faycag in change any if , aaie l 2012 al. et Sasaki A epttogether put we d n prevent and rds T E interactions. l e ntefu- the in red tl l v3.0 file style X ad and/or wards : ). aeoefea- one have fexorings. of old with collide Several Namouni may and au- ex- ; 2 Alvarado-Montes et al.

ap = 1. 0 au ap = 0. 1 au ap = 0. 02 au 1.8 3.0 Jupiter - No core Saturn -- 25 M core 1.7 ⊕

1.6 2.5

1.5 Saturn Jupiter 1.4 2.0 /R /R p p R R 1.3

1.2 1.5

1.1

1.0 1.0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 t [Gyr] t [Gyr]

Figure 1. Evolution of planetary radius for two sets of planets: Jupiter-like (left panel) and Saturn-like (right panel) planets (Fortney et al. 2007). Solid and dashed lines correspond to planets with no solid core and with a solid core of 25 M⊕. Evolution has been computed at diﬀerent distances from the host star: 0.02 (blue), 0.1 (green) and 1.0 au (red).

As opposed to solid planets, gas giants may change Furthermore, in the last 10 years improved and more gen- significantly during the first hundreds of Myrs (see eral models have been developed to describe the evolution of Fortney et al. 2007 and references therein). In particular, planets in extreme conditions, given that many of them have planetary radius and tidal-related properties (gyration ra- been discovered in those environments (Guillot et al. 2006; dius, love number, dissipation reservoir, etc.) may change Fortney et al. 2007; Garaud & Lin 2007). To illustrate and in relatively short time-scales. The change in these proper- test our models of exomoon-migration under the effect of ties, that among other effects, determines the strength of the an evolving planet, we will restrict to the well-known re- tidal interaction between the star, the planet and the moon, sults published by Fortney et al. (2007). These results en- may also have an important effect in the orbital evolution compass a wide range of planetary masses, compositions and of exomoons. distances to the host star. In this work we explore how the final fate of an ex- In Figure 1 we show the evolution of radius for two fam- omoon can be affected by the evolution of the physical ilies of planets: Jupiter and Saturn analogues, with different properties of its host planet. We aim at computing the compositions (core mass) and located at different distances time-scale of exomoon orbital evolution taking into account from a solar-mass star. The time-scale of planetary radius the change in planetary properties that may take place evolution strongly depends on planetary mass and distance during migration. For this purpose, we use the same ba- to the star. At close distances, namely aP ∼ 0.1 au, a Jupiter- sic analytical and numerical formalism applied in previ- like planet will change its radius by ∼ 30 per cent within the ous works (Barnes & O’Brien 2002; Sasaki et al. 2012), but first Gyr, while a Saturn-analogue will shrink almost by a plug into them the results of planetary evolution models factor of 2.5 (∼ 60 per cent) in the same time. On the other (Fortney et al. 2007) and recent analytical formulas devel- hand, although planetary radius at a given time strongly oped to calculate the tidal-related properties of fluid giant depends on composition, the fractional evolution as well as planets (Ogilvie 2013; Guenel et al. 2014; Mathis 2015). the time-scale remain the same for planets with different This paper is organized as follows: In Section 2 we core mass. present and describe the results of the planetary evolution Even though planetary radius and evolutionary time- model we adopted here, and the analytical formulas we used scales depend on the distance to the star, and therefore we to describe the tidal-related properties of a bi-layer fluid should model the evolution of our planets depending on their planet. Section 3 explains the physics of exomoon migration assumed orbital radius, we notice that for masses between and the formalism used to describe it. In Section 4 we ap- 0.3 and 3 MJupiter, planets with the same composition, have ply our model to study exomoon migration under different nearly the same radii, provided their distances are in the evolutionary scenarios. Finally, Section 5 summarizes our re- range aP = 0.1 − 1.0 au (see Fig. 3 of Fortney et al. 2007). sults and discusses the limitations, implications and future Planets much closer than 0.1 au are considerably larger. prospects of our model. Hereafter, we will use in all our models, the properties calculated for planets at 1 au. However, this implies that in our results, especially for planets migrating at distances ≪ 1 au, planetary radii and densities could be slightly under and 2 THE EVOLUTION OF GAS GIANTS overestimated, respectively. Models of thermal evolution and internal structure of giant A young gas planet, remains ‘inflated’ for up to a bil- planets have been available in literature for several decades. lion of years. The evolution of its radius comes mostly from

MNRAS 000, 1–9 (2017) Migration of exomoons around evolving planets 3

frequency-dependent coeﬃcients that measure the response of the planet to tidal stresses. Under simplifying assump- − 1 .5 Observed values (references in caption) tions (low planetary obliquity and moon orbit-planetary axis Analytical values (Mathis, 2015) relative inclination), the only non-negligible love number is Io 2 2 − 2 .0 k2 (ω). In physical terms k2 (ω) provide information about the dissipation of tidal energy inside the planet and the inter- change of rotational and orbital angular momentum among − 2 .5 the planet and the moon. In particular the frequency averaged imaginary part of 2 ) k (ω) k /Q p provides an estimation of , − 3 .0 Moon 2 2

/Q Ea rth Ura n u s 2 k +∞ +∞ 2 − 3 .5 dω |k2 (ω)| dω log( k2/Q = Im[k2(ω)] = (1) Sa tu rn ω Q2(ω) ω Ne p tu n e ¹−∞ ¹−∞ 2 − 4 .0 In recent years, several advances have been achieved in the understanding and description of tidal dissipation in the fluid interiors of gas giants (Ogilvie & Lin 2007; Ogilvie − 4 .5 2013; Guenel et al. 2014; Mathis 2015). In particular, analytical expressions for the tidal dissipation reservoir as a Jup ite r − 5 .0 function of the bulk properties of simplified bi-layer fluid − 0 .5 0 .0 0 .5 1 .0 planets have been obtained (Ogilvie 2013). For our investi- log(Rp/R ⊕) gation we will apply the formula adapted by Guenel et al. (2014) from Ogilvie (2013):

Figure 2. Values of k2/Q for several Solar System bodies: Jupiter and Io (Lainey et al. 2009); Saturn (Lainey et al. 2012); the Moon k 100π α5 1 − γ 5 1 − γ −2 2 = ǫ2 1 + α3 1 + α3 (2) (Dickey et al. 1994); the Earth (Ray, Eanes & Chao 1996; Kozai Q 63 1 − α5 γ 2 γ 1968) and Neptune (Trafton 1974). The tidal dissipation reservoir 2 2 is larger in solid bodies (Io, Moon and Earth) than in planets hav- Here ǫ ≡(Ω/Ωc) , where Ω is the planetary rotational Ω = 3 1/2 ing extensive gaseous envelopes (Neptune, Saturn and Jupiter). rate and c (GMp/Rp) is the critical rotation rate; α, β Red crosses show the values of k2/Q estimated with the analytical and γ are dimensionless parameters deﬁned in terms of the formula in Eq. (2), assuming values for α, β and γ as given by bulk properties of the planet as: Mathis (2015).

3 Rc Mc ρe α (1 − β) α = , β = and γ = = , (3) the evolution of its extended atmosphere. Its deep interior, Rp Mp ρc β(1 − α3) however, does not change significantly in the same time-scale where Rc (Rp), Mc (Mp) and ρc are the radius, the mass and neither in composition, nor in mass or radius. Hereafter we the density of the core, respectively; ρe is the density of the will assume that the mass and radius of the planetary solid (fluid) envelope. core remain constant during the relevant time-scales studied This formula is valid only under very specific conditions. in this work. We are assuming that the planet is made of two uniform A change in planetary radius, mean density and rota- layers: a fluid external one and a fluid or solid denser core. tional rate, would have a significant impact in other key me- In our models tidal dissipation occurs only in the turbu- chanical and gravitational properties of the planet. For the lent fluid envelope; for the sake of simplicity we are neglect- purposes pursued here we will focus on the evolution of the ing the inelastic tidal dissipation in the core (Guenel et al. so-called “tidal dissipation reservoir” (Ogilvie 2013), which 2014). Dissipation in the fluid layer arises from turbulent in the classical tidal theory is identified with the ratio k2/Q; friction of coriolis-driven inertial waves (this is the reason being k the (frequency-independent) love number and Q 2 of the strong dependence of k2/Q on the rotational rate ǫ). the so-called “tidal dissipation quality factor” (the ratio of The approximation in Eq. (2) also breaks down if centrifugal the energy in the equilibrium tide and the energy dissipated forces are significant and therefore it only applies in the case per rotational period). when the planet rotates very slowly, i.e. when ǫ ≪ 1. Usually, k2 and Q are estimated from observations of ac- Mathis (2015) estimated the values of α and β for tual bodies (planets and stars). In Figure 2 we present the Jupiter, Saturn, Uranus and Neptune that better match the measured value of k /Q for several Solar System objects. In 2 measured values of k2/Q (see Table 1 in Mathis 2015 and red the case of exoplanets, k2/Q is estimated using simplifying crosses in Figure 2 here). To that end, he assumed educated assumptions about their interior structures, which are usu- estimations of the unknown internal bulk properties of the ally assumed static (k /Q constant). For our purpose here, 2 Solar System giants, namely Rc, Mc and ρc. we need to estimate theoretically the value of k2/Q as a In order to apply Equation 2 to the purposes pursued function of the evolving bulk and interior properties of the here, we need to provide an estimate to the functions of planet. α(t), β(t) and ǫ(t) or equivalently Rc(t), Rp(t), Mc(t) and Ωp(t). For simplicity, we will assume that the liquid/solid core of the planet is already formed or evolves very slowly during m = Love numbers, kl (ω), are, in general, complex most of the envelope contraction, i.e. β(t) β(0). ǫ(t) will

MNRAS 000, 1–9 (2017) 4 Alvarado-Montes et al.

Prot = 8 h Prot = 15 h Prot = 22 h Papaloizou & Terquem 2006; Armitage 2010). At its ﬁnal − − orbit the planet harbors a regular moon (orbital motion in the same direction as planetary rotation) with mass Mm, and M⊕ − whose formation process and/or mechanism to survive mi- − gration are irrelevant here. The moon is orbiting the planet − at an initial nearly circular orbit, with a semi major axis = − parametrized as am(0) fRp(0). Thus, for instance f ≃ 4

) −

p for Saturn’s moon Enceladus and f ≃ 10 for Jupiter’s moon /Q

2 − Europa. k The star-planet-moon interaction raises a complex tidal − log ( − bulge on the planet, which gives rise to star-planet and moon-planet torques, that according to the constant time- − − lag model are given by (Murray & Dermott 2000): − 2 5 3 k2 GM∗ Rp − − τp−∗ ≈− sgn (Ωp − np) (5) − − − − − − − − 2 Q a6 log (t/Gyr) log (t/Gyr) p

2 5 3 k2 GMmRp Figure 3. Evolution of the tidal dissipation reservoir k /Q for τ ≈− sgn (Ω − n ) (6) 2 p−m Q 6 p m Jupiter and Saturn analogues orbiting a Solar-mass star at 1 au. 2 am Colors correspond to different rotational periods (shown at the The latter consideration does not lead to discontinuities top of the graph). Solid and dashed lines correspond to planets for vanishing tidal frequencies (e.g. synchronous rotation), with no core (β → ∞) and a liquid/solid core of 25M⊕, respectively. and it sheds light on a thorough analytical assessment of the tidal-related effects, free from assumptions on the eccentricity (which is crucial when it comes to close-in exoplanets, be computed consistently from the radius provided by the see Leconte et al. 2010). Still, the constant time-lag model planetary evolution models (Figure 1) and the instantaneous is a linear theory and taking into account nonlinear terms rotational rate of the planet, which changes due to the tidal can result in noticeable changes (Weinberg et al. 2012). interaction with the star and the moon. For α(t), we will In the linear regime the torques exerted by the moon assume an asymptotic value α(∞), similar to that of Solar and the star add up, τtot ≈ τp−∗ +τp−m, and produce a change System planets (Table 1 in Mathis 2015); the value at any in the rotational angular momentum of the planet Lp(t) = 2 2 time will be obtained from: Ip(t)Ωp(t), where Ip(t) = κ MpRp(t) is the planet’s moment of inertia, and κ2 is the gyration radius, given by the Newton Rp(∞) second law: α(t) = α(∞) (4) Rp(t) dLp ≈ τp−∗ + τp−m (7) In Figure 3 we show the result of applying Eq. (2) and dt the previously described prescription, to compute the evolu- The reaction to the moon-planet torque, τm−p = −τp−m tion of k /Q for Jupiter and Saturn analogues at 1 AU. For 2 modifies the orbital angular momentum of the moon Lm = illustration purposes, a set of different constant rotational Mmam(t)nm(t) as follows: rates is assumed. In both cases we have assumed asymptotic α values equal to that of Jupiter and Saturn respectively. dLm As expected k2/Q strongly depends on rotational rate. ≈−τ (8) dt p−m A change of Prot by a factor of 3, change k2/Q by almost one order of magnitude. As the planet contracts the value of α We are assuming that the orbit of the moon is not mod- 2 3 goes up as 1/Rp; at the same time ǫ goes down as Rp. For ified by stellar perturbations nor the tidal interaction of the 2 5 2 small values of α, k2/Q ∝ ǫ α ∝ Rp, the planet’s contraction moon with the planet. Although both cases may be rele- produces a net increase in tidal dissipation. vant in realistic scenarios (Sasaki et al. 2012), the inclusion of these effects will not modify considerably our general conclusions. 3 MOON ORBITAL MIGRATION Under the assumption of a relatively slow planetary radius variation, i.e. dIp/dt ≪ (Ip/Ωp)dΩp/dt, Eqs. (7,8) can Let us now consider an already formed close-in giant planet be rewritten as: with mass Mp, initial radius Rp(0) and rotation rate Ωp(0), around a star of mass M∗. It is also assumed that the Ω 3 2 planet is in a final nearly circular orbit with semi ma- d p 3 k Rp (GM∗) = − 2 sgn (Ω − n )+ a P 2 6 p p jor axis p (orbital period orb). In addition, we suppose dt 2 Q k G " Mpap that the planet has migrated from its formation place (9) M2 to its final orbit, in a time scale much shorter than its m n4 sgn (Ω − n ) 3 m p m thermal evolution (Tanaka et al. 2002; Bate et al. 2003; Mp #

MNRAS 000, 1–9 (2017) Migration of exomoons around evolving planets 5

Figure 4. Moon orbital migration under the interplay of the star-planet-moon tidal interactions. Upper panel: a schematic representation of the process. Lower panel: an actual integration of the tidal evolution diﬀerential equations, for a Saturn analogue, located at an orbit with Porb = 80 days around a Solar-mass star. The moon is an Enceladus-mass object with an initial orbit of am(0) = 2Rp(0). For obtaining this plot we assumed that the values of k2/Q and Rp remain constant (quasistatic scenario).

migrate outwards (see Figure 4). However, since the planetary rotational rate is also slowing down, under certain cir- M R5 dnm = 9 k2 m p 16/3 Ω t n (t ) = Ω (t ) − nm sgn ( p − nm). (10) cumstances and at a finite time sync, m sync p sync . At dt 2 Q 5/3 8/3 G Mp this stage, migration will stop momentarily until nm starts to outpace Ω (n > Ω , dn /dt > 0). Therefore, inward Ω p m p m Under general circumstances, p > np and Mp/M∗ ≫ migration is triggered and the moon will eventually collide, Ω Mm/Mp, and hence d p/dt < 0. Consequently the planet ro- or be obliterated by the planet inside the Roche radius. tation slows down monotonically until it finally “locks-down” We show in Figure 4 an schematic representation of this Ω 1 at around p ∼ np. process and an actual solution to Eqs. (9,10) for a typical On the other hand, the evolution of nm is more complex. planet-moon-star system. If the moon starts at a relatively large distance from the The described tidal-induced moon migration process Ω planet where nm(0) < p(0), then dnm/dt < 0. In addition, has three relevant time-scales: the orbital mean motion will slow down and the moon will • The synchronization time, tsync which can be defined by the condition: 1 its final rotational state actually depends, among other factors, on its orbital eccentricity and planetary interior properties, see e.g. Cuartas-Restrepo et al. 2016). nm(tsync) = Ωp(tsync) (11)

MNRAS 000, 1–9 (2017) 6 Alvarado-Montes et al.

• The decay time, tdecay given by: Quasistatic 6 nm(tsync + tdecay) = nRoche (12) Porb = 60. 0 d Porb = 70. 0 d Porb = 80. 0 d = −3/2 Here nRoche GMpaRoche is the orbital mean motion at the Roche radius. For the purpose of this work we will as- 5 p 1/3 sume aRoche ≈ 2(ρm/ρp) Rp (Roche 1849; Esposito 2002), where ρm is the mean density of the moon. • and the “round-trip” time, tround−trip: 4 p tround−trip = tsync + tdecay (13) R / m a 3 4 RESULTS It is usual that when solving Eqs. (9,10) all the properties in the right-hand side, including the tidal dissipation reservoir 2 2 k2/Ωp, planetary radius Rp and gyration radius κ (moment of inertia), be assumed as constants. This is the way the problem has been analyzed in literature (Barnes & O’Brien 1 2002; Sasaki et al. 2012). In a more realistic case we must 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 allow that these quantities evolve independently, or coupled t [Gyr] with orbital migration. However, in order to individualize the effect that each property has on moon orbital evolution, we will consider four different evolutionary scenarios we have Figure 5. Moon orbital semi major axis as function of time for labeled as quasistatic, unresponsive, dynamical and realistic. a warm Saturn-Superenceladus system. Solid, dashed and dotted lines correspond to planetary orbital periods of 60, 70 and 80 days, respectively. 4.1 Quasistatic migration In this case all of the relevant mechanical properties (k /Ω , 2 p larger orbital angular momentum and outwards/inwards mi- R and κ) are assumed nearly constant or varying very slowly p gration takes also longer (see left column in Figure 6). during moon migration. This is the most common scenario found in literature (Barnes & O’Brien 2002; Sasaki et al. 2012). Although unrealistic, the results obtained in this sce- 4.2 Unresponsive migration nario provide us with first order estimations of the time- scales of moon migration and its dependence on key proper- The first significant effect of planetary evolution on moon 2 ties of the system. migration is observed when we let that Rp and κ change in For illustration purposes, in Figure 5 we show the re- time. To disentangle the effect of planetary evolution from sults of integrating equations (9) and (10) for a particular other effects at play, we will assume that, despite the obvious case: a Saturn-mass planet orbiting a solar mass star with effect that a varying Rp will have in the tidal response of the planet, tidal dissipation reservoir k /Q remains constant. We Porb = 60 − 80 days (ap ≈ 0.3 − 0.4 au) and having a moon 10 2 unresponsive times the mass of Enceladus at am(0) = 3Rp. Hereafter, we call this scenario migration. will use this ‘warm Saturn-Superenceladus’ system to illus- In Figure 7 we show the result of integrating Eqs. (9,10) trate the results in each scenario and perform comparison for the same case studied in the quasistatic case, but includ- among them. It should be understood, however, that the ing now a variable planetary radius (lower plot). conclusions of our comparison are not substantially modi- The main effect that the evolution of planetary radius fied, if we use a system with different properties. Only the has on moon orbital migration, arises from the fact that an moon migration time-scales will change. Precisely, and in evolving young planet is effectively larger than a static one. 5 order to gain some insight on the dependence of the rele- Since tidal torques scale-up with radius as Rp, even a ∼ 30 vant time-scales of the properties in the system, we show in per cent larger radius, produce tidal torques ∼ 4 times larger Figure 6 contour plots of round-trip times in the quasistatic at earlier phases of planetary evolution as compared to those scenario. produced in the quasistatic case. This effect speeds up both Migration time-scales are very sensitive to planetary moon orbital migration and planetary rotational braking. distance (see right column in Figure 6). Since stellar torque As a result, migration timescales are reduced by almost a 4 factor of 2 (see Figure 7). on the planetary bulge goes as Porb (Eq. Equation 5), closer planets loss their rotational angular momentum faster, so that moons do not reach distant positions and synchroniza- 4.3 Dynamical migration tion between Ωp and nm is achieved much earlier. Time-scales are also very different among systems hav- One of the key features of our model is the consistent calcu- ing different planetary and moon masses. More massive plan- lation of the tidal dissipation on giant planets as a function ets have a larger rotational inertia and hence larger synchro- of their evolving properties. As we have already shown in nization times. On the other hand more massive moons have Section 3, turbulent friction of inertial waves sustained by

MNRAS 000, 1–9 (2017) Migration of exomoons around evolving planets 7

Figure 6. Contour plots of the total round-trip time-scale tround−trip by assuming diﬀerent properties for the planet-moon system. In all of the cases we consider M∗ = 1 M⊙.

Unresponsive Dynamical 7 P = 60. 0 d P = 70. 0 d P = 80. 0 d Porb = 60. 0 d Porb = 70. 0 d Porb = 80. 0 d orb orb orb

5 p p R R / /

m 4 m a a

1 1.4 1.3 ) /Q Saturn 1.2 2 k R /

1.1 log( R 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1 1 t [Gyr]

Figure 7. Upper plot: moon orbital semi major axis as function of Figure 8. Upper plot: moon orbital semi major axis as function time for a warm Saturn-Superenceladus system in the unrespon- of time for a warm Saturn-Superenceladus system. Lower plot: sive scenario. Lower plot: evolution of planetary radius. Solid, evolution of k2/Q. Solid, dashed and dotted lines correspond to dashed and dotted lines correspond to planetary orbital periods planetary orbital periods of 60, 70 and 80 days, respectively. of 60, 70 and 80 days, respectively.

Coriolis forces is the main driver of tidal dissipation in ﬂuid giant planets. As such, it will strongly depend on the ro- nario, assumes that the radius of the planet Rp and its inte- tational velocity of the planet, which varies as the planet rior structure remain almost constant whereas the rotational interact with the star and the moon. rate Ωp changes in time. In Figure 8 we show the eﬀect that Our third evolutionary scenario, the dynamical sce- a varying Ωp has on the tidal dissipation reservoir k2/Q and

MNRAS 000, 1–9 (2017) 8 Alvarado-Montes et al. the impact that those changes have on the moon orbital evolution. Realistic At the beginning, when the planet is rotating faster, Porb = 60. 0 d Porb = 70. 0 d Porb = 80. 0 d k2/Q has its largest value. In this phase moon migration and planetary rotational braking occur in a shorter time- scale than in the quasistatic case. As the planet breaks, k2/Q

falls by almost one order of magnitude in several hundreds p R of Myr. Less tidal energy is dissipated by inertial waves in /

m the liquid envelope of the planet, and moon migration stalls. a We have veriﬁed that independent of moon initial distance to the planet, planet or moon mass, or planetary distance to its host star, moon migration around a liquid 2 giant planet will never end in a course of collision or dis- 2 ruption of the moon inside the Roche radius. This result ) is in starking contrast with what was previously expected /Q 2

(Barnes & O’Brien 2002; Sasaki et al. 2012). However, it is k worth to note that final conclusions can be affected accord- log( ing to the dissipation mechanism adopted in the process, which can lead to different scenarios from those proposed Saturn 2 R in this work. For instance, the apparently strange behaviour / p of Saturn’s moons’ tidal evolution implies that Saturn’s in- R 2 2 ternal tidal dissipation may be strongly peaked in forcing frequency space, leading to nonlinear effects that are not considered for our purposes.

Figure 9. Upper plot: moon orbital semi major axis as function 4.4 Realistic migration of time for a warm Saturn-Superenceladus system in the unresponsive scenario. Middle plot: evolution of k2/Q. Lower plot: Finally, the realistic scenario is that on which all the proper- evolution of planetary radius. Solid, dashed and dotted lines cor- ties (i.e. Rp, κ, Ωp and k2/Q) vary as functions of time. The respond to planetary orbital periods of 60, 70 and 80 days, re- result of including all the effects on moon orbital evolution spectively. is shown in Figure 9. As in the dynamical case, tidal torques are large at the beginning, accelerating moon migration and planetary rota- 5 SUMMARY AND CONCLUSIONS tional braking. Once the moon has reached the synchronous distance it seems to stall. However, since the planet is still In this paper we have used the tidal model of contracting, and the moon is even migrating inwards, its Barnes & O’Brien (2002), recently extended and updated distance, as measured in planetary radii seems to be mono- by Sasaki et al. (2012), to study the orbital evolution of ex- tonically increasing. Although the increase in the relative omoons around close-in giant planets. As a novel feature we distance does not modify the strength of the gravitational have consistently included in those models, the evolution of or tidal forces on the moon, it contributes to monotonically the physical bulk properties of the planet and its evolving drive the moon away of a collision or its tidal disruption. response to tidal stresses. In large time-scales this would cause that moons pass along Our “evolutionary” moon migration model, relies on the the critical semi-major axis at ∼ 0.48 RH (Domingos et al. well-known results of close-in giant planet thermal evolution 2006) becoming their orbits unstable and prone to be ejected by Fortney et al. (2007), which predict the bulk properties of from planetocentric to heliocentric orbits. Therefore, this planets with different composition and at different distances situation leads to the possibility that under certain circum- from their host-star. To model the response of the planet to stances, the moon becomes a new dwarf rocky planet, some- tidal stresses and predict tidal dissipation of rotational en- thing that we call a ‘ploonet’. ergy and transfer of angular momentum towards the moon, It is noticeable that dynamical and realistic situations we use the recent analytical models of Ogilvie (2013). In seem to have a similar behaviour. Even so, the first one ex- that model the value of the frequency averaged ratio k2/Ωp hibits an inflection point at short time-scales which is not is calculated as a function of planetary bulk properties and prominent in the second one. Hence, the dynamical sce- rotational rate. nario tends to park the moon in a quasi-stationary orbit, For our numerical experiments, and in particular for the constraining any future orbital evolution. On the contrary, calculation of k2/Ωp, we assume that the interior of giant- the realistic case allows the moon to migrate outward in- planets can be modeled as constituted by two constant den- definitely. In other words, the spin-orbit synchronization is sity layers, an outer one, made of a low density liquid, and always present for short time-scales in situations where the a central liquid/solid core. We assumed that the dissipation planet’s size remains constant over time. The difference be- of tidal energy occurs via turbulent friction of inertial waves tween both cases lies in that the realistic scenario couples in the liquid outer layer. We neglected the contribution of the gradual change of Rp into the evolution of k2/Q, by tidal dissipation inside the core. means of equation (2), influencing the underlying dynamic We found that orbital migration of exomoons is signifi- and therefore the final fate of the moon. cantly modified when the evolution of planetary bulk prop-

MNRAS 000, 1–9 (2017) Migration of exomoons around evolving planets 9 erties is included. In the most studied case where the planet Cuartas-Restrepo P. A., Melita M., Zuluaga J. I., Portilla-Revelo bulk properties evolve very slowly with respect to moon mi- B., Sucerquia M., Miloni O., 2016, MNRAS, 463, 1592 gration time-scales, and tidal dissipation is completely inde- Dickey J. O., et al., 1994, Science, 265, 482 pendent of planetary rotational rate (quasistatic scenario), Domingos R. C., Winter O. C., Yokoyama T., 2006, MNRAS, moons migrate outwards and then inwards, facing a collision 373, 1227 Esposito L. W., 2002, Reports on Progress in Physics, 65, 1741 with the planet or tidal obliteration/disruption in relatively Fortney J. J., Marley M. S., Barnes J. W., 2007, ApJ, 659, 1661 short time-scales (several Gyrs). If only the planetary radius Garaud P., Lin D. N. C., 2007, ApJ, 654, 606 changes (unresponsive scenario), outwards then inwards mi- Gong Y., Zhou J., Xie J., Wu X., 2013, ApJ, 769, L14 gration is faster and the fate of the moons are similar. How- Guenel M., Mathis S., Remus F., 2014, A&A, 566, L9 ever if we let the tidal response to depend on rotational Guillot T., Santos N. C., Pont F., Iro N., Melo C., Ribas I., 2006, rate (as is the case if tidal energy is dissipated in the liquid A&A, 453, L21 outer envelope of the planet) and/or if the planet contracts Heller R., Pudritz R., 2015, ApJ, 806, 181 in a similar or shorter time-scale (i.e. dynamical and realis- Heller R., et al., 2014, Astrobiology, 14, 798 ´ tic scenarios, respectively), the moon never falls back into Kipping D. M., Forgan D., Hartman J., NesvornýD., Bakos G. A., the planet. This result is independent of the planet, star and Schmitt A., Buchhave L., 2013, ApJ, 777, 134 Kozai Y., 1968, PASJ, 20, 24 moon masses or their mutual distance. Lainey V., Arlot J.-E., Karatekin O.,¨ van Hoolst T., 2009, Nature, This is a completely unexpected, still fortunate, out- 459, 957 come of the effect of planetary evolution on moon orbital Lainey V., et al., 2012, ApJ, 752, 14 migration. If confirmed by further more detailed models, Leconte J., Chabrier G., Baraffe I., Levrard B., 2010, A&A, large and detectable regular exomoons may have survived 516, A64 orbital migration around already discovered exoplanets and Mathis S., 2015, in Martins F., Boissier S., Buat V., Cambrésy L., could be awaiting a future detection. Petit P., eds, SF2A-2015: Proceedings of the Annual meeting Interestingly, the dependence of an exomoon’s final fate of the French Society of Astronomy and Astrophysics. pp 283– arXiv:1510.05639 on the evolution of their host close-in giant planets, could 288 ( ) be used to constraint planetary evolutionary models, and/or Mayor M., Queloz D., 1995, Nature, 378, 355 Murray C. D., Dermott S. F., 2000, Solar System Dynamics (New to model the planet’s interior structure and its response to York: Cambridge Univ. Press) tidal stresses. In the forthcoming future, when missions such Namouni F., 2010, ApJ, 719, L145 as Transiting Exoplanet Survey Satellite (TESS), Charac- Ogilvie G. I., 2013, MNRAS, 429, 613 terizing Exoplanets Satellite (CHEOPS), the James Webb Ogilvie G. I., Lin D. N. C., 2007, ApJ, 661, 1180 Space Telescope (JWST) and Planetary Transits and Oscil- Papaloizou J. C. B., Terquem C., 2006, lations of stars (PLATO) (see Rauer et al. 2014 and refer- Reports on Progress in Physics, 69, 119 ences therein) hopefully provide us a definitive confirmation Rauer H., et al., 2014, Experimental Astronomy, 38, 249 (or rejection) of the existence of exomoons around close-in Ray R. D., Eanes R. J., Chao B. F., 1996, Nature, 381, 595 planets, these results will be confirmed. Roche E., 1849, Acad. Montpellier, 1, 243 Sasaki T., Barnes J. W., O’Brien D. P., 2012, ApJ, 754, 51 Tanaka H., Takeuchi T., Ward W. R., 2002, ApJ, 565, 1257 Trafton L., 1974, ApJ, 193, 477 ACKNOWLEDGEMENTS Weinberg N. N., Arras P., Quataert E., Burkart J., 2012, ApJ, 751, 136 We thank the referee J. W. Barnes, whose valuable comments allowed us to improve the manuscript. We also ap- This paper has been typeset from a T X/LAT X file prepared by preciate the useful comments to the initial versions of this E E the author. manuscript provided by R. Canup. J.A.A. is supported by the Young Researchers program of the Vicerrector´ıa de In- vestigación, J.I.Z. by Vicerrector´ıa de Docencia of the Uni- versidad de Antioquia (UdeA) and M.S. by Doctoral Pro- gram of Colciencias and the CODI/UdeA. This work is supported by Vicerrectoria de Docencia-UdeA and the Estrate- gia de Sostenibilidad 2016-2017 de la Universidad de An- tioquia. Special thanks to Nadia Silva for the review of the manuscript.

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