Tidal dissipation in a homogeneous spherical body. II. Three examples: Mercury, Io, and Kepler-10 b
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Tidal dissipation in a homogeneous spherical body.II. Three examples: Mercury, Io, and Kepler-10 b
Valeri V. Makarov and Michael Efroimsky
US Naval Observatory, 3450 Massachusetts Avenue NW, Washington DC 20392 [email protected] , [email protected]
ABSTRACT
In our recent study (Efroimsky & Makarov 2014), we derived from the first principles aformula for the tidal heating rate in a tidally perturbed homogeneous sphere. We compared ourresult with the formulae used in the literature, and we pointed out the differences. Now, usingthis result, we present three case studies – Mercury, the enigmatic Kepler-10 b, and a triaxial Io.A very sharp frequency dependence of k /Q near spin-orbit resonances yields a similarly sharpdependence of k /Q (and, therefore, of tidal heating) upon the spin rate. This indicates thatphysical libration may play a major role in the tidal heating of synchronously rotating planets.The magnitude of libration in the spin rate being defined by the planet’s triaxiality, the lattershould be a factor determining the dissipation rate. Other parameters equal, a synchronouslyrotating body with a stronger triaxiality should generate more heat than a similar body of a moresymmetrical shape. Further in the paper, we discuss possible scenarios where initially triaxialobjects melt and lose their triaxiality. Thereafter, dissipation in them becomes less intensive;so the bodies solidify. The tidal bulge becomes a new permanent figure, with a new triaxialitylower than the original. In the paper, we also derive simplified, approximate expressions for thedissipation rate in a rocky planet of the Maxwell rheology, with a not too small Maxwell time(longer than the inverse tidal frequency). The three expressions derived pertain to the casesof a synchronous spin, a 3:2 resonance, and a nonresonant rotation; so they can be applied tomost close-in super-Earth exoplanets detected thus far. In such bodies, the rate of tidal heatingoutside of synchronous rotation is weakly dependent on the orbital eccentricity and equator’sobliquity, provided both these parameters are small or moderate. According to our calculation,the rocky Kepler-10 b, which is one of the densest exoplanets known to date, could hardly survivethe great amount of tidal heating without being synchronised, circularised and also reshapedthrough a complete or partial melt-down.
1. Motivation and plan
In the work by Efroimsky & Makarov (2014),we derived from first principles a formula for thetidal dissipation rate in a homogeneous sphericalbody. When restricted to the special case of anincompressible body spinning synchronously, thatresult was compared to the commonly used expres- sion from Peale & Cassen (1978, Eqn. 31), andthe differences were pointed out. Now, using thetheoretical exposition from Efroimsky & Makarov(2014), we demonstrate how tidal dissipation canbe estimated for synchronous and asynchronousrocky planets.Section 2 serves to remind the said expressionfor the tidal dissipation rate. It is compared with1he analogous formulae from Kaula (1964) andPeale & Cassen (1978).Section 3 gives an overview of the popular sim-plified formulae derived from the theory by Peale& Cassen and explains the highly restrictive con-ditions, under which these formulae can be used.Section 4 presents the first example, Mercury.We show that tidal heating is not likely to haveplayed a major role in the history of this planet,despite its considerable eccentricity and the factthat Mercury is in the 3:2 spin-orbit resonance.Sections 5 addresses the second practical ex-ample, tidal heating in Io. We provide argumentsin favour of a hypothesis that the energy dampingrate in synchronous bodies may be sensitive to tri-axiality. This sensitivity stems from a very sharp,kink-shaped frequency-dependence of k /Q nearresonances, which is within the range of physical li-bration for significant values of triaxiality. Our hy-pothesis bears a qualitative character and shouldbe propped up by numerical modeling, which willbe presented elsewhere.Section 6 is devoted to the third example,Kepler-10 b, a very dense super-Earth that maysooner be classified as a super-Mercury (Selsis etal. 2013). Given the extreme proximity of theplanet to its host star (less than 0.017 AU), wepresume that the planet is experiencing a con-siderable tidal interaction and may, therefore, beoverheated. The mantle’s response in this caseis viscoelastic and may be approximated with theMaxwell model. Assuming finite values of eccen-tricity and equator obliquity, we estimate the rateof energy dissipation in Kepler-10 b, for the case ofsynchronism and for other rotational states. Tidalheating in this planet becomes so intense that thetemperature should be increasing by several de-grees per year, if the eccentricity is pumped up bythe companion planet. We complete this sectionby sketching possible scenarios of rotational andthermal evolution of such close-in planets subjectto extreme tides, including episodic melt-downand reshaping of their surfaces.In Section 7, we provide three simplified, ap-proximate expressions for the dissipation rate: onefor a synchronised planet, another for a planet ina nonresonant rotation, and a third for a planettrapped in the 3:2 spin-orbit resonance. Theseformulae are derived for a specific case when the rheology is viscoelastic (Maxwell, with no An-drade creep) and the Maxwell time is not too small(larger than the inverse tidal frequency).
2. Tidal dissipation of energy
Consider a planet of mass M that is tidallydisturbed by an external body of mass M ∗ . Asseen from the planet, the perturber describes anorbit parameterised by the Keplerian variables a, e, i, ω, Ω , M , which are: the semimajor axis,eccentricity, inclination, agrument of the pericen-tre, longitude of the node, and mean anomaly.In the frame of the planet, the external tide-raising potential can be expanded in a Fourier se-ries whose terms will contain sines and cosines of ω lmpq t . Here t is time and ω lmpq are the Fouriertidal modes. As explained, e.g., in Efroimsky &Makarov (2013), these are given by ω lmpq = ( l − p ) ˙ ω + ( l − p + q ) ˙ M + m ( ˙Ω − ˙ θ ) , (1) lmpq being integers, θ and ˙ θ being the rotationangle and spin rate of the disturbed body, and ˙ M being the perturber’s “anomalistic” mean motion.While the Fourier modes ω lmpq can assume eithersign, the resulting physical forcing frequencies arepositive definite: χ lmpq = | ω lmpq | . (2)In Efroimsky & Makarov (2014), we derive a gen-eral formula for the time-averaged damping rate.When the apsidal precession of the perturber, asseen from the perturbed body, is uniform, the rateis: h P i = G M ∗ a ∞ X l =2 (cid:18) Ra (cid:19) l +1 l X m =0 ( l − m )!( l + m )! (2 − δ m ) l X p =0 F lmp ( i ) ∞ X q = −∞ G lpq ( e ) ω lmpq k l ( ω lmpq ) sin ǫ l ( ω lmpq ) , (3)where k l ( ω lmpq ) and ǫ l ( ω lmpq ) are the dynamicalLove numbers and tidal phase lags.Being even functions of the tidal modes, the dy-namical Love numbers may as well be understoodas functions of the physical frequencies (2): k l ( ω lmpq ) = k l ( χ lmpq ) . (4)2he phase lags are odd functions of ω lmpq andhave the same sign as ω lmpq . So they may bewritten down as ǫ l ( ω lmpq ) = | ǫ l ( ω lmpq ) | Sgn ω lmpq = ǫ l ( χ lmpq ) Sgn ω lmpq , (5)where ǫ l ( χ lmpq ) are non-negative, because so are χ lmpq . All in all, we have: k l ( ω lmpq ) sin ǫ l ( ω lmpq )= k l ( χ lmpq ) sin ǫ l ( χ lmpq ) Sgn ω lmpq , (6)where k l ( χ lmpq ) sin ǫ l ( χ lmpq ) are positive definiteand are often denoted as k l /Q l .The frequency dependence k l ( χ lmpq ) sin ǫ l ( χ lmpq )is derived in the Appendix. It is a functional of theplanet’s rheology and also of its size and mass. Atlower frequencies, self-gravitation is playing a keyrole in tidal damping, so the tidal quality factorsdefined through 1 /Q l = sin ǫ l ( χ ) differ consider-ably from the seismic quality factor Q . However,they approach Q at higher frequencies where rhe-ological properties become more important thangravity (Efroimsky 2012a,b).
3. Limitations on a previously used for-mula for tidal dissipation
Jackson et al. (2008) estimated tidal dissipa-tion in 18 exoplanets, relying on the following ex-pression for the average damping rate: h P i = 3619 π ρ n R µ Q e , (7)where ρ is the mean density, µ is the rigidity,and Q is the tidal quality factor. The formula wasadopted from the paper by Peale et al. (1979) whoreferred to their preceding work (Peale & Cassen1978). We, however, failed to find an explicit pres-ence of this formula in Ibid .In other publications (e.g., Mardling 2007, Mur-ray & Dermott 1999, Segatz et al. 1988), a differ-ent expression is commonly used: h P i = 212 k Q G M ∗ R a n e , (8) at times accompanied with a reference to the samepaper by Peale & Cassen (1978). Insertion of theapproximate expression k ≈ ρ g R µ , (9)in the equation (7) transforms the latter into theequation (8), although with a different numericalfactor; namely, with 9 instead of 21 / i of the perturber’s orbit onthe equator of the perturbed body is setequal to zero;b. the terms of power 4 and higher in the ec-centricity e are neglected;c. only quadrupole ( l = 2 ) inputs are in-cluded; d. the consideration is limited to bodies rotat-ing synchronously ;Under the assumptions [a - c], only the terms with( lmpq ) = (201 , −
1) , (2011) , (220 , −
1) , (2201)are to be taken into account. From the formula(1), we see that for all these terms the physicalforcing frequency χ lmpq ≡ | ω lmpq | approximatelyassumes the same value n , provided the assump-tion [d] is also imposed, i.e., provided that ˙ θ = n .This way, in the case of synchronous spin , k /Q assumes the same values for all the four termstaken into account within this approximation.Now consider a situation where items [a] and[b] are relaxed, items [c] and [d] are kept, and anextra, highly restrictive item is added:e. the Constant Phase Lag (CPL) model oftides is adopted, so the inverse tidal qual-ity factor Q − lmpq ≡ sin | ǫ l ( ω lmpq ) | assumesthe same value for all Fourier modes ω lmpq . While l = 2 inputs are usually sufficient, sometimes termswith higher values of l can not be neglected. One suchcase is Phobos, whose orbital evolution is influenced bythe l = 3 and, perhaps, even the l = 4 terms (Bills etal. 2005). Another class of exceptions is constituted byclose binary asteroids. The topic was addressed by Taylor& Margot (2010), who took into consideration terms up to l = 6 . h P i = k Q G M ∗ R a n (cid:20) (cid:18) i − i (cid:19) + (cid:18)
212 + 152 i − i (cid:19) e + (cid:18) i − i (cid:19) e (cid:21) + O ( i ) + O ( e ) . (10)Importantly, for bodies with a significant i andsmall e the term 3 i / e / k /Q ≡ k sin ǫ is set frequencyindependent.
4. Case study I: Mercury
Of all the planets in the solar system, Mercuryis the only one captured into a 3:2 spin-orbit res-onance. It is the closest to the Sun and has thelargest orbital eccentricity. This makes one won-der if tidal heating could play any role in Mer-cury’s evolution and segregation.In the expansion (3) for the damping rate, aterm numbered with lmpq contains a multiplier ω lmpq . For this reason, when the planet is inan lmpq spin-orbit resonance, the input from the lmpq Fourier mode into tidal heating is zero. Forexample, the dominating (at small eccentricities) The expression (10) is valid for the CPL model (i.e., fora frequency independent k /Q ). An analogous formulafor the CTL model (with k /Q linear in frequency) waswritten down by Wisdom (2008). Naturally, the higher co-efficients in our formulae differ, although the leading termscoincide and contain the same coefficient 21 / τ M χ ≫ semidiurnal Fourier tidal mode 2200 contributesno heat when the rotator is in the exact 1:1 reso-nance. The physical meaning of this circumstanceis that a Fourier component of the tidal bulge,which moves with the same angular velocity asthe perturber, does not lag and, therefore, gen-erates no friction. The other components of thebulge, however, do lag and, thereby, do contributeto heating.One exception is the case of a synchronous ro-tation with e = 0 , a situation where tidal dissipa-tion ceases completely, the tidal bulge being at restwith respect to the perturbed body. Ultimately,any planet that happens to be a sole companionto its star, should come to this state of completecircularisation and synchronisation, which is theonly long-term equilibrium state (Hut 1980, Bam-busi & Haus 2012).However, Mercury (as well as several knownclose-in exoplanets) is a part of a multiple-planetsystem. The pull from its fellow planets pre-vents Mercury’s eccentricity from keeping too lowa value. Detailed numerical simulations demon-strate that Mercury’s eccentricity has varied overæons within a rather wide interval, mostly be-tween 0.1 and 0.3 (Correia & Laskar 2009), so itscurrent value (0.20563) is not extraordinarily highfor this planet. However, this significant eccentric-ity is not a very important factor in the thermalhistory of Mercury, because in the series (3) theleading term (the one with lmpq = 2200 ) is ofthe order of O ( e ) .Figure 1 illustrates the dependence of thedamping rate on the dimensionless spin rate ˙ θ/n .The left plot depicts a very narrow vicinity of theresonant frequency, and shows in detail the cleftcaused by the vanishing second-largest tidal term lmpq = 2201 . The cleft is hardly of any practi-cal significance, because the rotation rate of theplanet performs forced libration within a muchwider range than the one in the graph. The widthof this feature is defined mostly by the averageviscosity, or by the Maxwell time of the body.The right plot gives the same dependence for amuch wider interval of values of the spin rate, andfor three values of eccentricity, e = 0 . . . µ = 0 . · Paand Maxwell time τ M = 500 yr, which are closeto Earth’s values. Θ (cid:144) n L og H â E (cid:144) â t L W - - ´ ´ ´ ´ rate of rotation Θ (cid:144) n - â E (cid:144) â t W Fig. 1.— Time-averaged tidal dissipation rate dE/dt = h P i in a uniform Mercury captured intothe 3:2 spin-orbit resonance. Left: decimal loga-rithm of the dissipation rate versus the normalisedrotation frequency ˙ θ/n , in a close vicinity of theresonance, for e = 0 . e = 0 . e = 0 . e = 0 . Fe S layer at the top of the core (Padovan et al.2014). We would suppose that the actual rate ofdissipation can be an order of magnitude higherthan what is shown in Figure 1. Even with thisupgrade, however, the estimated rate of dissipa-tion is much smaller than the production of elec-tric power by the mankind. It is also very closeto the present-day tidal heating rate of the Moon,which is log ( dE/dt ) = log h P i = 9 . dE/dt = h P i being measured in Watts and thelogarithm being decimal. So tidal heating is un-likely to have made an impact on the formation ofMercury’s molten core.
5. Case study II: Io
The most famous manifestation of tidal dissi-pation is the volcanism of Io. That Io is subjectto intense tidal heating was pointed out by Pealeet al. (1979) in their cornerstone work which drewconsiderable attention to the problems of ther-mal balance in moons. Although the authors bril-liantly predicted the semi-molten state of Io’s in-terior, their estimate of damping rate may needre-examination. Back in 2012, the world annual electricity net generationwas about 22500
TW h . Qualitatively, our conclusion that tidal heating does notadd much to the energy budget agrees with the study bySchubert et al. (1988). In
Ibid. , thermal convection lastsfor 3 Gyr without tidal heating but can, under favourableconditions, be maintained for additional 225 Myr if tidesare taken into account. Quantitative comparison of our re-sults with those from
Ibid. is however impossible, becausethose authors employed an old model assuming that Mer-cury formed hot, with early differentiation of the iron core.This is no longer regarded probable – see, e.g., Noyelles etal. (2014) and references therein.Our conclusions are in a good agreement with theresults obtained by Bills (2002). Although Bills claimsthat tidal damping in Mercury is important, his formu-lae evidence the opposite. Estimating the tidal dampingrate, the author forgot to multiply the overall factor of n R / (2 G ) = 2 . × W by the sum of the seriesitself – which, very roughly, is of the order of k /Q . Withthat omission corrected, Bills’s estimate would become sev-eral orders of magnitude lower.
5o compute the dissipation intensity, we usedour equation (3), with Io’s inclination set to zero.With the maximal moment of inertia written as C = ξM R , the coefficient ξ was assumed tobe ξ = 0 . µ = 0 . × kg m − s − (Eckhardt1993). The least-known parameter, the Maxwelltime, was set to be τ M = 1 day, close to theexpected value for Titan (F. Nimmo, private com-munication). The Andrade time, τ A , was set toinfinity. Thus, it was assumed that the reactionof the material is purely Maxwell, with no An-drade creep (see the Appendix for details and ref-erences). The motivation for the latter decisioncomes from the fact that Io’s mantle is partiallymolten, so the friction in it is mainly viscoelastic,with no significant input from dislocation unpin-ning. - - ´ ´ ´ ´ ´ ´ rate of rotation Θ (cid:144) n - â E (cid:144) â t W Fig. 2.— Time-averaged rate of energy dissipationin Io, dE/dt = h P i , as a function of the spin rate (cid:5) θ , in the vicinity of the 1:1 spin-orbit resonance.Figure 2 illustrates how the heating depends onthe angular velocity (cid:5) θ in the vicinity of the 1:1spin-orbit resonance. The figure shows the damp-ing rate dE/dt = h P i plotted against the quan-tity ˙ θ/n − ≈ × W. This is signif-icantly larger than the original estimate by Pealeet al. (1979), but is somewhat smaller than theestimate (9 . ± . × W obtained fromastrometric observations by Lainey et al. (2009)who also used an extra assumption that the CPLmodel is applicable to Io. Given the intrinsic un-certainty of some of our parameters, we find thecoincidence up to a factor less than two to be agood match. The fact that the model reproduces(within a factor of two) the result from Lainey etal. (2009) may argue in favour of the Maxwell timebeing close to one day. For purely Maxwell rheol-ogy, the quality factor is inversely proportional to τ M if τ M n ≪
1, which is the case here. Therefore,setting τ M = 0 . τ M ≈ . average spin ratethat stays resonant, while the instantaneous spinrate undergoes variations over the period of aver-aging. The planet approaches a spin-orbit reso-nance relatively slowly, but is captured into res-onant rotation very quickly, typically within oneperiod of free libration (e.g., Makarov 2013). Inthe process of capture into a resonance (2+ q ) : 2 ,6he evolution of the angle γ ≡ θ − (1 + q/ M abruptly switches from circulation to oscillation,and the orbit-average spin rate (cid:5) θ assumes a near-resonant value. Immediately after the capture, themagnitude of free librations is close to the maxi-mal possible value, but these librations are quicklydamped by tidal friction. However, the forced li-brations do not go away because they are causedby the eccentricity. As a result, the instantaneousspin rate oscillates around the resonant value, in-sofar as the neighbouring planets’s gravity keepsthe residual eccentricity nonvanishing.To understand the importance of the librationterms of (cid:5) θ , note that, generally, these terms mul-tiplied by the harmonics of torque do not averageout to zero. So libration contributes to the powerexerted by the tidal torque and, thereby, to thedissipation. Therefore, in the presence of libra-tions, the energy dissipation rate is higher than itwould have been without libration. Unfortunately,the Fourier decomposition of the tidal and triax-ial torque is very complex, both for the Andrademodel and for its simplified version, the Maxwellmodel. It is not obvious whether a satisfactory an-alytical treatment of the problem can be obtained.For now, we resort to an approximate, qualitativereasoning described below.To estimate the role of physical librations inheating, we simulated the spin of Io subject toboth the triaxiality-caused torque and the tidaltorque, whose averages balance one another andmake the synchronous rotation state that of a sta-ble equilibrium. The formulae for these torquescan be looked up in our preceding paper (Makarovet al. 2012, equations 4 - 6). The simulationdemonstrates that the forced libration of Io ranges,approximately, from − . . θ − M , and within ± × − in ˙ θ/n − dE/dt = h P i curve in Figure 2. Within that vicinity, the curveis quite flat, and the variation of dissipation ratedue to libration is negligible. However, the am-plitude of the forced librations is sensitive to thetriaxiality parameter ( B − A ) /C (and, of course,to the eccentricity e ). In our calculation, we usedthe value ( B − A ) /C = 6 . × − borrowed from Anderson et al. (2001). If we increase ( B − A ) /C by a factor of two, we shall find the half-amplitudeof libration to increase to ≈ × − . Due to theconcavity of the dE/dt = h P i curve, the rate ofdissipation goes up by roughly a factor of two. Wesee that the shape of a moon plays a significantrole in its tidal heating.We conclude that, with the other parame-ters equal, less axially-symmetric (more triaxial)moons should be subject to a significantly strongerheating than their more rotationally symmetricpeers. Io represents a borderline case, obviouslybeing close to complete meltdown. It appearsentirely plausible that Io had a more elongatedshape in the past. Later, because of the excessivetidal heating, it melted down (or, rather, up) tothe surface and underwent a drastic reshaping.Acquiring a more symmetric shape helps a tidallyperturbed body to lower the heat production inthe state of synchronous rotation. The diminishedheat flux allows the crust to emerge. The uppermantle becomes colder and less prone to alter itsshape under varying tidal stresses. So the tidalbulge solidifies and becomes the new triaxial fig-ure. Speculatively, Io could have gone throughseveral such seesaw variations, having graduallyreshaped itself to more symmetrical forms, es-pecially if the rise of dissipation was assisted byepisodical boosts in eccentricity or inclination.The above reasoning is qualitative, so it re-quires further numerical confirmation. Results ofnumerical modeling of this situation will be re-ported elsewhere.
6. Case study III: Kepler-10 b
Kepler-10b was the first confirmed terres-trial planet discovered outside the Solar System(Batalha et al. 2011). It is located remarkablyclose to its host star, the semimajor axis beingonly 2 . × m, which is less than 0.017 AU.Among the super-Earths discovered with the sen-sitive Kepler photometer, Kepler-10 b stands out The influence of librations upon tidal heating of Enceladuswas studied analytically by Wisdom (2004). He considereda very special case where the libration period was aboutthree times longer than the orbital period, so the directemployment of the formula for the time-averaged dampingrate was legitimate, at least for qualitative estimates. Alsonote that in
Ibid the CTL (constant time lag) model wasused.
7s one of the smallest and densest bodies knownoutside the Solar system. With an estimated massof 4 . M earth and the radius 1 . R earth ( Ibid. ),the mean density of the planet comes up to 8640kg m − , which is almost 60% greater than themean density of the Earth, the densest planetin the solar system. While the remarkable factthat the Earth is four to five times denser thanJupiter was known already to Sir Isaac Newton(1687), here we are dealing with a planet consid-erably more massive than the Earth and severaltimes more dense than gas giants. This leaveslittle doubt that the planet is terrestrial, unlikethe distinct category of “hot Jupiters” which aremore massive but have mean densities between 0.3and 3 densities of Jupiter. The mean density ofthe Earth interior is equal to the local density atapproximately 3500 km radius, where the core-mantle boundary is located. The greater densityof Kepler-10 b may very well indicate that the rel-ative radius of its molten core (the actual radiusof the core, divided by the overall radius of theplanet) is larger than the relative radius of themolten core of the Earth. If this is the case, thenKepler-10 b may be classified, in terms of its in-ternal composition, as a super-massive Mercury. Following Peale & Cassen (1978), we speculatethat the core can boost tidal damping by a factorof a few to several. However, we shall not attemptto take this extra boost into account, because it isnot large enough to change our conclusions.
Presently, we possess observational data neitheron the rotation of Kepler-10 b , nor on its obliq-uity. The eccentricity of Kepler-10 b could notbe determined in Batalha et al. (2011), becausethe signal detected in the follow-up spectroscopicobservations of the host star was too weak for aconfident estimation. A recent analysis carried outby Fogtmann et al. (2014) indicates that the ec- It should be noted that our understanding of terrestrialexoplanets does not stand only on comparisons with thedensity of the Earth, as the compressibility of the mantlehas to be taken into account for the large pressures reachedinside massive planets. Various works have addressed thepossible internal structure of these objects in general and ofKepler-10b in particular (e.g., Grasset et al. 2009, Valenciaet al. 2010, Zeng & Sasselov 2013). centricity is extremely small. Although the value0 . +0 . − . provided in Ibid . is consistent with theeccentricity being zero, it should be interpreted asan upper limit. Setting e = 0 is not an option,because the orbit is likely to be excited by a moremassive neighbour, the planet Kepler-10c.Under regular circumstances, tidal dissipationof the orbital kinetic energy in a two-body systemis wont to damp both the eccentricity and obliq-uity. Important exceptions are: Multiple-planet systems, where mutual in-teractions between the planets can pump upboth the eccentricity and obliquity of the in-ner planet (Correia et al. 2012; Greenberget al. 2013). Situations where either a close-in planet orthe star rotates faster than the orbital mo-tion in the prograde sense. In particular,if the star rotates faster than n , the tidalbulge on it leads the direction to the planet.An increase in both e and a ensues (see,e.g., Murray and Dermott 1999). The lagon the star may be small, but it is enoughto keep the planet’s eccentricity nonvanish-ing. A slow tidal dissipation in the star alsomeans it can retain its fast rotation for along time, no matter how massive the close-in planets happen to be. The described situ-ation is analogous to the Earth-Moon systemwhose eccentricity and semi-major axis areboth increasing.Thus, finite residual eccentricities and obliquitiesshould not be unusual for close-in planets. Thepresence of the more massive and distant planetKepler-10 c with an orbital period of 43.3 days(Fressin et al. 2011; Dumusque et al. 2014), makesit likely that the inner planet is neither completelycircularised nor aligned. So we consider smallresidual values of e and i . Somewhat arbitrar-ily, we chose two cases: one of e = 0 .
001 and i = 0 .
001 , another of e = 0 .
001 and i = 0 . k /Q is defined by two ma-jor physical circumstances, the self-gravitation ofthe planet and the rheology of its mantle. A rhe-ological law (i.e., an equation interconnecting thestrain and the stress) contains contributions fromelasticity, viscosity and inelastic processes (mainly,dislocation unjamming). Together, these threefactors render a so-called Andrade creep (Efroim-sky 2012 a, 2012 b). It should be noted that amantle behaves as the Andrade body at higher fre-quencies only, and changes its behaviour towardthe Maxwell model at lower frequencies. Thishappens because, at frequencies below a certainthreshold, only elasticity and viscosity contributeto the rheological response of the mantle. Abovethe threshold, dislocation unpinning (unjamming)plays a considerable role. The value of the thresh-old frequency is highly sensitive to the tempera-ture of the mantle, as can be seen from formula(17) in Karato and Spetzler (1990). The formulaindicates that, for realistic binding energies, a 10to 20 % increase in temperature can increase thethreshold frequency by an order or two of magni-tude. Given that for the Earth the threshold isof the order of 1 yr − , we see that for overheatedplanets the threshold may be as high as 1 day − .It would be even higher for higher temperaturesof the mantle.Speaking of the planet Kepler-10 b , we assumethat, owing to intensive tidal heating, its man-tle should contain a lot of partial melt and thushave a low average viscosity. The Maxwell time,therefore, is likely to be much shorter than thoseof the Earth or Mercury. It should be closer tothe Maxwell times for icy satellites, which is be-lieved to be of the order of days. With an orbitalperiod about one day, Kepler-10 b should expe-rience tides at frequencies of the order 1 day − ,these frequencies likely being below the Andrade-Maxwell threshold. So the Andrade mechanism oftidal friction (unpinning of dislocations) is likelyto be less significant for this planet, allowing us to use a purely Maxwell model. Armed with theseconsiderations, we now have to build the so-calledquality functions k l ( ω lmpq ) sin ǫ l ( ω lmpq ) stand-ing in the expression (3) for the damping rate. Θ (cid:144) n L og H â E (cid:144) â t L W Fig. 3.— Time-averaged rate of energy dissipation dE/dt = h P i in Kepler-10 b, as a function of thedimensionless rotation rate ˙ θ/n , in the vicinityof the 1:1 spin-orbit resonance. The two curves(one computed for τ M = 10 days, e = 0 .
001 , i = 0 .
001 , another for τ M = 10 days, e = 0 .
001 , i = 0 . Each term of the series (3) contains a qualityfunction. These are calculated by the below for-mula (12), with the expression (13) built in. Theresult is presented in Figure 3 which depicts thedependence of tidal damping upon the spin rate ofKepler-10 b (assuming it has a rocky mantle). Forthis computation, however, we assumed a rathershort Maxwell time of 10 days, taking into accountthat the mantle may have a lot of partial melt init. A small residual eccentricity of 0.001 was alsoaccepted, and two values of i were explored: 0.001and 0.0001. In the past, several other rheological models were employedin the literature (e.g., Henning et al. 2009, Heller et al.2011, Henning & Hurford 2014). k /Q vanishes in the zero-frequency limit. More generally, an lmpq term ofthe series (3) vanishes when the tidal mode ω lmpq goes through zero, while outside the resonance theinput from this term is relatively flat. More sub-tle variations of tidal dissipation rate around theresonance are concealed in this figure by the loga-rithmic scale.Both the perceived flatness of the curve out-side the main resonances and the apparent weakdependence on i are explained by the approxi-mate equation (14) derived in Section 7, for thespecial case of τ M n ≫ i = 10 − , we obtain the following estimatesfor the rate of dissipation at the exact 1:1 reso-nance: log( dE/dt ) = 17 .
30 , 15 .
36 , and 13 .
30 for e = 10 − , 10 − , and 10 − , respectively. Therate of dissipation increases by almost exactly twoorders of magnitude for each order of magnitudeincrease in e , as expected from the equation (14)when the O ( i ) terms in it are small. In the tidalregime in question, when τ M n ≫ τ M , as can be seenfrom the equation (A9). Therefore, the dissipationrate is less strongly dependent on τ M than on e ,and the inevitable uncertainty in the former pa-rameter is relatively less restrictive. The absolutevalues in Figure 3 can be used only for very gen-eral guidance and comparison with the previousestimates for Mercury and Io, but the character ofthe curves is valid for a significant range of thesecritical parameters.Thus, in synchronous spin-orbit resonance,Kepler-10 b will dissipate less energy, by roughlyfive orders of magnitude, than in any other rota-tion state, including the 1:2 and 3:2 resonances.The ensuing implications for the destiny of suchclose-in planets are dramatic. If a planet doesnot succeed in falling into the state 1:1, and getscaptured into a higher spin-orbit resonance, the rate of tidal dissipation in the planet becomes sohigh that its temperature should be growing byseveral degrees per year. This should be enoughto quickly melt the planet to the surface and makeit a ball of magma. In a very close vicinity of thehost star, planets rotating synchronously may re-main solid for a longer time than asynchronousplanets. Still, even synchronised planets may notbe able to survive for longer than ∼ One possible scenario for a close-in terrestrialplanet is the following. If the orbital eccentricityand obliquity are not excited by a third body, andthe star does not pump up these parameters bythe transfer of angular momentum from its ownrotation, the orbit should relatively quickly cir-cularise, and the obliquity should decrease. Thiswould drive the tidal dissipation down to smallvalues. As we explained above, in the space of pa-rameters there exists a dip wherein the tidal dis-sipation rate is minimal. This is the synchronousrotation with a zero or near-zero obliquity. In thisregime, the damping rate is by orders of magni-tude lower than in a non-resonant state or in ahigher resonance. In the presence of a non-zeroresidual eccentricity, the planet should also be al-most perfectly spherical in order to get a respitefrom the excessive tidal heating through libration(see Section 5).In multiple systems, the eccentricity and obliq-uity of close-in planets can be excited by externalinteractions. In this situation, a young planet getscompletely molten even if it is synchronised – soit loses its permanent figure before the orbit circu-larisation and obliquity decrease take place. Re-siding at the bottom of the energy dissipation dip( e ≈ i ≈ For the planet’s heat capacity we adopted a value of1200 J kg − K − from Bˇehounkov´a et al. (2011).
7. Analytic approximations for a warmMaxwell planet
Introduced as a function of the Fourier tidalmode ω lmpq , the product k l ( ω lmpq ) sin ǫ l ( ω lmpq )can be also written down as a function of the pos-itive definite forcing frequency χ lmpq ≡ | ω lmpq | : k l ( ω lmpq ) sin ǫ l ( ω lmpq )= k l ( χ lmpq ) sin ǫ l ( χ lmpq ) Sgn ω lmpq = k l ( χ lmpq ) Q l ( χ lmpq ) Sgn ω lmpq , (11)see Section 2.For a homogeneous planet obeying the Maxwellrheological law, the frequency dependence of k l /Q l = k l ( χ ) sin ǫ l ( χ ) is furnished by the ex-pression ( Maxwell ) k l ( χ ) sin ǫ l ( χ )= 32 A l ( χ τ M ) − ( 1 + A l ) + ( χ τ M ) − , (12)derived in the Appendix. Here τ M is theMaxwell time, χ = χ lmpq ≡ | ω lmpq | is the forc-ing frequency corresponding to an lmpq tidalmode, and A l are dimensionless factors reflectingthe interplay of self-gravitation and rheology intidal response. Being interested in the principal(quadrupole) part of the expansion (3), we needthe expression for A : A ≡ µ ρ R = 192 µ RG ρ M = 57 µ π G ρ R = 578 π G ρ R J , (13) ρ , g, R , M being the planet’s mean density, sur-face gravity, radius and mass; G being the New-ton gravitational constant; and µ and J = 1 /µ being the unrelaxed rigidity and compliance, re-spectively. For an Earth-sized planet, A ≈ l = 2 part, and expanding it over e and i , we arrive at an expression for the dissipationrate, written as a series over powers of e and i . In the special case of a warm but not com-pletely molten super-Earth or icy satellite, wecan simplify the series further by assuming that τ M χ ≫ dE/dt = h P i in a synchro-nised planet: h P i = 32 GM ∗ R a A τ M (1 + A ) (cid:20)(cid:18) i − i (cid:19) + (cid:18) − i + 2716 i (cid:19) e + (cid:18) i − i (cid:19) e (cid:21) + O ( e ) + O ( i ) , (14)and in a non-resonant planet: h P i = 32 GM ∗ R a A τ M (1 + A ) (cid:20)(cid:18)
34 + 34 i − i (cid:19) + (cid:18)
274 + 94 i − i (cid:19) e + (cid:18) i − i (cid:19) e (cid:21) + O ( e ) + O ( i ) , (15)and in a planet trapped in the 3:2 resonance: h P i = 32 GM ∗ R a A τ M (1 + A ) (cid:20)(cid:18)
34 + 34 i − i (cid:19) + (cid:18) − i − i (cid:19) e + (cid:18) − i + 1773512 i (cid:19) e (cid:21) + O ( e ) + O ( i ) , (16)11here M ∗ is the mass of the star. As ever, P isthe power exerted by the tidal stresses, and h . . . i denotes time averaging over one or several cycles oftidal flexure. Insofar as the truncation of O ( e ) + O ( i ) is legitimate (conservatively, for e . . dE/dt = h P i must scale as 3 / i and21 / e . Accordingly, h P i in such planetsscales as either 3 / i or 21 / e , whicheveris greater.2. Tidal dissipation in non-resonant planets isvirtually independent of e or i .3. Likewise, the dissipation rate at the 3:2 res-onance is virtually independent of e or i .The latter conclusion may look somewhat coun-terintuitive, but it is easily propped up by the fol-lowing observation. In the series (3) for the damp-ing rate, the semidiurnal ( lmpq = 2200) termis the largest and it scales with both e and i as O (1) . The second-largest term (the one with lmpq = 2201) turns out to be proportional to3 n − θ , whereby it vanishes in the 3:2 spin-orbitresonance. Hence, in this resonance, we are leftwith the obliquity- and eccentricity-independentsemidiurnal term, as well as many terms that aremuch smaller. In Figure 3, the two curves (corre-sponding to the case of e = 0 .
001 , i = 0 .
001 andto that of e = 0 .
001 , i = 0 . / e , the obliquity-dependentterms being less important.
8. Conclusions
We have demonstrated that tidal dissipation isconsiderably more involved a topic than was as-sumed in many studies conducted after the sem-inal work by Peale & Cassen (1978). The com-monly accepted in the literature approximate for-mula (8) for the damping rate follows from theequation (31) in Peale and Cassen (1978), pro-vided that the inclination (or obliquity) is set zeroand higher-order terms in the eccentricity are ne-glected. It can also be derived from a more gen-eral expression, our formula (3), under an extraassumption that the rotator is synchronised. On the examples of Mercury, Io, and Kepler-10 b, we addressed a broad range of issues emerg-ing from the so-revised theory of tidal dissipation.The main practical highlights are:1. Like Mercury, close-in exoplanets of ter-restrial composition may be captured into stable,long-term asynchronous resonances, such as 3:2 or2:1. In such states, the planets have a net ro-tation with respect to the mean direction to thestar. The tidal bulge runs across their surface,which results in a dissipation rate that is higher,by orders of magnitude, than the dissipation ratein a synchronised planet. This conclusion is for-tified by our expressions (14), (15), and (16) forthe damping rate in a planet, in the cases whenit is synchronised, or nonresonant, or in a 3 : 2spin-orbit resonance, respectively. These formulaewere derived for a planet that is described withthe Maxwell rheology and is sufficiently close-in(so that τ M χ ≫ τ M is the Maxwelltime and χ is the principal tidal frequency).2. Planet-planet orbital interactions play a cru-cial role in defining the ultimate fate of those rockyplanets that managed to get close to their stars. Ifa considerable eccentricity is secularly excited bythe outer companions, both the orbital evolutionrate and the tidal heating become boosted by afew to several orders of magnitude. Our prelim-inary calculations show that such planets shouldbe liquefied, even when they are settled in the ab-solute energy minimum (the 1:1 resonance, with azero or near-zero inclination).A planet can, however, survive in the rockystate, provided there is no significant planet-planet orbital interaction pumping up its eccen-tricity or the obliquity. For such survivors, thetidal dissipation in the host star may become animportant factor. Specifically, if the rotation ofthe star is prograde and is faster than the or-bital motion, it will pump up the eccentricity andmay also lead to a finite obliquity that, in turn,will perturb the orbit inclination (Teyssandier etal. 2013). All these circumstances will channelthe kinetic energy into the heating of the close-inplanet, resulting in its liquefaction. It appearsthat most of the host stars with transiting close-ingiant exoplanets rotate slower than these planets’ n (Matsumura et al. 2010, Table 1).3. We have hypothesised that the tidal damp-ing rate can be considerably boosted by phys-12cal librations. The hypothesis stems from thefollowing considerations. An lmpq term of theexpression (3) for the damping rate contains amultiplier k l ( χ lmpq ) sin ǫ l ( χ lmpq ) that dependson the physical frequency χ lmpq . This depen-dence is extremely sharp near resonances, i.e.,in closest vicinities of the zeroes of the fre-quency. As obvious from the expression (2) forthe frequency, we can interpret the multipliers k l ( χ lmpq ) sin ǫ l ( χ lmpq ) as functions of the rota-tion rate (cid:5) θ . Their dependence on (cid:5) θ will also bevery sharp when a resonance is near (i.e., when (cid:5) θ is very close to ( l − p + q ) n/m ). Due to the sharpform of this dependence, even a tiny deviation of (cid:5) θ from a resonant value will change the effectivevalue of k /Q considerably. This situation is bestillustrated by Figure 3, where the dependence ofthe average dissipation rate upon (cid:5) θ is depicted ina close vicinity of the 1:1 spin-orbit resonance.The sensitivity of the energy damping rate tothe values of (cid:5) θ indicates the key role played bythe physical libration in the tidal heating process.Although physical libration does not change the mean value of the spin rate (which stays resonant),the libration yields variations of the instantaneous value of ˙ θ . We have provided qualitative ar-gumentation showing that these variations shouldincrease the overall rate of heat production. How-ever, our physical arguments are not yet rigorousproof. The latter needs to be obtained throughaccurate numerical simulations.4. The magnitude of libration in the spinrate being defined by the planet’s triaxiality, thelatter should be a significant factor determin-ing the dissipation rate at spin-orbit resonances.Other parameters being equal, a body with a morepronounced triaxiality should generate more heatthan a similar body of a more symmetrical shape.On the other hand, we surmise that a feedbackmay also exist, in that the rate of tidal heatingmay change the shape of close-in planets throughrepeated episodes of complete melt-down. Acknowledgments
The authors deeply thank both referees (PatrickTaylor and an anonymous referee) for their de-tailed and very thoughtful report on earlier ver- sions of this work. The authors are also indebtedto James G. Williams for reading the manuscriptand offering very important comments. All thesecolleagues have helped the authors greatly to im-prove the quality of the paper.13 ppendix
A. How rheology and self-gravitation determine thefrequency dependencies of Love numbers and phase lags
The time-averaged dissipation rate in a homogeneous planet is given by the expression (3), providedthe apsidal precession of the star, as seen from the planet, is uniform. An lmpq term of that expressioncontains a quality function k l ( ω lmpq ) sin ǫ l ( ω lmpq ) . Interplay of self-gravitation and rheological propertiesof the planet makes the forms of these functions nontrivial, although some qualitative features of thesedependencies are generic and invariant of rheology and size.As demonstrated, e.g., in Efroimsky & Makarov (2014), a quality function of a Fourier mode ω lmpq canalways be written down as a function of the appropriate physical frequency χ lmpq = | ω lmpq | : k l ( ω lmpq ) sin ǫ l ( ω lmpq ) = k l ( χ lmpq ) sin ǫ l ( χ lmpq ) Sgn ω lmpq . (A1)The following was derived in Efroimsky (2012 a, b) for a homogeneous spherical body: k l ( χ ) sin ǫ l ( χ ) = 32 ( l − − A l J I m (cid:2) ¯ J ( χ ) (cid:3)(cid:0) R e (cid:2) ¯ J ( χ ) (cid:3) + A l J (cid:1) + (cid:0) I m (cid:2) ¯ J ( χ ) (cid:3) (cid:1) . (A2)Here χ is a shortened notation for the frequency χ lmpq , while the factors A l are given by A l ≡ (2 l + 4 l + 3) l g ρ R µ = 3 (2 l + 4 l + 3)4 l π G ρ R µ = 3 (2 l + 4 l + 3)4 l π G ρ R J , (A3) ρ , g, and R being the density, surface gravity, and radius of the body; and G being the Newton gravitationalconstant. The unrelaxed elastic modulus and its inverse, the unrelaxed compliance, are denoted with µ and J , respectively. The complex compliance ¯ J ( χ ) of the mantle is a Fourier image of the kernel ˙ J ( t − t ′ ) ofthe integral equation 2 u γν ( t ) = ˆ J ( t ) σ γν = Z t −∞ (cid:5) J ( t − t ′ ) σ γν ( t ′ ) dt ′ (A4)interconnecting the present-time deviatoric strain tensor u γν ( t ) with the values assumed by the deviatoricstress σ γν ( t ′ ) over the time t ′ ≤ t . The Fourier transform of (A4) reads as:2 ¯ u γν ( χ ) = ¯ J ( χ ) ¯ σ γν ( χ ) , (A5)¯ u γν ( χ ) and ¯ σ γν ( χ ) being the strain and stress in the frequency domain. The complex compliance ¯ J ( χ )contains contributions from elasticity, viscosity and inelastic processes (mainly, dislocation unjamming).Together, these three factors render the Andrade creep:¯ J ( χ ) = J + β ( iχ ) − α Γ (1 + α ) − iηχ (A6a)= J + β ( iχ ) − α Γ (1 + α ) − i J ( χ τ M ) − , (A6b)Γ denoting the Gamma function; η being the mantle viscosity; τ M ≡ η/µ = ηJ being the Maxwell time; α and β being a dimensionless and dimensional Andrade parameters. The parameter β has fractionaldimensions, which makes it impractical; so it was suggested in Efroimsky (2012 a, 2012 b) to rewrite thecompliance as ¯ J ( χ ) = J (cid:2) i χ τ A ) − α Γ (1 + α ) − i ( χ τ M ) − (cid:3) , (A6c)14ith the parameter τ A defined through β = J τ − α A . (A7)In Ibid., τ A was christened the Andrade time .Below some threshold frequency (Karato and Spetzler 1990, Eqn. 17), dislocation unjamming becomesless efficient, and the rheology of the mantle becomes purely viscoelastic. This is why at low frequencies it islegitimate to treat the mantle as the Maxwell body. Mathematically, this is expressed through the Andradetime rapidly growing as the frequency goes beneath the said threshold; so at lower frequencies the complexcompliance becomes simply ( Maxwell ) ¯ J ( χ ) = J − iηχ = J (cid:2) − i ( χ τ M ) − (cid:3) . (A8)Insertion of this formula into the expression (A2) yields: ( Maxwell ) k l ( χ ) sin ǫ l ( χ ) = 32 ( l − A l ( χ τ M ) − ( 1 + A l ) + ( χ τ M ) − . (A9)In Section 6, we use this formula to model dissipation in the planet Kepler-10 b.15 EFERENCES
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