Coupled Orbital-Thermal Evolution of the Early Earth-Moon System with a Fast-Spinning Earth

Icarus 281 (2017) 90–102

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Icarus

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Coupled orbital-thermal evolution of the early Earth-Moon system with a fast-spinning Earth

∗ ZhenLiang Tian a, , Jack Wisdom a, Linda Elkins-Tanton b a Massachusetts Institute of Technology, Cambridge, MA 02139, USA b School of Earth and Space Exploration, Arizona State University, Tempe, AZ 85281, USA

a r t i c l e i n f o a b s t r a c t

Article history: Several new scenarios of the Moon-forming giant impact have been proposed to reconcile the giant im- Available online 6 September 2016 pact theory with the recent recognition of the volatile and refractory isotopic similarities between Moon and Earth. Two scenarios leave the post-impact Earth spinning much faster than what is inferred from Keywords:

Moon the present Earth-Moon system’s angular momentum. The evection resonance has been proposed to drain

Planetary dynamics the excess angular momentum, but the lunar orbit stays at high orbital eccentricities for long periods in Thermal histories the resonance, which would cause large tidal heating in the Moon. A limit cycle related to the evec- Solid body tides tion resonance has also been suggested as an alternative mechanism to reduce the angular momentum, which keeps the lunar orbit at much lower eccentricities, and operates in a wider range of parameters. In this study we use a coupled thermal-orbital model to determine the effect of the change of the Moon’s thermal state on the Earth-Moon system’s dynamical history. The evection resonance no longer drains angular momentum from the Earth-Moon system since the system rapidly exits the resonance. Whereas the limit cycle works robustly to drain as much angular momentum as in the non-thermally-coupled model, though the Moon’s tidal properties change throughout the evolution. ©2016 Elsevier Inc. All rights reserved.

1. Introduction are evidence that the lunar material was predominantly derived from the proto-Earth, unless the impacting planet had isotopic The giant impact theory for lunar formation ( Hartman and signatures very close to those of the proto-Earth ( Mastrobuono- Davis, 1975; Cameron and Ward, 1976 ) can account for the large Battisti et al, 2015 ; Kaib and Cowan, 2015 ). However, the standard angular momentum of the Earth-Moon system, the late formation model indicates that the Moon should have mostly been derived of the Moon, the Moon’s deﬁciency of iron and volatiles, and the from the impactor, given the dynamical constraint that the angular presence of a magma ocean in the early lunar history. Simulations momentum of the Earth-Moon system at the time of lunar forma- of the impact and accretion processes led to a well accepted stan- tion should be close to the present value. dard model of lunar formation ( Canup and Asphaug, 2001 ; Canup, Several new impact scenarios ( Cuk and Stewart, 2012 ; 20 04, 20 08 ), in which a Mars-sized body collided with the proto- Canup, 2012 ; Reufer et al., 2012 ) have been proposed to reconcile Earth, forming an Earth-Moon system with the angular momen- this contradiction. While these models can produce a Moon with tum equal to the present value. This model ﬁt well with multi- essentially the same materials that make up the terrestrial mantle, ple lines of observations, until the relatively recent recognition of they all assume impact angular momenta much higher than the the isotopic similarities of the elements oxygen ( Wiechert et al., current angular momentum of the Earth-Moon system. The Reufer 2001 ), potassium ( Huang et al., 2016 ), chromium ( Lugmair and et al. scenario assumes that the excess angular momentum is re- Shukolyukov, 1998 ), tungsten ( Touboul et al., 2007 ) and titanium moved through the escape of a portion of the impactor mass from ( Zhang et al., 2012 ) between lunar samples and the terrestrial the system just after the impact. The other two assume an Earth- mantle. These isotopic similarities, especially of highly refractory Moon system with an initial angular momentum much higher than elements like Ti, which is not likely to have equilibrated during the current value. With dynamical simulations that begin with a a post-impact equilibration phase ( Pahlevan and Stevenson, 2007 ), fast-spinning Earth, it was suggested ( Cuk and Stewart, 2012 ) that the excess angular momentum can be transferred to the Earth’s orbit around the Sun through an orbital resonance between the

∗ Corresponding author. precession of the lunar pericenter and the Earth’s orbit around E-mail addresses: [email protected] (Z. Tian), [email protected] (J. Wisdom). the Sun. Upon the system’s entry into this resonance, called the http://dx.doi.org/10.1016/j.icarus.2016.08.030 0019-1035/© 2016 Elsevier Inc. All rights reserved. Z. Tian et al. / Icarus 281 (2017) 90–102 91 evection resonance, the period of precession of the lunar pericen- 2. Model ter is locked to one year, the angle between the direction of the lunar pericenter and the direction of the Sun from the Earth (the The computational scheme is composed of two interacting sub- evection angle) begins to oscillate about either π/2 rad or 3 π/2 rad, systems, the thermal evolution of the Moon and the orbital evolu- and the eccentricity of the lunar orbit is raised. While the system is tion of the Earth, Moon, and Sun. The thermal system tracks the captured in the evection resonance, the eccentricity is maintained changes of the Moon’s thermal profile, structure, and tidal proper- at a high value, the semi-major axis of the lunar orbit decreases, ties over time. The orbital system tracks the locations and veloci- and the Earth despins, all contributing to the loss of angular mo- ties of the Earth, Moon, and Sun, as well as the Earth and Moon’s mentum from the Earth-Moon system. With this orbital mecha- rotational state. nism to remove the excess angular momentum, the two giant im- At every time step, the thermal system reads the values of lunar pact scenarios work well to produce an Earth-Moon system that orbital eccentricity and semi-major axis from the orbital system, could meet both the chemical and angular momentum constraints. because they are needed to calculate the heating of the Moon. The However, passage through the evection resonance would cause thermal system determines the values of the Moon’s tidal potential large tidal heating in the Moon. To illustrate the problem, con- Love number, k 2 , and dissipation factor, Q , which are then used by sider the heating in the Moon with fixed parameters. The reso- the orbital system for calculating the tidal accelerations. nance excites the eccentricity of the lunar orbit to as high as 0.5, In the thermal model, we begin with a lunar structure that is and keeps the system trapped in it for a period of ∼10 0,0 0 0 years, suggested by magma ocean solidification studies: a thin (5 km), when the Moon is just about 4–7 Earth radii ( R e ) from the Earth. solid anorthositic flotation lid overlies a liquid magma ocean 5 21 G m 2e 2R n k ∼ Take the rate of tidal heating of the Moon to be E M 2 of 100 km in depth, which overlies a solid interior. According 2 a 6 Q to some solidification models (for example, Elkins–Tanton et al., (Peale and Cassen, 1978), where G, mE , RM , e, n, a, k2 , Q are the gravitational constant, mass of the Earth, radius of the Moon, the 2011 ), an original magma ocean as deep as 10 0 0 km, without a lunar orbit’s eccentricity, mean motion, semi-major axis, degree- lid on the surface, can solidify 80% in just several thousand years, 2 potential Love number, and dissipation factor of the Moon. Ap- as heat efficiently escapes the magma ocean to space by radia- ρ 3 gRM μ ρ tion. After this point, plagioclase begins to crystalize and buoy- proximate k2 with 19 μ , where , , g are the Moon’s rigidity, density and surface acceleration. Taking the time spent in the antly segregates from the magma to form a lid at the surface. The resonance to be 10 0,0 0 0 years, e = 0.5, Q = 100, ρ = 3.34 ×10 3 kg presence of the lid switches the heat loss regime from radiation m − 3 , μ= 6.5 ×10 10 N m − 2 , g = 1.62 m s − 2 , the Earth-Moon dis- from a free liquid surface to the significantly less efficient thermal tance to be 6.5 Earth radii, the Moon’s specific heat capacity to be conduction through the solid lid. Then it takes tens of millions of

1256 J kg −1 C − 1 , and the lunar mass to be 7.438 ×10 22 kg, the tem- years (My) to solidify the remaining magma. During the first sev- perature of the Moon will be raised by ∼10,0 0 0 °C after the sys- eral thousand years, tidal dissipation in the Earth dominates the tem’s passage through the evection resonance. Such an increase in orbital evolution of the Earth-Moon system, since the dissipation temperature would have completely vaporized the Moon! So the in the Moon without a solid lid overlying the magma ocean would assumption of fixed parameters is inadequate. It is necessary to be very small. The orbital change is mainly an increase in the lu- take into account how the large tidal heating in the Moon changes nar orbit’s semi-major axis, with little change in eccentricity and the parameters and consequently the dynamical evolution. the system’s angular momentum. Since several thousand years is ∼ With new orbital simulations that use a more complete tidal short compared to the 100,000 years the system spends in the model, and are run over a greater range of parameters, an alterna- evection resonance or limit cycle, and nothing significant happens tive mechanism was found to be capable of removing the Earth- during this period, we choose to start the evolution assuming 80%

Moon system’s angular momentum while avoiding the drawbacks of the magma ocean has already solidified and that a thin plagio- of the evection resonance ( Wisdom and Tian, 2015 ). In this mech- clase lid has been formed. anism, which is a limit cycle related to the evection resonance, the For radiogenic heating in the Moon, we assume that all the lunar eccentricity oscillates in a range with an upper bound typi- heat-generating isotopes are retained in the magma during magma cally near 0.1, and the evection angle circulates from 0 to 2 π rad ocean solidification. We infer the amount of heating accord- with the same period as the oscillation of eccentricity. During the ing to chondritic abundances of radiogenic isotopes, assuming system’s capture in the limit cycle, the semi-major axis of the lu- the simulated evolution begins 60 My after the formation of nar orbit decreases, and the Earth despins, so the angular momen- calcium-aluminum-rich inclusions, as suggested by tungsten iso- tum of the Earth-Moon system is also reduced, as in the evection tope chronologies (Touboul et al., 2007). We find that radiogenic resonance. Compared with the evection resonance, this mechanism heating is not a significant contribution to the energy budget. excites the system to orbital eccentricities smaller by a factor of 3– The orbital system’s set-up and evolution are the same as in

5, even using the upper bound of eccentricity oscillation. Since the Wisdom and Tian (2015), and we brieﬂy outline it here. We evolve rate of tidal heating is proportional to the square of eccentricity, the locations and velocities of the Earth, Moon, and Sun using the the Moon is therefore much less seriously heated by tides through N-body symplectic mapping method (Wisdom and Holman, 1991). the limit cycle. The limit cycle has an additional and signiﬁcant ad- We also evolve the rigid body rotational states of the Earth and vantage that it occurs over a much wider range of tidal parameter Moon, and include the spin-orbit interactions of the Earth with the than the evection resonance. lunar orbit and the Earth’s orbit around the Sun, and the Moon’s

How well do these two mechanisms (the evection resonance spin-orbit interactions with the lunar orbit around the Earth. In and the evection limit cycle) work when considering the conse- Wisdom and Tian (2015) we used both the quaternion equations quences of tidal heating? Heating of a body must induce changes of motion (Sussman and Wisdom, 2015) and the Lie-Poisson algo- in the body’s tidal properties, and thus affect the orbital evolution rithm (Touma and Wisdom, 1993) for the rigid body dynamics, and by changing the tidal accelerations. The purpose of this study is the two approaches produced the same results. In this study we to determine the manner and extent of the effect of the thermal use the Lie-Poisson integrator. evolution of the Moon on the orbital evolution that starts with a The full form of the Darwin–Kaula tide (Kaula 1964) is used fast-spinning Earth, and to evaluate the effectiveness of the limit for computing the tidal accelerations of the Earth and Moon. cycle and the evection resonance in removing the Earth-Moon sys- The tidal potential is expanded to a sum of terms of differ- tem’s excess angular momentum. ent frequencies. Each frequency component of the tidal poten-

tial, Vlmpq , is speciﬁed by the combination of numbers (l,m,p,q), 92 Z. Tian et al. / Icarus 281 (2017) 90–102 as is the phase lag for each frequency, which is denoted The expression for the azimuthally averaged rate of volumetric ε as lmpq . We adopt a constant-Q model (Q being frequency- tidal heating in the lid is derived based on Peale and Cassen(1978), = / · independent rather than time-independent) with lmpq 1 Q and we correct and outline it in the appendix. It is sign ( m ( ˙ − θ˙ ) + ( l − 2 p ) ω˙ + ( l − 2 p + q ) n ) , where ˙ and ω˙ de- 2 2 2 −6 −1 −2 2 2 2 H = μG m E n R M a Q g k 2 (21 / 5 · α0 + 42 / 5 · α1 note the rate of change of the perturber’s longitude of ascending − / · α α + α 2 + / · α 2 ) node and argument of pericenter, n denotes the orbital mean mo- 252 5 1 2 126 2 252 5 3 tion, θ˙ denotes the rotation rate of the deforming body. The con- ×( e 2 + 403 / 56 e 4 + 1207 / 42 e 6 + 336953 / 4032 e 8 + . . . ) , stant Q model is consistent with what is known about the fre- (5) quency dependence of the Q of the present day Moon ( Williams = − 2 α et al., 2005, 2014 ). A disadvantage of the constant Q model is that where g 1.62 m s is the surface acceleration, and the i are the tidal phase lags exhibit discontinuities near commensurabili- functions of the layer’s position and the lid thickness, which are α ties, which presumably only approximate the actual behavior. An presented in the appendix. In this expression, a, Q, k2 , e and i all alternate choice could have been the popular Mignard tidal model evolve over time.

( Mignard, 1979 ), but in this model the frequency dependence of The dissipation factor, Q, for a layer at temperature T is given by the Ojakangas and Stevenson (1986) formula the Q is not consistent with what is observed. We feel the advan- n tages of the constant Q model outweigh its disadvantages. 1 1 1 1 T The evolution of the thermal system is governed by a set of ( T ) = + − , (6) Q Q Q Q T processes including tidal heating and thermal diffusion in the lid, max min max p partial melting and melt migration in the lid, crystallization and where T p is the melting temperature of plagioclase. We take = = release of latent heat and radiogenic heating in the magma ocean, Tp 1550 °C. We set Qmin 20, and let Qmax vary with the runs. Ex- and exchange of heat between the magma ocean and the lid. We perimentally, the parameter n ranges from 20 to 30 ( Ojakangas and = illustrate each of them below. Stevenson, 1986); we use 25. The choice of Qmin 20 is somewhat arbitrary. The current Q of the Moon is approximately 20 ( Williams 2.1. Tidal heating and thermal diffusion in the lid et al., 2005, 2014 ). We take an effective 1/ Q m of the Moon for use in the orbital The contribution of tidal heating and conduction of heat to the system to be the average of the 1/ Q of the individual lid layers. lid’s temperature evolution is described by The potential Love number k2 as a function of the lid thickness ∂T 2 κ ∂T ∂ ∂T H is given in the appendix. The rigidity of the shell is taken to be = + κ + , (1) μ = × 9 − 2 ∂ ∂ ∂ ∂ ρ 6.5 10 N m . This is an order of magnitude smaller than t r r r r l Cp the rigidity used by Peale and Cassen (1978) based on seismic ve- where T is the local temperature, r is the radius, t is the time, locities in today’s cold Moon. We use a lower rigidity because of κ ρ is the thermal diffusivity, l is the density, Cp is the specific the high temperatures in the lid during the early epoch, though we heat capacity, H is the local volumetric tidal heating rate. We use do not explicitly take into account the temperature dependence of = − 1 − 1 κ = − 6 2 − 1 ρ = Cp 1256 J kg °C and 10 m s . We take l 2927 kg the rigidity. To some extent the choice of rigidity is arbitrary and − 3 m , the average grain density of the crust ( Wieczorek et al., offset by the uncertainty in the values of the tidal Q s of the early 2013 ). The current actual crustal density with porosity is 2550 kg Moon and Earth. m − 3 , but the porosity is the result of billions of years of accu- mulative impact cratering, which was not the case for the earliest 2.2. Partial melting and melt migration in the lid period of the Moon. We also made simulations using the current − 3 density (2550 kg m ) and find that the results are insensitive of Tidal heating strongly affects the solid lid, and in these models the choice between the two densities. The first two terms on the raises parts of the lid above its melting temperature. Melting por- right side constitute Fourier’s law written in spherical coordinates, tions of the lid creates plagioclase melt migration upward in the with the approximation that the heating and temperature are az- lid, and can result in plagioclase melt eruption and re-solidification imuthally homogeneous. on the surface. This process has the critical effect of changing the In the computation, first we parameterize the depth of each relationship of age of the plagioclase with depth in the lid: Origi- layer using y , which varies from 0 (at the surface) to 1 (at the bot- nally, the youngest plagioclase is at the bottom of the lid where it tom of the lid). So has floated from the remaining magma ocean, but if remelted and = − δ · , erupted, then the youngest plagioclase is on the surface. r RM l y (2) Whenever some part of the lid is heated to T p , the plagioclase where δ is the thicknesses of the lid. Let T’ ( t, y ) = T ( t, r ), and note l there begins to melt. The melt fraction depends on the amount of δ that l is varying with time, Eq. (1) becomes heating after the temperature reaches T p . For example, after the 2 ∂T y δ˙ 2 κ ∂T κ ∂ T H lid has been subjected to tidal heating and thermal diffusion for = l − + + (3) ∂t δ r δ ∂y δ2 ∂ y 2 ρ C some time, a particular lid layer could reach a temperature that l l l l p exceeds T by a positive value, T , according to Eq. (4) . But phys- = δ p Then we discretize the lid to 100 layers of thickness r l y ically, after the layer temperature reaches T p , the further amount ( y = 1/100). Denote the temperature of the i th layer to be T i , Eq. of heating, E , would result in melting of layer rather than fur-

(3) becomes ther raising the layer temperature. Let Vlayer is the volume of the i i +1 i −1 i +1 i i −1 d T y δ˙ 2 κ T − T κ T − 2 T + T layer, L is the latent heat in melting and crystallization of plagio- = l − + p = ρ dt δ r δ 2 y δ2 y 2 clase, then E Vlayer l Cp T. The volume of the melted solid is l l l V = E /( L ρ ), so the melt fraction is H melt p l + (4) ρ f = V /V = (C / L ) T . (7) l Cp melt melt layer p p We evolve the lid temperatures according to Eq. (4) . Note that We start the computation of melting in the bottom layer. Once this equation does not include the thermal effects of partial melt- some melt is produced, it begins to rise through the lid to the ing and melt migration, which is treated separately and will be surface, thermally interacting with the lid layers that it travels elaborated later. through. When the melt meets a layer that is also heated to T p Z. Tian et al. / Icarus 281 (2017) 90–102 93 and melted to some fraction, the two bodies of melt join and As the magma ocean solidifies, both plagioclase and mafic min- migrate upward together. When the melt meets a cold layer, or erals crystallize from the magma. We assume constant fractional a layer with a negative T , some fraction crystallizes there, re- crystallization such that the volume of the crystallized plagioclase leasing the latent heat and increasing the layer’s temperature. is of a fixed proportion, f , of the volume of all crystallized solids. Meyer et al. (2010) found that the change in this fraction value Upon crystallization, the plagioclase floats to join the lid, while the from 0 to 1 does not affect the results qualitatively. We take the mafic crystals precipitate on the solid interior. Let R denote the ra- δ = π 3 − −δ 3 fraction to be 1 in this study. One exception is when the cold layer dius of the Moon, Vl ( l ) 4 /3(R (R l ) ) denote the volume 3 is very near T p . In this situation, the full amount of crystallization of the lid, V s ( δs ) = 4 π/3 δs denote the volume of the solid inte- ˙ ˙ will raise the layer temperature above Tp . Therefore, in this case rior, Vl and Vs denote the rates of change of the volumes, we have we take the volume of melt crystallization to be the amount that = − + , is just enough to raise the layer temperature to Tp . V˙ f V˙ l V˙ s (11) When the body of melt reaches the surface, it is instantly quenched and assumes the equilibrium surface temperature. Then V˙ = f V˙ + V˙ , (12) any hollow in each layer caused by melting and migration is filled l l s with material from the neighboring overlying layer or layers, with the heat content of the filling material mixed with that of the layer = π − δ 2δ˙ , V˙ l 4 ( R l ) l (13) to be filled, thus updating the temperatures of each layer.

We take the latent heat in the melting and crystallization of − ˙ = πδ 2δ˙ , plagioclase to be 4.187 × 10 5 J kg 1, and the equilibrium surface Vs 4 s s (14) temperature to be 280°K. Putting these equations together, we find δ˙ = − ˙ − ˙ / , 2.3. Magma ocean model l Er Ec S1 f D (15) The magma ocean is assumed to be convecting and follows an ˙ δs = − E˙ r − E˙ c S 2 ( 1 − f ) /D, (16) adiabatic temperature profile; for the Moon this adiabatic temperature varies little with pressure and we approximate the temper- ˙ = − δ ˙ − ˙ − / , ature as constant with depth. Therefore, instead of tracking the Tf DTs ( s ) Er Ec S2 ( 1 f ) D (17) full temperature profile of the magma ocean, we only track this 2 2 where S = 4 πδs , S = 4 π( R − δ ) , D = −C p ρ S V ( δ , δs ) DT s ( δs ) constant temperature, T , and the thicknesses of the lid ( δ ), the 1 2 l f 2 f l f l − + ρ δ (1 f) L f S1 S2 , the volume of the magma ocean is Vf ( l , magma ocean ( δ ), and the radius of the solid interior ( δs ), which f 3 3 δs ) = 4 π/3(( R −δ ) − δs ). And δ can always be specified by δ =R are related to Tf in the process of magma ocean solidification. l f f − δ − δ The thermal evolution of the magma ocean is described by l s . We tabulate the parameter values in Table 1. − − ρ = ρ , E˙ r E˙ c L f V˙ f Cp f Vf T˙ f (8)

3. Results where E˙ r is the rate of radiogenic heating of the magma ocean, E˙ is the rate at which heat is conducted from the magma ocean c We made simulations over a range of parameters. The variation to the lid at the base of the lid, the third term on the left hand of tidal parameters can be characterized by the A parameter side is the rate of production of latent heat of solidiﬁcation as 2 5 the volume of the magma ocean (Vf ) decreases, and the right k Q m R A = 2M E E M , (18) hand side is the rate of change of the heat content of the magma 2 5 QM k2 E mM R E ρ = − 3 ocean due to its changing temperature. We take f 3000 kg m .

˙ which is a measure of the relative dissipation in the Earth and Meyer et al. (2010) included an additional Vf related term on the in the Moon. m is the mass of the Moon, R is the radius of the right hand side to represent the change of the magma ocean’s heat M E Earth. content due to volume decrease during solidification. However, this Fig. 1 shows the angular momentum of the Earth-Moon sys- is compensated by the increase in heat content of the crystals set- tem when the Earth-Moon distance reaches 8.5 R , at which point tling out as their volume increases. Here we correct this error. E the system would have exited the limit cycle or the evection res- We compute the radiogenic heating by extrapolating the chon- onance if previously captured by either, versus the initial A pa- dritic abundances of 235 U, 238 U, 4° K, and 232Th back to the time rameter of the system. It also shows the maximum eccentricity of formation of the Moon. We then multiply by the heat produc- reached in each run. For comparison, we also show the results for tion per mass, the density, and the volume of the magma ocean a non-thermally-coupled orbital model on the same figure. When and sum over the four isotopes. The half-lives, current abundances, the system is set such that the A parameter computed from the and specific heat productions are given by Turcotte and Schu- initial thermal state is small, in the range for evection resonance bert (2002) . in the non-thermally-coupled model, the system does not lose an- The solidus of the magma ocean is parameterized to fit the bulk gular momentum to a noticeable amount. The maximum eccentric- lunar mantle solidus of Longhi (2003) . We use ity is 1.5–2 times smaller than that of the non-thermally-coupled −4 2 T s ( r ) = 2134 − 0 . 1724 r − 1 . 3714 × 10 r , (9) counterpart, which indicates that the system does not stay long enough in the evection resonance for the eccentricity to be fully where T is the solidus in Kelvin and r is the radius in km. T = T s f s raised. On the other hand, when the initial A parameter falls in δ ˙ δ˙ ( s ), so Tf can be related to s : the range for the occurrence of the limit cycle, almost the same angular momentum is lost in the thermally coupled model as in T˙ = D T ( δ ) · δ˙ . (10) f s s s the non-thermally-coupled model, and the maximum eccentricity Note that the solidus temperature of the magma ocean at the is also similar. base of the lid, T s ( δs ), is lower than T p . This is because the pure First consider a case where the system is captured into the mineral has a higher melting temperature than the multiphase evection resonance. Fig. 2 shows the details of evolution for a run melt has. with the initial A = 1.74. The system gets captured in the evection 94 Z. Tian et al. / Icarus 281 (2017) 90–102

Table 1 Parameter values.

Orbital constants Initial orbital conditions − 6 c m E / m Sun 3.0034398 × 10 Semi-major axis 5 R E − 8 c m M / m Sun 3.6949711 × 10 Day length 2.53 hr R E 1737 6371 km Earth’s obliquity 0 R M 6371 1737 km Moon’s obliquity 0

C / ( m · R 2 ) 0.3308 Lunar inclination 0 p E E −1 ω p 2 π rad day Lunar eccentricity 0.001 J 2p 0.00108263 Thermal constants Initial thermal conditions

T p 1550 °C(1823.15 °K) δl 5 km −1 −1 C p 1256 J kg °K δf 110 km 5 −1 L p 4.187 × 10 J kg Tidal parameters − 6 2 −1 κ 1.0 × 10 m s k 2E 0.299 −3 ρl 2927 kg m Q E 400 −3 ρ 30 0 0 kg m k 2M Determined by MO structure. f f a 0.4 b 9 μ 6.5 × 10 Pa Q M Determined by lunar temperature proﬁle, and Q , Q Mmax . lid Mmin Moon gravity 1.62 m s −2

Moon surface temperature 280 °K Q Mmin 20 Q Mmax calculated according to initial A parameter of each run.

a f is the plagioclase crystal proportion in magma ocean solidiﬁcation. b μ lid is the rigidity of the lid. c The combination of 5 R E and 2.53 hr is equivalent to the combination of 3.5 R E and 2.5 hr. We use 5 R E and 2.53 hr in order to save some computation time.

Fig. 1. Angular momentum of the Earth-Moon system when the Earth-Moon distance reaches 8.5 R E (left) and the maximum eccentricity in each run (right) versus: (black points) the initial A parameter, for thermal-orbital coupled simulations; (red crosses) the A parameter, for non-thermally-coupled simulations. The unit of angular momentum -3 1/2 is L s = C ( Gm E R E ) , where G is the gravitational constant, and C is Earth’s largest moment of inertia (current value). The black line is the current value of the system’s angular momentum. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.) resonance at t = 38 thousand years (ky). Then the evection angle rence of the resonance as observed in the non-thermally-coupled starts to oscillate around π/2 rad, and the lunar orbital eccentric- model ( Fig. 1 ). Then as A decreases to within the range of evec- ity rises rapidly. As tidal heating gets elevated, lid temperatures tion resonance ( ∼5), the system enters the evection resonance a gradually increase, Q of the Moon decreases, and the A parame- second time. The eccentricity growth is less rapid this time, and ter grows gradually. After a time lag of ∼12 ky, during which the the duration is short ( ∼20 ky), which are consistent with the non- excess heat builds up in the Moon, the increase of lid tempera- thermally coupled model when A is near 5 ( Fig. 1 , right panel, tures is accelerated, and the A parameter grows rapidly from less red crosses). Compared to the corresponding run with the non- than 5 to ∼110. The increase of A is mainly due to the decrease thermally-coupled model, in which the system stays in the evec- ∼ ∼ in QM , though k2M also changes. Some layers’ temperatures reach tion resonance for 600 ky and maintains an eccentricity of 0.34 T p , so partial melting and melt migration occur. At the same time throughout the resonance, the system in this run attains a maxi- of this rapid growth of the lid temperatures and the A parame- mum eccentricity of just 0.2, and stays in the evection resonance ter, the system escapes the evection resonance, and the eccentric- for only 30 ky. Therefore, the amount of angular momentum loss ity plunges to zero. The system’s exit from the resonance is due to is negligible. the A parameter growing out of the range suitable for the occur- Z. Tian et al. / Icarus 281 (2017) 90–102 95

Fig. 2. The evolution of the system’s A parameter (blue in upper left plot), the lunar orbit’s eccentricity (black in upper left plot), the evection angle (upper right plot), the semi-major axis (in Earth radii, blue in lower left plot), the system’s angular momentum (black in lower left plot), and the temperatures of selected (the 10th, 20th, …100th) lid layers (lower right plot) for a thermal-orbital coupled simulation with an initial A parameter being 1.74. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

The orbital and thermal processes in this run, such as the quick large as when the limit cycle begins, the A parameter does not in- termination of the evection resonance and brief periods of partial crease. This is because the lid is getting increasingly more efficient melting and melt migration in the lid, are typical among the cases at emitting heat, both to space and to the magma ocean, after the in which the system encounters the evection resonance. short initial phase of A increase. The detailed behavior of A is ex- Now consider a case where the system is captured into the plained below. limit cycle. Fig. 3 shows the details of evolution for a run with In detail, in the beginning of the limit cycle, the large eccen- the initial A = 14. The system not only enters the limit cycle, but tricities cause elevated tidal heating in the lid, decreasing Q of the remains in the limit cycle for a long time and a significant amount Moon and thus increasing the A parameter. But soon a maximum of angular momentum is drained. Upon entry into the limit cycle A value is reached, as the result of the combination of four nega- at t = 37 ky, the lunar orbital eccentricity rapidly grows, as in the tive feedbacks: (a) as A increases, both bounds of the eccentricity’s evection resonance. But the eccentricity oscillates between ∼0.01 oscillation decrease, leading to less rapid deposit of tidal heat in and ∼0.09, a range whose upper bound is considerably smaller the lid; (b) as the temperatures of lid layers near the surface in- than what would be achieved in the evection resonance (0.3–0.5 in crease while the surface temperature stays constant at the equilib- the non-thermally coupled model). Since the rate of tidal heating rium temperature, the negative temperature gradient at the surface is approximately proportional to e 2 , heating in the Moon is signif- and thus the heat flux to space increase; (c) as the temperatures of icantly less severe, and the growth of the A parameter is limited. the bottom layers rise above the magma ocean temperature, heat As the A parameter increases, the system is still retained in the flux switches direction (Fig. 4) so that now the lid loses some heat limit cycle, and the upper bound of eccentricity is lowered, which to the magma ocean; (d) as the heat flux at the lid base switches is in agreement with the trend observed in the non-thermally- direction, according to Eq. (15) , the lid begins to thin, while the coupled simulations. Then the A parameter reaches a peak value temperatures of each layer continues to increase, so the effects of and this is when the upper bound of eccentricity stops decreasing. (b) and (c) get strengthened. After that, the A parameter begins to decrease, and both bounds After A reaches the maximum, the lid continues to thin as the of the eccentricity shift upward with the change of A parameter. bottom layers continue to be hotter than the magma ocean. So In the later half of the limit cycle, though the eccentricity gets as even though the lid temperatures have stopped increasing, (d) con- 96 Z. Tian et al. / Icarus 281 (2017) 90–102

Fig. 3. The evolution of the same parameters as in Fig. 2 , for a thermal-orbital coupled simulation with an initial A parameter being 14.

tinues at work, strengthening (b). Then the rate of heat loss from see that at all Q E values there is no significant angular momen- the lid, now dominated by (d) and (b), exceeds the rate of heat de- tum decrease in the evection resonance region ( A in 1–3), consis- posit in the lid, and the lid temperatures begin to decrease, caus- tent with what we observed for the Q E = 400 case. The results also ing the A parameter to decrease, too. But as A decreases, the range show that when a smaller Q E (300) is used, the limit cycle is still of eccentricity oscillation gets to higher values, thus slowing down encountered, but over a smaller range of parameters. When Q E is the rate of decrease of A . further decreased to 200, no observable angular momentum de- In the later half the limit cycle, two opposing mechanisms work crease occurs, indicating the absence of the system’s capture in the simultaneously: (i) the thinning of the lid leads to an increasing limit cycle. This trend illustrates the importance of Q E in the Earth- rate of heat loss from the lid mainly through (d) and (b), and (ii) Moon system’s early evolution: Q E controls the rate of Earth-Moon the resultant decrease of A leads to an increasing rate of tidal heat- separation, and the limit cycle, as well as resonances, capture the ing in the lid (through increase eccentricities) which slows down system with a higher chance when the Moon is separating from the decrease of A , acting as a negative feedback on (i). The result the Earth at a lower rate. A small Q E means larger tidal disturb- is that A approaches an asymptotic value, as shown in Fig. 3. ing forces to the system, which makes it harder for the limit cy- The system remains in the limit cycle in the whole process even cle to occur and to be maintained. Zahnle et al. (2007) preferred a though the A parameter changes, allowing the continued draining large Q E , which can better facilitate angular momentum loss in our of angular momentum from the Earth-Moon system. This robust- model. ness is partly attributed to the range of A parameter that allows These results also demonstrate the importance of the A param- the occurrence and maintenance of the limit cycle being much eter in the early evolution of the Earth-Moon system. As long as wider than that for the evection resonance. the Earth-Moon separation is sufficiently slow (by assuming a sufficiently large Q E ), the system enters the limit cycle at the same

A parameter values. Thus the detailed values of Q and k2 are less

4. Discussion and conclusion important. With the orbital evolution of the Earth-Moon system being cou-

We also carried out simulations with QE being 200, 300 and pled with the thermal evolution of the Moon, and beginning with

500, and other parameters kept the same. Results are shown in a fast spinning Earth, the system cannot lose its excess angular Fig. 5 as angular momentum versus the initial A parameter. We Z. Tian et al. / Icarus 281 (2017) 90–102 97

Earth-Moon system’s angular momentum to a value that is near the current value over a wide range of parameters. Therefore, we conclude that if the Moon formed in a giant impact scenario that leaves the early Earth spinning much faster than is deduced from the current Earth-Moon’s angular momentum, the limit cycle is a viable mechanism to have removed the excess angular momentum.

Appendix: tidal heating in a synchronously rotating, two-layered satellite

The issue of tidal heating in a shell has been studies by pre- vious investigators ( Peale and Cassen, 1978 ; Peale et al., 1979 ; Wahr et al., 2006 ; Meyer et al., 2010 ; Beuthe, 2013 ). For conve- nience, we derive the expression of degree-2 tidal heating in synchronously rotating, two-layered satellite. The solid exterior layer of the satellite has a much higher shear rigidity than the liquid interior layer ( μE μI ). Both layers are incompressible and have the same density. We use the constant-Q Darwin–Kaula formulation to model the tide. In part 1, we derive the equations of the equilib- Fig. 4. The evolution of the lid thickness (black curve, left scale) and temperature rium displacement in the satellite and the boundary conditions. In gradients at the top (blue solid curve, right scale) and the bottom (blue dashed part 2, we solve the displacement ﬁeld and derive the equilibrium curve, right scale) of the lid, for the same simulation as in Fig. 3 . The lid thickness is normalized with the initial thickness. (For interpretation of the references to colour strain. In part 3, we derive tidal heating as a function of position in this ﬁgure legend, the reader is referred to the web version of this article.) in the satellite, the angle-averaged heating at a given radius, and the total heating in the outer solid layer of the satellite. 1: equation of displacement, boundary conditions To establish the equation of the equilibrium displacement, we need to analyze the forces in the satellite in the presence of a gravitational perturber. The forces are:

− (i) hydrostatic pressure ( p0 , p0 is positive when the pressure is compressive); (ii) satellite’s self gravitational acceleration when unperturbed ( −g r rˆ ); (iii) stress caused by tidal distortion ( X); (iv) forces from the total disturbing potential ( ∇U, U is deﬁned after Eq. (a1) ).

(i), (ii) satisfy (as when the satellite is unperturbed):

−∇ p 0 − ρg r rˆ = 0 (i), (ii), (iii), (iv) satisfy the equilibrium equation ( Love 1944 , pp.257, eq. 23):

−∇ p 0 + ∇ · X − ρg r rˆ + ρ∇U = 0 the two equations give

∇ · X + ρ∇U = 0 .

Fig. 5. The system’s angular momentum when the Earth-Moon distance reaches In Cartesian coordinates, the x component is: 8.5 R E , versus the initial A parameter, for the cases of Q E = 200 (green), Q E = 300 = = = = = (red), QE 400 (black), QE 500 (blue). The QE 300, QE 400 and QE 500 points ∂ x X xx + ∂ y X xy + ∂ z X xz + ρ×∂ x U = 0 (a1) are offset by –0.05, –0.10 and –0.15 on the y-axis respectively to improve clarity. The lines of current angular momentum are also offset accordingly. (For interpretation U, the total disturbing potential, is the sum of the perturber’s of the references to colour in this ﬁgure legend, the reader is referred to the web disturbing potential and the potential associated with the satellite’s version of this article.) distortion. It can be spherical harmonically expanded. According to Love 1944 (pp.257, eq. 22),

= U n Un n momentum through the evection resonance. When the system en- = n V n +3g / ( 2n + 1 )× n ×( r / R ) , counters the evection resonance, the eccentricity tends to be ex- where ࢞ is the degree- n radial displacement at the satellite’s sur- cited to large values (0.3–0.5), tidal heating in the lid becomes too n face, g is the surface gravitational acceleration, r is the radius of intense to be eﬃciently conducted out, and partial melting and the point of evaluation, and R is the satellite’s radius. melt migration occur. The severe heating during the resonance in- n duces rapid changes in the Moon’s tidal properties and thus in V n = GM r / r P n ( cosS ), the tidal accelerations, forcing the system to exit the resonance where G is the gravitational constant, M is the mass of the planet, soon after it enters the resonance. In contrast, due to the lower S is the angle between the direction of the perturber from the cen- eccentricities and the effects of feedback mechanisms that reduce ter of the satellite and the direction to the point of evaluation, r’ is the net heating in the lid, the limit cycle mechanism works sta- the distance between the satellite and the planet. Deﬁne bly over extended periods of time, and successfully reduces the 98 Z. Tian et al. / Icarus 281 (2017) 90–102

0 −n-1 n V n = GMr’ R , then The complete solution is 0 n V n = V n (r/R) P n (cosS), and the corresponding surface radial displacement takes the form −→ −→ −→ u n =u n+ +u n − ࢞ = ࢞ 0 n n Pn(cosS). So −→ = 2∇ 2 + un ± An ±r Pn ± Bn ±rPn ± = 0 + / ( + ) · 0 Un Vn 3g 2n 1 n −∇ ( n ± + r · ∇ n ± ) (a10) n ×( r / R ) P n ( cosS ). (a2)

With the deﬁnition of the degree n potential Love number k n Now we specify the boundary conditions. as At the satellite’s deformed surface, total traction vanishes:

U n =V n +k n ×V n ( where r ≤ R ), ( − )| + · ∂ ( − ) | = , we have Xrr p0 r=R [ r Xrr p0 ] r=R 0 + · ∂ = , = / ( + ) 0/ 0 . Xr θ | r=R r Xr θ | = 0 kn 3g 2n 1 n Vn (a3) r R X φ = + · ∂ r X φ = 0 , Now we relate (1) to the displacement ﬁeld. Since r r R r r=R

X ij = −pδ ij +2 μe ij , (a4) where is the sum of . Note that p | = = 0, ∂ p = – where p is the mean pressure due to tidal distortion, and in rect- n 0 r R r 0 ρg. X = –p δ + 2 μe , p = O( μ /R), e = O( /R), so X = O( /R), angular coordinates, ri ri ri ri ri ࢞∂ = 2 ࢞ 2 r Xri |r = R O( /R) . Neglect terms of O( R/R) and higher, we e ij = 1 / 2 ∂ i u j + ∂ j u i , get we can rewrite (1) as

X rr | r=R + ρg = 0 , (a11) −∂ x ( p − ρU ) + μ( ∂ x ∂ x + ∂ y ∂ y +∂ z ∂ z )u x + μ∂ x ( ∂ x u x +∂ y u y + ∂ z u z ) = 0. (a5) | = . Xr θ r=R 0 (a12) The incompressible assumption gives −→ ∇ · = u 0 (a6) The Xr φ equation is neglected because it gives the same con-

straint as the Xr θ equation. Deﬁne P = p − ρU , (a7) The total traction is continuous at the boundary between the two layers: then (5) and (6) give −→ −∇ P + μ∇ 2 u = 0 . (a8) ( − )| + · ∂ ( − ) | Xrr p0 r=R [ r Xrr p0 ] r=b − μ Since is constant in each layer, (a2), (a6) and (a8) imply that = ( − )| + · ∂ ( − ) | , Xrr p0 r=R [ r Xrr p0 ] r=b+ P is a harmonic function (i.e., ∇ 2 P = 0). Then (a8) can be expanded X θ | = −+ · ∂ X θ | as: r r b r r r=b − = | + · ∂ | . 2 −→ Xr θ r=b+ r Xr θ r=b+ −∇ P n ± + μ∇ u n ± = 0 , (a9) where We assume the two layers have the same density, so

= 0 ( / )n ( )= −ρ , ∂ = ∂ = ρ ࢞ 2 Pn+ Pn+ r R Pn cosS pn+ Un r p0 |r = b - r p0 |r = b + – g. Neglect of O( R/R) and higher terms − − = 0 ( / ) n 1 ( )= gives: Pn − Pn − r R Pn cosS pn − in the outer layer, and X rr | = −=X rr | , (a13) = 0 ( / )n ( ), r b r=b+ Pn+ PIn r R Pn cosS P − = 0 n | = | . Xr θ r=b − Xr θ r=b+ (a14) in the inner layer. A particular solution is given by Love 1944 (p. 258): The displacement components are also continuous at the −−→ = 2∇ + u Pn± An ±r Pn ± Bn ±r Pn ± boundary: where A n ±, B n ± satisfy −1 | = | , ( 4 k ±+2 )A n ±+2B n ±=μ , ur r=b − ur r=b+ (a15) 2 k ±A n ±+ ( k ±+3 )B n ±= 0 , k + = n , k −= −( n + 1 ), −1 −1 −1 ( A 2+ = 5 / 42 μ ,B 2+ = −2 / 21μ ,A 2 −= 0 ,B 2 −= 1 / 2μ ) , where μ is the rigidity of whichever zone is under consideration. u θ | = −= u θ | . (a16) We use μE to denote the rigidity of the outer layer and μI for the r b r=b+ inner layer. Following Peale and Cassen 1978 , we take the homogeneous so- = lution to be We also have being the radial displacement at r R: −−→ u Hn± = ∇ × ( r × ∇ n ± ) u r | r=R = . (a17) where n ± is any harmonic function. The harmonic coeﬃcients 0 0 0 n + , n- and In are deﬁned similarly to those for Pn ±. Z. Tian et al. / Icarus 281 (2017) 90–102 99

μ ࢞ 0 = −1 μ 0 μ 0 2: solution of displacement and strain We note that R 2 5/3 (g Rk2 V2 ). Since 2 + , μ 0 2 0 2 0 μ ࢞ 0 We can transform the conditions on the traction to conditions 2- , R P2 + , R P2- are all ratios of R 2 , they are also ra- −1 μ 0 η on the displacement using the relation: tios of (g Rk2 V2 ), and the ratios are functions of , as shown −→ in (a20) . −1 −1 1 μ r X r = −μ (P + ρU) r + ∇( r · u ) + ( r · ∇) u − u , (a18 ) Substitution of these coeﬃcients into (a10) gives the displace-

ment solution: where X r = X · rˆ . Use of (a2) , (a10) and (18) in boundary conditions E 0 E 0 (a11) ∼(a17) leads to a set of linear equations for μ + , μ , − − 2 2- = 1 α ( 1 ) , 2 0 2 0 μI 0 2 0 μE ࢞ 0 ur Rg k2 1 r V2 R P2 + , R P2- , I2 , R PI2 , and R 2 . The matrix of coef- −1 −1 ﬁcients is u θ = R g k 2 α2 ( r ∂ θ V 2 ) , −1 −1 −1 ρ u φ = R g k α ( r sin θ · ∂ φV ) , (a23) − − / − 2 gR ρ 2 0 2 2 2 12 48 1 7 3 0 0 5 μE R V2 −6 −16 8 / 21 1 / 2 0 0 0 0 where − η2 η−3 −η4 / − η−1 η2 1 η4 12 48 7 3 12 7 0 0 − 8 − −8 − η2 − η 3 η4 / η 1 η2 η4 − −5 6 16 21 1 2 6 21 0 0 α = / ξ 1 ( − η − / η ) − − 1 2 3 96 56 57 2 789 r1 − η2 − η3 η4/ / η 1 η2β 1 η4β − 6 6 7 1 2 6 7 0 0 + ( η + η + η ) 3 − 160 34 76 56 76 789 r − η2 η−3 5 η4 η2 β 5 η4 β 1 3 2 0 3 0 0 + η + η + η 42 42 ( 96 012 96 34 180 56 ) −6 −6 1 / 7 1 / 2 0 0 −1 0 + ( − η − η ) 2 , 3 36 012 96 34 r1 (a24 ) − − (a19) α = / ξ 1 ( η + / η ) 5 2 2 3 32 56 19 2 789 r1 + η + η + η where η = b/R, b is the radius of the boundary between two layers, ( 48 012 48 34 90 56 )

β = μE μI β + ( − η − η ) 2 . and / . Since the inner layer is liquid, we take the limit – 30 012 80 34 r1 > ∞ . Then the two lines in (a19) dominated by β terms imply that μI 0 = 2 0 = I2 R PI2 0. So we ignore these two variables, and here- The equilibrium strain is the spatial derivative of the equilib- after use μ to refer to μE since μI does not appear. The solutions = ∂ + ∂ rium displacement. In Cartesian coordinates, eij ½( i uj j ui ). In of the other variables are: arbitrary curvilinear coordinates ( α, β, γ ), the relation between −1 μ0 = μR 0( 5 ξ ) strain and displacement is 2+ 2 ·( −32η 012 −32η 34 −60η 56 ) , −1 = ∂ + ∂ ( / )+ ∂ ( / ) μ0 = μ 0 ( ξ ) eαα hα αuα hαhβ uβ β 1 hα hγ hαuγ γ 1 hα 2 − R2 5 ·( 32η 56 +19 / 2η 789 ), e ββ = h β ∂ β u β +h β h γ u γ ∂ γ 1 /h β +h αh β u α∂ α 1 /h β (a20) 2 0 0 −1 R P = μR ( 5 ξ ) 2+ 2 e γγ = h γ ∂ γ u γ +h γ h αu α∂ α 1 /h γ + h β h γ u β ∂ β 1 /h γ ·( −504η 012 −1344η 34 ) , −1 R 2P 0 = μR 0( 5 ξ ) e αβ = 1 / 2[h α/ h β · ∂ α h β u β + h β / h α · ∂ β ( h αu α )] 2 − 2 ·( 640η +304η +304η ), 34 56 789 e βγ = 1 / 2[h β / h γ · ∂ β h γ u γ +h γ / h β · ∂ γ h β u β ] and e γα = 1 / 2[h γ / h α · ∂ γ ( h αu α )+ h α/ h γ · ∂ α h γ u γ ] (a25)

μ 0 = ρ 2 0( ξ )/ ξ · ρ / μ + λ , R 2 R V2 5 [2 gR 24 ] (a21) = ∇α = ∇β = ∇γ where where hα | |, hβ | |, hγ | |. In spherical coordinates (r, θ, φ), ξ = 24 η012 +64η 34 +64η 56 +19η 789 , −1 −1 −1 λ = 19 η012 −56η 34 +56η 56 −19η 789 , h r = 1 , h θ = r , h φ = r sin θ. (a26) 2 η012 = 1 + η+η , 3 4 Substitution of (a23) and (a26) in (a25) gives η34 = η +η , 5 6 η = η +η , −1 56 e rr = Rg k 2 · A α0 , 7 8 9 −1 η789 = η +η + η . e θθ= Rg k 2 · (Aα 1 +Bα 2 ) , − e φφ= Rg 1k · (Aα +Cα ) , From (a3) and (a21) 2 1 2 (a27) = −1 · α , er θ Rg k2 D 3 3 ξ = −1 · α , 0 0 2 2 eθφ Rg k2 E 2 k = 3g / 5 / V = . (a22 ) 2 2 2 = −1 · α , ξ + 12 λμ/ ( ρgR ) er φ Rg k2 H 3

where 1 Proof: From (a25) , (a26) , (a4) and (a7) , we have X =-(P + ρU) + 2 μ∂ u , rr r r −2 −3 =μ ∂ -1 + -1 ∂ A =r V = GMr P ( cosS ), Xr θ (r r (r uθ ) r θ ur ), 2 2 =μ -1 -1 θ∂ + ∂ -1 Xr φ (r sin φ ur r r (r uφ )). B =∂ θ ∂ θ A , −→ −2 Since X r = X rr rˆ+ X θ θˆ+ X φ φ, we get C = sin θ · ∂ φ∂ φA + cot θ · ∂ θ A , −→ r r ˆ (a28) -1 -1 =∂ , X r + (P + ρU)rˆ= 2 μ∂ r u r rˆ+ μ[r ∂ r (r u θ ) + r ∂ θ u r ] θˆ D θ A -1 -1 -1 −1 + μ[r sin θ∂ φ u r + r ∂ r (r u φ )] φ. = θ · ∂ ( ∂ − θ · ), ˆ E sin φ θ A cot A ∂ -1 = -1 + ∂ μ-1 − Apply the relation: r r(r ur) -r ur rur, and multiply both sides with r, we H = sin 1θ · ∂ φA , get: −→ -1 -1 μ r X r + μ (P + ρU) r -1 -1 -1 = r[ ∂ r u r rˆ+ r ∂ θ u r θˆ+ r sin θ∂ φ u r φˆ] 2 -1 -1 -1 + r u r rˆ+ r [ ∂ r (r u r )rˆ+ ∂ r (r u θ ) θˆ+ ∂ r (r u φ ) φˆ] 2 -1 = r( ∇ u r + u r ∇ r) + r ∂ r (r u ) = ∇ + ∂ 3 2 2 (r ur ) -u r r u When η= 0, α1 =8/3 -r 1 , α2 = 4/3 -5/6 r 1 . Note that k 2 α1 is equivalent to = ∇( · ) + ∇ r u -u (r )u.(a18) is proven. k 2 ’ α2 in Peale & Casen (1978) and Meyer (2010) , and that k 2 α2 is equivalent to / 2 η= ξ = λ= = 3 2 α When 0, we get 24, 19, k2 1+( 19 / 2 ) μ/ ( ρgR ) . their k2 ’ 1 . 100 Z. Tian et al. / Icarus 281 (2017) 90–102

which are all independent of r, and spherical harmonic addition theorem, α = ∂ ( r · α ) −3 2 0 r 1 A = GMr ( = ) c m P 2m c os θ − − m 0 = / ξ 1 ( η + η ) 5 2 3 384 56 114 789 r1 − ×P ( cos θ ) cos m φ − φ , + ( − η − η − η ) 3 2m 320 34 152 56 152 789 r1 + ( 96 η012 + 96 η34 + 180 η56 ) = δ + where cm (2 – 0m )(2 – m)!/(2 m)!. + − η − η 2 , ( 108 012 288 34 )r We then transform r , θ and φ to the planet’s orbital ele- 1 (a29 4,5) α = / ∂ α + α 3 1 2( r r 2 1 ) ments to study the satellite’s tidal response at different disturb- − − = / ξ 1 ( − η − η ) 5 ω 2 3 128 56 38 789 r1 ing frequencies. Let , , M, f, i, a denote the planet’s longitude − + ( η + η + η ) 3 of ascending node, argument of pericenter, mean anomaly, true 80 34 38 56 38 789 r1 + η + η + η ( 48 012 48 34 90 56 ) anomaly, inclination relative to the satellite’s equatorial plane, and

+ ( − η − η ) 2 . semi-major axis, respectively. Note that these elements are with 48 012 128 34 r1 χ φ = ψ respect to the inertia frame of reference ( 0 ), so ’ – , where ψ χ χ 3: tidal heating distribution, total heatin g is the longitude of the basis meridian of 1 measured in 0 . The transform is based on As seen in (a27) and (a4) , both the stress and the equilibrium strain depend on r’ and S (the angle between the line to the planet cos P cos θ · m φ and the line to the point of evaluation). At a given fixed point in lm sin the satellite, r’ and S change over time due to the planet’s orbital l = F (i) (a32) motion and the satellite’s rotation, so the stress and strain vary, (p=0) lmp cos too. Dissipation occurs due to the anelastic flexation of the satel- × ( − ψ) + ( − )(ω + ) + κ , [m l 2p f (l −m) ] lite, which is equivalent to a phase delay applied to the elastic de- sin ∗ formation. We denote the phase-delayed strain as e ij . The rate of 0 , x is even energy dissipation per unit volume is where κ ={ π , x − , 2 x is odd ˙ = ∗= − δ ∗+ μ ∗ W Xij e˙ ij p ij e˙ ij 2 eij e˙ ij ∗ = = − ∂ ( ∇ · ∗ ) + μ Flmp ( i) p t u 2 eij e˙ ij (a30) ∗ = 2 μe e˙ , p ij ij min − floor ( l m ) 2 ( 2l − 2t )! l −m −2t sin i where the summation convention is used. The incompressible 2l −2t t=0 t! ( l − t )!2 ⎛ ⎞ ⎛ ⎞ assumption is used. In spherical coordinates, substitution of l − m − 2t + s m − s ⎝ ⎠ ⎝ ⎠ (a27) into (a30) gives: + min l s − m −2t m m [ m+p+t ] c p − t − c − s ˙ = μ 2 2 2 × cos i W 2 R g k2 ( − − ) s=0 s 0 l m 2t ! c= max − − + ∗ ∗ [ p t m s ] 2 2 − × A A˙ α + 2A A˙ α c −floor ( l m ) 0 1 ( −1 ) 2 ∗ ∗ ∗ ∗ + A B˙ + B A˙ + A C˙ + C A˙ α1 α2 (a31) ( Kaula 1961 ), and the Hansen expansion ∗ ∗ ∗ 2 + B B˙ + C C˙ + 2E E˙ α 2 m ∗ ∗ 2 REM cos ∞ m , n cos + 2 D D˙ + H H˙ α . ( nf ) = −∞ X ( e ) ( kM ) (a33) 3 a sin k= k sin Now we can see the separation of W˙’s dependence on radius θ φ θ φ and on the angles. The r-dependence is completely expressed in So A(r’, ’, ’, , ) is transformed to: α θ φ ˙ ∗ ˙ ∗ i , and the - and -dependences are expressed in AA , AB …A, 2 2 ∞ A = = = = −∞ A , B… functions are determined by the planet’s distance and orienta- m 0 p 0 q 2mpq ∗ ∗ −3 tion relative to the satellite, and A˙ , B˙ . . . are determined by the A 2 mpq = GMa planet’s orbital motion, the satellite’s rotation, and the phase lags ×c m P 2m ( cos θ )F 2 mp ( i )G 2 pq ( e ) in the satellite’s tidal responses. ∗ × − φ − ψ + ν + κ , We take A A˙ as an example to study the form of the angle- cos m m 2 mpq m dependent part of W . Let ( θ , φ ), ( θ, φ) denote the colatitude ˙ where and longitude of the planet and the point of evaluation. φ’ and φ are measured from a particular meridian of the satellite, which is ν = + ( − ) ω + ( − + ) , lmpq m l 2p l 2p q M χ an instantaneous inertia frame of reference ( 1 ). According to the − − , − ( ) = l 1 l 2p( ). Glpq e X(l−2 p+ q ) e

4 η= α = 2 α = 2 κ = κ When 0, 0 8/3 -3r1 , 3 4/3 -4/3r1 . Note that (l-m) m when l is 3, 5… 5 α As a test, we compute values k2 i at particular parameter values, and compare The phase delayed form is the result with Peale & Cassen (1978) and Meyer et al . (2010) . Take ρ= 3.34, g = 162, ∗ − R = 1.738e 8, μ= 6.5e 11 (in cgs). = 3 A2 mpq GMa For η= 0.5, -5 -3 2 × ( θ ) ( ) ( ) k 2 α0 =9.138e-3 r 1 -3.233e-2 r 1 + 9.084e-2 -0.1135 r 1 , cm P2m cos F2 mp i G2 pq e -5 -3 2 k 2 α1 =-2.284e-3r 1 + 1.617e-2r 1 + 9.084e-2 -3.784e-2r 1 , -5 2 × cos −m φ − m ψ + ν2 mpq + κm − ε 2 mpq . k 2 α2 =7.615e-4 r 1 + 4.542e-2 -3.154e-2 r 1 , -5 -3 2 k 2 α3 =-3.046e-3r 1 + 8.083e-3r 1 + 4.542e-2 -5.046e-2r 1 . ∗ For η= 0.95, The frequency of A lmpq due to the orbital motion of the per- -5 -3 2 k 2 α0 =0.8874 r 1 -1.174 r 1 + 0.776 8 -0.86 88 r 1 , turber and the rotation of the satellite is -5 -3 2 k 2 α1 = −0.2219 r 1 + 0.5871 r 1 + 0.7768 -0.2896 r 1 , α = -5 + 2 = − ψ˙ + ν k2 2 7.395e-2 r1 0.3884 -0.2413 r1 , flmpq m ˙ lmpq -5 -3 2 k 2 α3 =-0.2958 r 1 + 0.2936 r 1 + 0.3884 -0.3861 r 1 . = − ψ˙ + ˙ + ( − ) ω + ( − + ) ˙ . These agree with k 2 ’ α0 , k 2 ’ α2 , k 2 ’ α1 , k 2 ’ α3 in Peale & Cassen (1978) and Meyer m m l 2p ˙ l 2p q M et al . (2010) , except for a typo and two errors in P&C 1978. Z. Tian et al. / Icarus 281 (2017) 90–102 101

In completely damped 1:1 spin-orbit resonance, ψ˙= ˙+ ω˙+ M˙ , so

= ( − − ) ω + ( − + − ) . flmpq l 2p m ˙ l 2p q m M˙

ω ˙ ≈ + When /n is small, flmpq (l-2p q-m)n. So

∗ ˙ A2 mpq A2 m p q

− 2 2 6 = G M a c m c m P 2m ( cos θ )P 2m ( cos θ )

×F 2 mp ( i )F 2m p ( i )G 2 pq ( e )G 2p q ( e ) × −f 2m p q × cos −m φ − m ψ + ν2 mpq + κm × sin −m φ − m ψ + ν2m p q + κm − ε 2m p q

− = 2 2 6 ( θ ) ( θ ) G M a cm cm P2m cos P2m cos η Fig. A.1. The dependence of W˙total (scaled with (W˙total )η = 0 ) on .

×F 2 mp ( i )F 2m p ( i )G 2 pq ( e )G 2p q ( e ) ˙ ∗ . . . , ˙ ∗ ×( −1 ) 2 − 2p + q − m n · 1 / 2 AB AB θφ-avg …canbe computed in a similar way. Then with (a31) , we get the angle-averaged tidal heating: × − + φ − + ψ 2 2 2 −6 −1 −2 2 sin m m m m W˙ θφ− = μG M nR a Q g k avg 2 × / α2 + / α2 + ν + ν + κ + κ − ε 21 5 0 42 5 1 2m p q 2mpq m m 2m p q (a38) − / α α + α2 + / α2 252 5 1 2 126 2 252 5 3 + − − φ − − ψ sin m m m m × e 2 + 403 / 56 e 4 + o e 4 . + ν − ν + κ − κ − ε (a34) 2m p q 2mpq m m 2m p q The total heating rate in the outer layer (up to e 2 ) is The secular terms are those with m + m’ being even, and ˙ = π 2 ˙ Wtotal 4 r Wθφ−a v g dr ·η, 2 − 2p − m = 2 − 2 p − m , q = q [R R] − − − ( . ., = ) , = 2 · λ/ξ · πμ 2 2 5 2 6 1 2. 6 i e f2 mpq f2 m p q k2 28 4 G M nR e a Q g (a39 ) or Using parameters appropriate for Io 7, (a22) and (a39) give − − = − − − , = − 2 2p m 2 2p m q q η the relationship between W˙total /(W˙total )η= 0 and , as shown ( . ., = − ) . i e f2 mpq f2 m p q in Fig. A.1 , which agrees well with Peale et al. (1979) and Meyer et al. (2010) . Apply the form of phase lag in the constant-Q Darwin-Kaula tidal model References −1 ε = Q sign ( f ) (a35) 2mpq 2mpq Beuthe, M., 2013. Spatial patterns of tidal heating. Icarus 223, 308–329. doi: 10.1016/ j.icarus.2012.11.020 .

to the expression, and ignore the periodic terms, we get the Cameron, A.G.W. , Ward, W.R. , 1976. the origin of the moon. Proc. Lunar Planet. Sci. secular part of (a34) : Conf. 7, 120 . Canup, R.M., Asphaug, E., 2001. Origin of the moon in a giant impact near the end of the earth’s formation. Nature 412, 708. doi: 10.1038/35089010 . A 2 mpq A˙ ∗ 2 m p q Canup, R.M., 2004. Simulations of a late lunar-forming impact. Icarus 168, 433. − 2 2 6 = G M a c m c m P 2m ( cos θ )P 2 m ( cos θ ) doi:10.1016/j.icarus.2003.09.028. Canup, R.M., 2008. Lunar-forming collisions with pre-impact rotation. Icarus 196, × ( ) ( ) ( ) ( ) F2mp i F2m p i G2pq eG2p q e − 518. doi:10.1016/j.icarus.2008.03.011. ×1 / 2 2 − 2 p + q − m n Q 1sign (2 − 2 p + q − m ) ⎧ Canup, R.M., 2012. Forming a moon with an Earth-like composition via a giant im- cos m − m φ pact. Science 338, 1052. doi: 10.1126/science.1226073 . ⎪ ⎪ (a36) Cuk, M., Stewart, S.T., 2012. Making a moon from a fast-spinning Earth: a giant im- ⎪ m + m even, f = f ⎨⎪ 2mpq 2m p q pact followed by resonance despinning. Science 338, 1047. doi:10.1126/science. ( − )m + φ 1225542. × 1 cos m m Elkins-Tanton, L., Burgess, S., Yin, Q.-Z., 2011. The lunar magma ocean: reconciling ⎪ m + m e en, f = − f the solidiﬁcation process with lunar petrology and geochronology. EPSL 304, ⎪ v 2mpq 2m p q ⎪ 326. doi: 10.1016/j.epsl.2011.02.004 . ⎩⎪ 0 Hartmann, W.K., Davis, D.R., 1975. Satellite-sized planetesimals and lunar origin. ( ) else Icarus 24. doi: 10.1016/0019- 1035(75)90070- 6 . Huang, S. , Petaev, M.I. , Wang, W. , Lock, J. , Wu, Z. , Stewart, S.T. , Jacobsen, S.B. , 2016. The angle-averaged value is Lunar origin beyond the hot spin stability limit: stable isotopic fractionation. LPSC XXXVII. . ˙ ∗ AA θφ−avg Kaib, N., Cowan, N.B., 2015. The feeding zones of terrestrial planets and insights into

moon formation. Icarus 252, 161–174. doi:10.1016/j.icarus.2015.12.042. Kaula, W.M., 1961. Analysis of gravitational and geometric aspects of geodetic uti- −1 ∗ = ( 4 π ) A A˙ · sin θdθdφ lization of satellites. Geophy. J. Int. 5, 104–133. doi: 10.1111/j.1365-246X.1961. [ 0 , 2 π] [ 0 ,π] tb00417.x . 2 2 −6 −1 2 4 = G M a nQ · 21 / 10 e + 403 / 56e + . . . . (a37) 6 η= λ= ξ = = 2 πμE 2 2 5 2 -6 -1 -2 When 0, we get 19, 24, W˙total k2 133/6 4 G M nR e a Q g .

7 R = 1.821e8, ρ= 3.53, g = 179.71, μ= 6.5e11, in cgs. 102 Z. Tian et al. / Icarus 281 (2017) 90–102

Kaula, W.M., 1964. Tidal dissipation by solid friction and the resulting orbital evo- Touboul, M., Klein, T., Bourdon, B., Palme, H., Weiler, R., 2007. Late formation and lution. Rev. Geophys. 2, 661–685. doi: 10.1029/RG0 02i0 04p0 0661 . prolonged differentiation of the moon inferred from w isotopes in lunar metals. Lugmair, G.W., Shukolyukov, A., 1998. Early solar system timescales according Nature 450, 1206. doi: 10.1038/nature06428 .

to 53Mn- 53Cr systematics. Geochim. Cosmochim. Acta 62, 2863. doi: 10.1016/ Touma, J., Wisdom, J., 1993. Lie–Poisson integrators for rigid body dynamics in the S0 016-7037(98)0 0189-6 . solar system. Astron. J. 107, 1189–1202. doi: 10.1086/116931 . Longhi, J., 2003. A new view of lunar ferroan anorthosites: postmagma ocean pet- Turcotte, D.L. , Schubert, G. , 2002. Geodynamics. Cambridge University Press, Cam- rogenesis. J. Geophys. Res. 108, 1. doi: 10.1029/20 02JE0 01941 . bridge . Love, A.E.H. , 1944. A Treatise on the Mathematical Theory of Elasticity, 4th Ed Dover Wahr, J.M., Zuber, M.T., Smith, D.E., Lunine, J.I., 2006. Tides on Europa, and the publications, New York . thickness of europa’s icy shell. J. Geophys. Res. 111, E12005. doi: 10.1029/ Mastrobuono-Battisti, A., Perets, H.B., Raymond, S.N., 2015. A primordial origin for 20 06JE0 02729 . the compositional similarity between the earth and the moon. Nature 520, 212. Wiechert, U., Halliday, A.N., Lee, D.-C., Snyder, G.A ., Taylor, L.A ., Rumble, D., 2001. doi: 10.1038/nature14333 . Oxygen isotopes and the moon-forming giant impact. Science 294, 345. doi: 10. Meyer, J., Elkins-Tanton, L., Wisdom, J., 2010. Coupled thermal-orbital evolution of 1126/science.1063037 . the early moon. Icarus 208, 1–10. doi: 10.1016/j.icarus.2010.01.029 . Wieczorek, M.A., et al., 2013. The crust of the moon as seen by GRAIL. Science 339, Mignard, F., 1979. The evolution of the lunar orbit revisited. I. Moon Planets 20, 671–675. doi: 10.1126/science.1231530 . 301–315. doi: 10.10 07/BF0 0907581 . Williams, J.G. , Boggs, D.H. , Ratcliff, J.T. , 2005. Lunar ﬂuid core and solid-body tides. Ojakangas, G.W., Stevenson, D.J., 1986. Episodic volcanism of tidally heated satellites LPSC XXXVI 1503 . with application to io. Icarus 66, 341. doi: 10.1016/0019-1035(86)90163-6 . Williams, J.G., et al., 2014. Lunar interior properties from the GRAIL million. JGR Pahlevan, K., Stevenson, D.J., 2007. Equilibration in the aftermath of the lunar- Planets 119, 1546–1578. doi: 10.10 02/2013JE0 04559 . forming giant impact. EPSL 262, 438. doi: 10.1016/j.epsl.2007.07.055 . Wisdom, J., Holman, M., 1991. Symplectic maps for the N-body problem. Astron. J. Peale, S.J., Cassen, P., 1978. Contribution of tidal dissipation to lunar thermal history. 102, 1528–1538. doi: 10.1086/115978 . Icarus 36, 245. doi: 10.1016/0019-1035(78)90109-4 . Wisdom, J., Tian, Z., 2015. Early evolution of the earth–moon system with a fast- Peale, S.J., Cassen, P., Reynolds, R.T., 1979. Melting of io by tidal dissipation. Science spinning earth. Icarus 256, 138. doi: 10.1016/j.icarus.2015.02.025 . 203, 892–894. doi: 10.1126/science.203.4383.892 . Zahnle, K., Arndt, N., Cockell, C., Halliday, A., Nisbet, E., Selsis, F., Sleep, N.H., Reufer, A, Meier, M.M.M., Benz, W., Wieler, R., 2012. A hit-and-run giant impact sce- 2007. Emergence of a habitable planet. Space Sci. Rev. 129, 35–78. doi: 10.1007/ nario. Icarus 221, 296. doi: 10.1016/j.icarus.2012.07.021 . s11214- 007- 9225- z . Sussman, G.J. , Wisdom, J. , 2015. Structure and Interpretation of Classical Mechanics, Zhang, J., Dauphas, N., Davis, A.M., Leya, I., Fedkin, A., 2012. The proto-Earth as a second edition MIT Press ISBN:9780262028967 . signiﬁcant source of lunar material. Nat. Geosci. 5, 240. doi: 10.1038/ngeo1429 .