Bodily tides near spin-orbit resonances
aa r X i v : . [ a s t r o - ph . E P ] A ug Extended version of a paper published in “ Celestial M echanics & Dynamical Astronomy ” , V ol. , pp. −
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Bodily tides near spin-orbit resonances
Michael Efroimsky
US Naval Observatory, Washington DC 20392 USAe-mail: michael.efroimsky @ usno.navy.mil
Abstract
Spin-orbit coupling can be described in two approaches. The first method, known as the “MacDonald torque” , is often combined with a convenient assumption that the quality factor Q is frequency-independent. This makes the method inconsistent, for the MacDonald theorytacitly fixes the rheology of the mantle by making Q scale as the inverse tidal frequency.Spin-orbit coupling can be treated also in an approach called “the Darwin torque” .While this theory is general enough to accommodate an arbitrary frequency-dependenceof Q , this advantage has not yet been fully exploited in the literature, where Q is oftenassumed constant or is set to scale as inverse tidal frequency, the latter assertion makingthe Darwin torque equivalent to a corrected version of the MacDonald torque.However neither a constant nor an inverse-frequency Q reflect the properties of realisticmantles and crusts, because the actual frequency-dependence is more complex. Hence it isnecessary to enrich the theory of spin-orbit interaction with the right frequency-dependence.We accomplish this programme for the Darwin-torque-based model near resonances. Wederive the frequency-dependence of the tidal torque from the first principles of solid-statemechanics, i.e., from the expression for the mantle’s compliance in the time domain. Wealso explain that the tidal torque includes not only the customary, secular part, but also anoscillating part.We demonstrate that the lmpq term of the Darwin-Kaula expansion for the tidal torquesmoothly passes zero, when the secondary traverses the lmpq resonance (e.g., the principaltidal torque smoothly goes through nil as the secondary crosses the synchronous orbit).Thus we prepare a foundation for modeling entrapment of a despinning primary into aresonance with its secondary. The roles of the primary and secondary may be played, e.g.,by Mercury and the Sun, correspondingly, or by an icy moon and a Jovian planet.We also offer a possible explanation for the unexpected frequency-dependence of thetidal dissipation rate in the Moon, discovered by LLR. We continue a critical examination of the tidal-torque techniques, begun in Efroimsky & Williams(2009), where the empirical treatment by and MacDonald (1964) was considered from the view-point of a more general and rigorous approach by Darwin (1879, 1880) and Kaula (1964). Re-ferring the Reader to Efroimsky & Williams (2009) for proofs and comments, we begin with aninventory of the key formulae describing the spin-orbit interaction. While in
Ibid. we employedthose formulae to explore tidal despinning well outside the 1:1 resonance (and in neglect of theintermediate resonances), in the current paper we apply this machinery to the case of despinningin the vicinity of a spin-orbit resonance. 1lthough the topic has been popular since mid-sixties and has already been addressed inbooks, the common models are not entirely adequate to the actual physics. Just as in thenonresonant case discussed in
Ibid. , a generic problem with the popular models of libration or ofcapture into a resonance is that they employ wrong rheologies (the work by Rambaux et al. 2010being the only exception we know of). Above that, the model based on the MacDonald torquesuffers a defect stemming from a genuine inconsistency inherent in the theory by MacDonald(1964).As explained in Efroimsky and Williams (2009) and Williams and Efroimsky (2012), theMacDonald theory, both in its original and corrected versions, tacitly fixes an unphysical shapeof the functional dependence Q ( χ ) , where Q is the dissipation quality factor and χ is the tidalfrequency (Williams & Efroimsky 2012). So we base our approach on the developments by Darwin(1879, 1880) and Kaula (1964), combining those with a realistic law of frequency-dependence ofthe damping rate.Since our main purpose is to lay the groundwork for the subsequent study of the process offalling into a resonance, the two principal results obtained in this paper are the following:(a) Starting with the realistic rheological model (the expression for the compliance in the timedomain), we derive the complex Love numbers ¯ k l as functions of the frequency χ , and write downtheir negative imaginary parts as functions of the frequency: − I m (cid:2) ¯ k l ( χ ) (cid:3) = | k l ( χ ) | sin ǫ l ( χ ) .It is these expressions that appear as factors in the terms of the Darwin-Kaula expansion of tides.These factors’ frequency-dependencies demonstrate a nontrivial shape, especially near resonances.This shape plays a crucial role in modeling of despinning in general, specifically in modeling theprocess of falling into a spin-orbit resonance.(b) We demonstrate that, beside the customary secular part, the Darwin torque contains ausually omitted oscillating part. Linearity of tide means that: (a) under a static load, deformation scales linearly, and (b) underundulatory loading, the same linear law applies, separately, to each frequency mode. The latterimplies that the deformation magnitude at a certain frequency should depend linearly upon thetidal stress at this frequency, and should bear no dependence upon loading at other tidal modes.Thence the dissipation rate at that frequency will depend on the stress at that frequency only.
At a point ~R = ( R, λ, φ ), the potential due to a tide-raising secondary of mass M ∗ sec , located at ~r ∗ = ( r ∗ , λ ∗ , φ ∗ ) with r ∗ ≥ R , is expandable over the Legendre polynomials P l (cos γ ) : W ( ~R , ~r ∗ ) = ∞ X l =2 W l ( ~R , ~r ∗ ) = − G M ∗ sec r ∗ ∞ X l =2 (cid:18) Rr ∗ (cid:19) l P l (cos γ )= − G M ∗ sec r ∗ ∞ X l =2 (cid:18) Rr ∗ (cid:19) l l X m =0 ( l − m )!( l + m )! (2 − δ m ) P l m (sin φ ) P l m (sin φ ∗ ) cos m ( λ − λ ∗ ) , (1)where G = 6 . × − m kg − s − is Newton’s gravity constant, and γ is the angular separationbetween the vectors ~r ∗ and ~R pointing from the primary’s centre. The latitudes φ, φ ∗ are reck-oned from the primary’s equator, while the longitudes λ, λ ∗ are reckoned from a fixed meridian.Under the assumption of linearity, the l th term W l ( ~R , ~r ∗ ) in the secondary’s potentialcauses a linear deformation of the primary’s shape. The subsequent adjustment of the primary’spotential being linear in the said deformation, the l th adjustment U l of the primary’s potential2s proportional to W l . The theory of potential requires U l ( ~r ) to fall off, outside the primary, as r − ( l + ) . Thus the overall amendment to the potential of the primary amounts to: U ( ~r ) = ∞ X l =2 U l ( ~r ) = ∞ X l =2 k l (cid:18) Rr (cid:19) l +1 W l ( ~R , ~r ∗ ) , (2) R now being the mean equatorial radius of the primary, ~R = ( R , φ , λ ) being a surface point, ~r = ( r , φ , λ ) being an exterior point located above it at a radius r ≥ R . The coefficients k l ,called Love numbers, are defined by the primary’s rheology.For a homogeneous incompressible spherical primary of density ρ , surface gravity g, andrigidity µ , the static Love number of degree l is given by k l = 32 ( l −
1) 11 + A l , where A l ≡ (2 l + 4 l + 3) µ l g ρ R = 3 (2 l + 4 l + 3) µ l π G ρ R . (3)For R ≪ r , r ∗ , consideration of the l = 2 input in (2) turns out to be sufficient. These formulae apply to static deformations. However an actual tide is never static, exceptin the case of synchronous orbiting with a zero eccentricity and inclination. Hence a realisticperturbing potential produced by the secondary carries a spectrum of modes ω l mpq (positiveor negative) numbered with four integers l mpq as in formula (105) below. The perturbationcauses a spectrum of stresses in the primary, at frequencies χ l mpq = | ω l mpq | . Although in alinear medium strains are generated exactly at the frequencies of the stresses, friction makeseach Fourier component of the strain fall behind the corresponding component of the stress.Friction also reduces the magnitude of the shape response – hence the deviation of a dynamicalLove number k l ( χ ) from its static counterpart k l = k l (0) . Below we shall explain that formulae(2 - 3) can be easily adjusted to the case of undulatory tidal loads in a homogeneous planet orin tidally-despinning homogeneous satellite (treated now as the primary, with its planet playingthe role of the tide-raising secondary). However generalisation of formulae (2 - 3) to the case ofa librating moon (treated as a primary) turns out to be highly nontrivial. As we shall see, thestandard derivation by Love (1909, 1911) falls apart in the presence of the non-potential inertialforce containing the time-derivative of the primary’s angular velocity.The frequency-dependence of a dynamical Love numbers takes its origins in the “inertia” ofstrain and, therefore, of the shape of the body. Hence the analogy to linear circuits: the l th components of W and U act as a current and voltage, while the l th Love number plays, up to afactor, the role of impedance. Therefore, under a sinusoidal load of frequency χ , it is convenientto replace the actual Love number with its complex counterpart¯ k l ( χ ) = | ¯ k l ( χ ) | exp [ − i ǫ l ( χ ) ] , (4) ǫ l being the frequency-dependent phase delay of the reaction relative to the load (Munk &MacDonald 1960, Zschau 1978). The “minus” sign in (4) makes U lag behind W for a positive ǫ l . (So the situation resembles a circuit with a capacitor, where the current leads voltage.)In the limit of zero frequency, i.e., for a steady deformation, the lag should vanish, and soshould the entire imaginary part: I m (cid:2) ¯ k l (0) (cid:3) = | ¯ k l (0) | sin ǫ l (0) = 0 , (5) Special is the case of Phobos, for whose orbital evolution the k and perhaps even the k terms may berelevant (Bills et al. 2005). Another class of exceptions is constituted by close binary asteroids. The topic isaddressed by Taylor & Margot (2010), who took into account the Love numbers up to k . The case of a permanently deformed moon in a 1:1 spin-orbit resonance falls under this description too.Recall that in the tidal context the distorted body is taken to be the primary. So from the viewpoint of thesatellite its host planet is orbiting the satellite synchronously, thus creating a static tide. k l (0) = R e (cid:2) ¯ k l (0) (cid:3) = | ¯ k l (0) | cos ǫ l (0) , (6)and equal to the customary static Love number:¯ k l (0) = k l . (7)Solution of the equation of motion combined with the constitutive (rheological) equationrenders the complex ¯ k l ( χ ), as explained in Appendix D.1. Once ¯ k l ( χ ) is found, its absolute value k l ( χ ) ≡ | ¯ k l ( χ ) | (8)and negative argument ǫ l ( χ ) = − arctan I m (cid:2) ¯ k l ( χ ) (cid:3) R e (cid:2) ¯ k l ( χ ) (cid:3) (9)should be inserted into the l th term of the Fourier expansion for the tidal potential. Things getsimplified when we study how the tide, caused on the primary by a secondary, is acting on thatsame secondary. In this case, the l th term in the Fourier expansion contains | k l ( χ ) | and ǫ l ( χ )in the convenient combination k l ( χ ) sin ǫ l ( χ ) , which is exactly − I m (cid:2) ¯ k l ( χ ) (cid:3) .Rigorously speaking, we should say not “the l th term”, but “the l th term s ”, as each l cor-responds to an infinite set of positive and negative Fourier modes ω l mpq , the physical forcingfrequencies being χ = χ l mpq ≡ | ω l mpq | . Thus, while the functional forms of both | k l ( χ ) | andsin ǫ l ( χ ) depend only on l , both functions take values that are different for different sets of num-bers mpq . This happens because χ assumes different values χ l mpq on these sets. Mind thoughthat for triaxial bodies the functional forms of | k l ( χ ) | and sin ǫ l ( χ ) may depend also on m, p, q . Beside the standard assumption U l ( ~r ) ∝ W l ( ~R , ~r ∗ ) , the linearity condition includes the re-quirement that the functions k l ( χ ) and ǫ l ( χ ) be well defined. This implies that they dependsolely upon the frequency χ , and not upon the other frequencies involved. Nor shall the Lovenumbers or lags be influenced by the stress or strain magnitudes at this or other frequencies.Then, at frequency χ , the mean (over a period) damping rate h (cid:5) E ( χ ) i depends on the valueof χ and on the loading at that frequency, and is not influenced by the other frequencies: h ˙ E ( χ ) i = − χE peak ( χ ) Q ( χ ) (10)or, equivalently: ∆ E cycle ( χ ) = − π E peak ( χ ) Q ( χ ) , (11)∆ E cycle ( χ ) being the one-cycle energy loss, and Q ( χ ) being the so-called quality factor.If E peak ( χ ) in (10 - 11) is agreed to denote the peak energy stored at frequency χ , theappropriate Q factor is connected to the phase lag ǫ ( χ ) through Q − energy = sin | ǫ | . (12)and not through Q − energy = tan | ǫ | as often presumed (see Appendix B for explanation).4f E peak ( χ ) is defined as the peak work , the corresponding Q factor is related to the lag via Q − work = tan | ǫ | − (cid:16) π − | ǫ | (cid:17) tan | ǫ | , (13)as demonstrated in Appendix B below. In the limit of a small ǫ , (13) becomes Q − work = sin | ǫ | + O ( ǫ ) = | ǫ | + O ( ǫ ) , (14)so definition (13) makes 1 /Q a good approximation to sin ǫ for small lags only.For the lag approaching π/ Q energy = 1 , while definition (13) furnishes Q work = 0 . The latter is not surprising, as in thesaid limit no work is carried out on the system.Linearity requires the functions ¯ k l ( χ ) and therefore also ǫ l ( χ ) to be well-defined, i.e., to beindependent from all the other frequencies but χ . We now see, the requirement extends to Q ( χ ) .The third definition of the quality factor (offered by Golderich 1963) is Q − Goldreich = tan | ǫ | .However this definition corresponds neither to the peak work nor to the peak energy. Theexisting ambiguity in definition of Q makes this factor redundant, and we mention it here only asa tribute to the tradition. As we shall see, all practical calculations contain the products of theLove numbers by the sines of the phase lags, k l sin ǫ l , where l is the degree of the appropriatespherical harmonic. A possible compromise between this mathematical fact and the historicaltradition of using Q would be to define the quality factor through (12), in which case the qualityfactor must be equipped with the subscript l . (This would reflect the profound difference betweenthe tidal quality factor s and the seismic quality factor – see Efroimsky 2012.) This section offers a squeezed synopsis of the basic facts from the linear solid-state mechanics.A more detailed introduction, including a glossary and examples, is offered in Appendix A.
Mechanical properties of a medium are furnished by the so-called constitutive equation or con-stitutive law, which interrelates the stress tensor S with the strain tensor U defined as U ≡ h ( ∇ ⊗ u ) + ( ∇ ⊗ u ) T i , (15)where u is the vector of displacement.As we shall consider only linear deformations, our constitutive laws will be linear, and willbe expressed by equations which may be algebraic, differential, integral, or integro-differential.The elastic stress ( e ) S is related to U through the simplest constitutive equation ( e ) S = B U , (16) B being a four-dimensional matrix of real numbers called elasticity moduli .A hereditary stress ( h ) S is connected to U as ( h ) S = ˜ B U , (17) Deriving this formula in Appendix to Efroimsky & Williams (2009), we inaccurately termed E peak ( χ ) aspeak energy. However our calculation of Q was carried out in understanding that E peak ( χ ) is the peak work . B being a four-dimensional integral-operator-valued matrix. Its component ˜ B ijkl acts on anelement u kl of the strain not as a mere multiplier but as an integral operator, with integrationgoing from t ′ = −∞ through t ′ = t . To furnish the value of σ ij = P kl ˜ B ijkl u kl at time t , theoperator “consumes” as arguments all the values of u kl ( t ′ ) over the interval t ′ ∈ ( − ∞ , t ] .The viscous stress is related to the strain through a differential operator A ∂∂t : ( v ) S = A ∂∂t U , (18) A being a four-dimensional matrix consisting of empirical constants called viscosities.In an isotropic medium, each of the three matrices, B , ˜ B , and ˜ A , includes two terms only.The elastic stress becomes: ( e ) S = ( e ) S volumetric + ( e ) S deviatoric = 3 K (cid:18) I Sp U (cid:19) + 2 µ (cid:18) U − I Sp U (cid:19) , (19)with K and µ being the bulk elastic modulus and the shear elastic modulus , correspondingly, I standing for the unity matrix, and Sp denoting the trace of a matrix: Sp U ≡ P i U ii .The hereditary stress becomes: ( h ) S = ( h ) S volumetric + ( h ) S deviatoric = 3 ˜ K (cid:18) I Sp U (cid:19) + 2 ˜ µ (cid:18) U − I Sp U (cid:19) , (20)where ˜ K and ˜ µ are the bulk-modulus operator and the shear-modulus operator , accordingly.The viscous stress acquires the form: ( v ) S = ( v ) S volumetric + ( v ) S deviatoric = 3 ζ ∂∂t (cid:18) I Sp U (cid:19) + 2 η ∂∂t (cid:18) U − I Sp U (cid:19) , (21)the quantities ζ and η being termed as the bulk viscosity and the shear viscosity , correspondinglyThe term 13 I Sp U is called the volumetric part of the strain, while U − I Sp U is calledthe deviatoric part. Accordingly, in expressions (161 - 163) for the stresses, the pure-trace termsare called volumetric , the other term being named deviatoric .If an isotropic medium is also incompressible, the relative change of the volume vanishes:Sp U = 0 , and so does the expansion rate: ∇ · v = ∂∂ t Sp U = . Then the volumetric part ofthe strain becomes zero, and so do the volumetric parts of the elastic, hereditary, and viscousstresses. The incompressibility assumption may be applicable both to crusty objects and to largeicy moons of low porosity. At least for Iapetus, the low-porosity assumption is likely to be correct(Castillo-Rogez et al. 2011). One approach to linear deformations is to assume that the elastic, hereditary and viscous devia-toric stresses simply sum up, each of them being linked to the same overall deviatoric strain: ( total ) S = ( e ) S + ( h ) S + ( v ) S = B U + ˜
B U + A ∂∂t U = (cid:18) B + ˜ B + A ∂∂t (cid:19) U . (22)An alternative option, to be used in section 5.3 below, is to start with an overall deviatoricstress, and to expand the deviatoric strain into elastic, viscous, and hereditary parts: U = ( e ) U + ( h ) U + ( v ) U , ( e ) U = 1 µ S , ( v ) U = 1 η Z t S ( t ′ ) dt ′ , ( h ) U = ˜ J S , (23)6 J being an integral operator with a time-dependent kernel.An even more general option would be to assume that both the strain and stress are comprisedby components of different nature – elastic, hereditary, viscous, or more complicated (plastic).Which option to choose – depends upon the medium studied. The rich variety of materials offeredto us by nature leaves one no chance to develop a unified theory of deformation.As different segments of the continuum-mechanics community use different conventions onthe meaning of some terms, we offer a glossary of terms in Appendix A. Here we would onlymention that in our paper the term viscoelastic will be applied to a model containing not onlyviscous and elastic terms, but also an extra term responsible for an anelastic hereditary reaction.(A more appropriate term viscoelastohereditary would be way too cumbersome.) In the general case, loading varies in time, so one has to deal with the stress and strain tensors asfunctions of time. However, treatment of viscoelasticity turns out to be simpler in the frequencydomain, i.e., in the language of complex rigidity and complex compliance. To this end, the stress σ γν and strain u γν in a linear medium can be Fourier-expanded as σ γν ( t ) = ∞ X n =0 σ γν ( χ n ) cos [ χ n t + ϕ σ ( χ n ) ] = ∞ X n =0 R e (cid:20) σ γν ( χ n ) e i χn t + i ϕσ ( χn ) (cid:21) (24a)= ∞ X n =0 R e h ¯ σ γν ( χ n ) e i χnt i , (24b) u γν ( t ) = ∞ X n =0 u γν ( χ n ) cos [ χ n t + ϕ u ( χ n ) ] = ∞ X n =0 R e (cid:20) u γν ( χ n ) e i χn t + i ϕu ( χn ) (cid:21) (25a)= ∞ X n =0 R e h ¯ u γν ( χ n ) e i χnt i , (25b)where the complex amplitudes are:¯ σ γν ( χ ) = σ γν ( χ ) e i ϕ σ ( χ ) , ¯ u γν ( χ ) = u γν ( χ ) e i ϕ u ( χ ) , (26)while the initial phases ϕ σ ( χ ) and ϕ u ( χ ) are chosen in a manner that sets the real amplitudes σ γν ( χ n ) and u γν ( χ n ) non-negative.We wrote the above expansions as sums over a discrete spectrum, as the spectrum generatedby tides is discrete. Generally, the sums can, of course, be replaced with integrals over frequency: σ γν ( t ) = Z ∞ ¯ σ γν ( χ ) e i χt dχ and u γν ( t ) = Z ∞ ¯ u γν ( χ ) e i χt dχ . (27)Whenever necessary, the frequency is set to approach the real axis from below: I m ( χ ) → − .4 Should we consider positive frequencies only? At first glance, the above question appears pointless, as a negative frequency is a mere abstraction,while physical processes go at positive frequencies. Mathematically, a full Fourier decompositionof a real field can always be reduced to a decomposition over positive frequencies only.For example, the full Fourier integral for the stress can be written as σ γν ( t ) = Z ∞−∞ ¯ s γν ( ω ) e i ωt dω = Z ∞ h ¯ s γν ( χ ) e i χt + ¯ s γν ( − χ ) e − i χt i dχ , (28)where we define χ ≡ | ω | . Denoting complex conjugation with asterisk, we write: σ ∗ γν ( t ) = Z ∞ h ¯ s ∗ γν ( − χ ) e i χt + ¯ s ∗ γν ( χ ) e − i χt i dχ . (29)The stress is real: σ ∗ γν ( t ) = σ γν ( t ) . Equating the right-hand sides of (28) and (29), we obtain¯ s γν ( − χ ) = ¯ s ∗ γν ( χ ) , (30)whence σ γν ( t ) = Z ∞ h ¯ s γν ( χ ) e i χt + ¯ s ∗ γν ( χ ) e − i χt i dχ = R e Z ∞ s γν ( χ ) e i χt dχ . (31)This leads us to (27), if we set ¯ σ γν ( χ ) = 2 ¯ s γν ( χ ) . (32)While the switch from σ γν ( t ) = R ∞−∞ ¯ s γν ( ω ) e i ωt dω to the expansion σ γν ( t ) = R ∞ ¯ σ γν ( ω ) e i χt dχ makes things simpler, the simplification comes at a cost, as we shall see in a second.Recall that the tide can be expanded over the modes ω l mpq ≡ ( l − p ) ˙ ω + ( l − p + q ) ˙ M + m ( ˙Ω − ˙ θ ) ≈ ( l − p + q ) n − m ˙ θ , (33)each of which can assume positive or negative values, or be zero. Here l , m , p , q are some integers, θ is the primary’s sidereal angle, ˙ θ is its spin rate, while ω , Ω, M and n are the secondary’speriapse, node, mean anomaly, and mean motion. The appropriate tidal frequencies, at whichthe medium gets loaded, are given by the absolute values of the tidal modes: χ lmpq ≡ | ω lmpq | .The positively-defined forcing frequencies χ lmpq are the actual physical frequency at whichthe lmpq term in the expansion for the tidal potential (or stress or strain) oscillates.The motivation for keeping also the modes ω lmpq is subtle: it depends upon the sign of ω lmpq whether the lmpq component of the tide lags or advances. Specifically, the phase lag betweenthe lmpq component of the perturbed primary’s potential U and the lmpq component of thetide-raising potential W generated by the secondary is given by ǫ lmpq = ω lmpq ∆ t lmpq = | ω lmpq | ∆ t lmpq sgn ω lmpq = χ lmpq ∆ t lmpq sgn ω lmpq , (34)where the time lag ∆ t lmpq is always positive.While the lag between the applied stress and resulting strain in a sample of a medium isalways positive, the case of tides is more complex: there, the lag can be either positive ornegative. This, of course, in no way implies whatever violation of causality (the time lag ∆ t lmpq is always positive). Rather, this is about the directional difference between the planetocentricpositions of the tide-raising body and the resulting bulge. For example, the principal componentof the tide, lmpq = 2200 , stays behind (has a positive phase lag ǫ ) when the secondary isbelow the synchronous orbit, and advances (has a negative phase lag ǫ ) when the secondary8s at a higher orbit. To summarise, decomposition of a tide over both positive and negativemodes ω lmpq (and not just over the positive frequencies χ lmpq ) does have a physical meaning,as the sign of a mode ω lmpq carries physical information.Thus we arrive at the following conclusions:1. As the fields emerging in the tidal theory – the tidal potential, stress, and strain – are allreal, their expansions in the frequency domain may, in principle, be written down using thepositive frequencies χ only.2. In the tidal theory, the potential (and, consequently, the tidal torque and force) containcomponents corresponding to the tidal modes ω lmpq of both the positive and negative signs.While the lmpq components of the potential, stress, and strain oscillate at the positivefrequencies χ lmpq = | ω lmpq | , the sign of each ω lmpq does carry physical information: it dis-tinguishes whether the lagging of the lmpq component of the bulge is positive or negative(falling behind or advancing). Accordingly, this sign enters explicitly the expression for theappropriate component of the torque or force. Hence a consistent tidal theory should bedeveloped through expansions over both positive and negative tidal modes ω lmpq and notjust over the positive χ lmpq .3. In order to rewrite the tidal theory in terms of the positively-defined frequencies χ lmpq only, one must inserts “by hand” the extra multiplierssgn ω lmpq = sgn h ( l − p + q ) n − m ˙ θ i (35)into the expressions for the lmpq components of the tidal torque and force.4. One can employ a rheological law (constitutive equation interconnecting the strain andstress) and a Navier-Stokes equation (the second law of Newton for an element of a vis-coelastic medium), to calculate the phase lag ǫ lmpq of the primary’s potential U lmpq relativeto the potential W lmpq generated by the secondary. If both these equations are expanded,in the frequency domain, via positively-defined forcing frequencies χ lmpq only, the resultingphase lag, too, will emerge as a function of χ lmpq : ǫ lmpq = ǫ l ( χ lmpq ) . (36)Within this treatment, one has to equip the lag, “by hand”, with the multiplier (35).As we saw above, the lag (36) is the argument of the complex Love number ¯ k l ( χ lmpq ) . Solutionof the constitutive and Navier-Stokes equations renders the complex Love numbers, from whichone can calculate the lags. Hence the above item [4] may be rephrased in the following manner:4 ′ . Under the convention that U lmpq = U ( χ lmpq ) and W lmpq = W ( χ lmpq ) , we have: U lmpq = ¯ k l ( χ lmpq ) W lmpq when ω lmpq > , i.e. , when ω lmpq = χ lmpq , (37a) U lmpq = ¯ k ∗ l ( χ lmpq ) W lmpq when ω lmpq < , i.e. , when ω lmpq = − χ lmpq , (37b)asterisk denoting the complex conjugation.9his ugly convention, a switch from ¯ k l to ¯ k ∗ l , is the price we pay for employing only the positivefrequencies in our expansions, when solving the constitutive and Navier-Stokes equations, to findthe Love number. In other words, this is a price for our pretending that W lmpq and U lmpq arefunctions of χ lmpq – whereas in reality they are functions of ω lmpq .Alternative to this would be expanding the stress, strain, and the potentials over the positiveand negative modes ω lmpq , with the negative frequencies showing up in the equations. With theconvention that U lmpq = U ( ω lmpq ) and W lmpq = W ( ω lmpq ) , we would have U lmpq = ¯ k l ( ω lmpq ) W lmpq , for all ω lmpq . (38)All these details can be omitted at the despinning stage, if one keeps only the leading term ofthe torque and ignores the other terms. Things change, though, when one takes these other termsinto account. On crossing of an lmpq resonance, factor (35) will change its sign. Accordingly,the lmpq term of the tidal torque (and of the tidal force) will change its sign too. The stress cannot be obtained by means of an integral operator that would map the past historyof the strain, U ( t ′ ) over t ′ ∈ ( − ∞ , t ] , to the value of S at time t . The insufficiency ofsuch an operator is evident from the presence of a time-derivative on the right-hand side of (18).Exceptional are the cases of no viscosity (e.g., a purely elastic material).On the other hand, we expect, on physical grounds, that the operator ˆ J inverse to ˆ µ is anintegral operator. In other words, we assume that the current value of the strain depends onlyon the present and past values taken by the stress and not on the current rate of change of thestress. This assumption works for weak deformations, i.e., insofar as no plasticity shows up. Sowe assume that the operator ˆ J mapping the stress to the strain is just an integral operator.Since the forced medium “remembers” the history of loading, the strain at time t mustbe a sum of small installments 12 J ( t − t ′ ) dσ γν ( t ′ ) , each of which stems from a small change dσ γν ( t − τ ) of the stress at an earlier time t ′ < t . The entire history of the past loading results,at the time t , in a total strain u γν ( t ) rendered by an integral operator ˆ J ( t ) acting on the entirefunction σ γν ( t ′ ) and not on its particular value (Karato 2008):2 u γν ( t ) = ˆ J ( t ) σ γν = Z ∞ J ( τ ) (cid:5) σ γν ( t − τ ) dτ = Z t −∞ J ( t − t ′ ) (cid:5) σ γν ( t ′ ) dt ′ , (39)where t ′ is some earlier time ( t ′ < t ), overdot denotes d/dt ′ , while the “age variable” τ = t − t ′ is reckoned from the current moment t and is aimed back into the past. The so-defined integraloperator ˆ J ( t ) is called the compliance operator , while its kernel J ( t − t ′ ) goes under the nameof the compliance function or the creep-response function .Integrating (39) by parts, we recast the compliance operator into the form of2 u γν ( t ) = ˆ J ( t ) σ γν = J (0) σ γν ( t ) − J ( ∞ ) σ γν ( −∞ ) + Z ∞ (cid:5) J ( τ ) σ γν ( t − τ ) dτ (40a)= J (0) σ γν ( t ) − J ( ∞ ) σ γν ( −∞ ) + Z t −∞ (cid:5) J ( t − t ′ ) σ γν ( t ′ ) dt ′ . (40b)The quantity J ( ∞ ) is the relaxed compliance . Being the asymptotic value of J ( t − t ′ ) at t − t ′ → ∞ , this parameter corresponds to the strain after complete relaxation. The load in theinfinite past may be assumed zero, and the term − J ( ∞ ) σ γν ( −∞ ) may be dropped10he second important quantity emerging in (40) is the unrelaxed compliance J (0) , which isthe value of the compliance function J ( t − t ′ ) at t − t ′ = 0 . This parameter describes theinstantaneous reaction to stressing, and thus defines the elastic part of the deformation (therest of the deformation being viscous and hereditary). Thus the term containing the unrelaxedcompliance J (0) should be kept. The term, though, can be absorbed into the integral if we agreethat the elastic contribution enters the compliance function not as J ( t − t ′ ) = J (0) + viscous and hereditary terms , (41)but as J ( t − t ′ ) = J (0) Θ( t − t ′ ) + viscous and hereditary terms , (42)the Heaviside step-function Θ( t − t ′ ) being unity for t − t ′ ≥ t − t ′ < δ ( t − t ′ ) , we can write (40b) simply as2 u γν ( t ) = ˆ J ( t ) σ γν = Z t −∞ (cid:5) J ( t − t ′ ) σ γν ( t ′ ) dt ′ , with J ( t − t ′ ) containing J (0) Θ( t − t ′ ) . (43)Equations (39), (40), (43) are but different expressions for the compliance operator ˆ J acting as2 u γν = ˆ J σ γν . (44)Inverse to the compliance operator is the rigidity operator ˆ µ defined through σ γν = 2 ˆ µ u γν . (45)Generally, ˆ µ is not just an integral operator, but is an integro-differential operator. So it cannottake the form of σ γν ( t ) = 2 R t −∞ ˙ µ ( t − t ′ ) u γν ( t ′ ) dt ′ . However it can be written as σ γν ( t ) = 2 Z t −∞ µ ( t − t ′ ) ˙ u γν ( t ′ ) dt ′ , (46)if we permit the kernel µ ( t − t ′ ) to contain a term η δ ( t − t ′ ) , where δ ( t − t ′ ) is the delta-function.After integration, this term will furnish the viscous part of the stress, 2 η ˙ u γν .The kernel µ ( t − t ′ ) goes under the name of the stress-relaxation function . Its time-independent part is µ (0) Θ( t − t ′ ) , where the unrelaxed rigidity µ (0) is inverse to the unrelaxedcompliance J (0) and describes the elastic part of deformation. Each term in µ ( t − t ′ ) , whichneither is a constant nor contains a delta-function, is responsible for hereditary reaction.For more details on the stress-strain relaxation formalism see the book by Karato (2008). Let us introduce the complex compliance ¯ J ( χ ) and the complex rigidity ¯ µ ( χ ) , which are, bydefinition, the Fourier images not of the J ( τ ) and µ ( τ ) functions, but of their time-derivatives: Z ∞ ¯ J ( χ ) e i χτ dχ = (cid:5) J ( τ ) , where ¯ J ( χ ) = Z ∞ (cid:5) J ( τ ) e − i χτ dτ . (47) Expressing the stress through the strain, we encountered three possibilities: the elastic stress was simplyproportional to the strain, the viscous stress was proportional to the time-derivative of the strain, while thehereditary stress was expressed by an integral operator ˜ µ . However, when we express the strain through thestress, we place the viscosity into the integral operator, so the purely viscous reaction also looks like hereditary.It is our convention, though, to apply the term hereditary to delayed reactions other than purely viscous . Recall that it is the time-derivative of J ( τ ) that is the kernel of the integral operator (43). Hence, to arriveat (50), we have to define ¯ J ( χ ) as the Fourier image of (cid:5) J ( τ ) . Z ∞ ¯ µ ( χ ) e i χτ dχ = ˙ µ ( τ ) , where ¯ µ ( χ ) = Z ∞ ˙ µ ( τ ) e − i χτ dτ , (48)the integrations over τ spanning the interval [ 0 , ∞ ) , as both kernels are nil for τ < Z ∞ ¯ u γν ( χ ) e i χt dχ = Z ∞ ¯ σ µν ( χ ) ¯ J ( χ ) e i χt dχ , (49)which leads us to: 2 ¯ u γν ( χ ) = ¯ J ( χ ) ¯ σ γν ( χ ) . (50)Similarly, insertion of (27) into (46) leads to the relation¯ σ γν ( χ ) = 2 ¯ µ ( χ ) ¯ u γν ( χ ) , (51)comparison whereof with (50) immediately entails:¯ J ( χ ) ¯ µ ( χ ) = 1 . (52)Writing down the complex rigidity and compliance as¯ µ ( χ ) = | ¯ µ ( χ ) | exp [ i δ ( χ ) ] (53)and ¯ J ( χ ) = | ¯ J ( χ ) | exp [ − i δ ( χ ) ] , (54)we split (52) into two expressions: | ¯ J ( χ ) | = 1 | ¯ µ ( χ ) | (55)and ϕ u ( χ ) = ϕ σ ( χ ) − δ ( χ ) . (56)From the latter, we see that the angle δ ( χ ) is a measure of lagging of a strain harmonic moderelative to the appropriate harmonic mode of the stress. It is evident from (53 - 54) thattan δ ( χ ) ≡ − I m (cid:2) ¯ J ( χ ) (cid:3) R e (cid:2) ¯ J ( χ ) (cid:3) = I m [ ¯ µ ( χ ) ] R e [ ¯ µ ( χ ) ] . (57) The developments presented in this section will rest on a very important theorem from solid-state mechanics. The theorem, known as the correspondence principle , also goes under the nameof elastic-viscoelastic analogy . The theorem applies to linear deformations in the absence ofnonconservative (inertial) forces. While the literature attributes the authorship of the theoremto different scholars, its true pioneer was Sir George Darwin (1879). One of the corollaries ensuingfrom this theorem is that, in the frequency domain, the complex Love numbers are expressed via12he complex rigidity or compliance in the same way as the static Love numbers are expressed viathe relaxed rigidity or compliance.As was pointed out much later by Biot (1954, 1958), the theorem is inapplicable to non-potential forces. Hence the said corollary fails in the case of librating bodies, because of thepresence of the inertial force − ˙ ~ω × ~r ρ , where ρ is the density and ~ω is the libration angularvelocity. So the standard expression (3) for the Love numbers, generally, cannot be employed forlibrating bodies.Subsection 4.1 below explains the transition from the stationary Love numbers to their dy-namical counterparts, the so-called Love operators. We present this formalism in the frequencydomain, in the spirit of Zahn (1966) who pioneered this approach in application to a purely vis-cous medium. Subsection 4.2 addresses the negative tidal modes emerging in the Darwin-Kaulaexpansion for tides. Employing the correspondence principle, in subsection 4.3 we then writedown the expressions for the factors | ¯ k l ( χ ) | sin ǫ l ( χ ) = − I m [¯ k l ( χ )] emerging in the expansionfor tides. Some technical details of this derivation are discussed in subsections 4.4 - 4.5.For more on the correspondence principle and its applicability to Phobos see Appendix D. A homogeneous incompressible primary, when perturbed by a static secondary, yields its formand, consequently, has its potential changed. The l th spherical harmonic U l ( ~r ) of the resultingincrement of the primary’s exterior potential is related to the l th spherical harmonic W l ( ~R , ~r )of the perturbing exterior potential through (2).As the realistic disturbances are never static (except for synchronous orbiting), the Lovenumbers become operators: U l ( ~r , t ) = (cid:18) Rr (cid:19) l +1 ˆ k l ( t ) W l ( ~R , ~r ∗ , t ′ ) . (58)A Love operator acts neither on the value of W at the current time t , nor at its value at anearlier time t ′ , but acts on the entire shape of the function W l ( ~R , ~r ∗ , t ′ ) , with t ′ belongingto the semi-interval ( −∞ , t ) . This is why we prefer to write ˆ k l ( t ) and not ˆ k l ( t, t ′ ) .Being linear for weak forcing, the operators must read: U l ( ~r , t ) = (cid:18) Rr (cid:19) l +1 Z τ = ∞ τ =0 k l ( τ ) (cid:5) W l ( ~R , ~r ∗ , t − τ ) dτ = (cid:18) Rr (cid:19) l +1 Z t ′ = tt ′ = −∞ k l ( t − t ′ ) (cid:5) W l ( ~R , ~r ∗ , t ′ ) dt ′ (59a)or, after integration by parts: U l ( ~r , t ) = (cid:18) Rr (cid:19) l +1 [ k l (0) W ( t ) − k l ( ∞ ) W ( −∞ ) ] + (cid:18) Rr (cid:19) l +1 Z ∞ ˙ k l ( τ ) W l ( ~R , ~r ∗ , t − τ ) dτ (59b)= (cid:18) Rr (cid:19) l +1 [ k l (0) W ( t ) − k l ( ∞ ) W ( −∞ )] + (cid:18) Rr (cid:19) l +1 Z t −∞ ˙ k l ( t − t ′ ) W l ( ~R , ~r ∗ , t ′ ) dt ′ (59c)= − (cid:18) Rr (cid:19) l +1 k l ( ∞ ) W ( −∞ )+ (cid:18) Rr (cid:19) l +1 Z t −∞ ddt [ k l ( t − t ′ ) − k l (0) + k l (0)Θ( t − t ′ ) ] W l ( ~R , ~r ∗ , t ′ ) dt ′ . (59d) The centripetal term is potential and causes no troubles, except for the necessity to introduce a degree-0Love number. k l (0) W ( t ) and − k l ( ∞ ) W ( −∞ ) . Of the latter term, we can get rid by setting W ( −∞ ) nil,while the former term may be incorporated into the kernel in exactly the same way as in (41 -43). Thus, dropping the unphysical term with W ( −∞ ) , and inserting the elastic term into theLove number not as k l (0) but as k l (0) Θ( t − t ′ ) , we simplify (59d) to U l ( ~r , t ) = (cid:18) Rr (cid:19) l +1 Z t −∞ ˙ k l ( t − t ′ ) W l ( ~R , ~r ∗ , t ′ ) dt ′ , (60)with k l ( t − t ′ ) now including, as its part, k l (0) Θ( t − t ′ ) instead of k l (0) .Were the body perfectly elastic, k l ( t − t ′ ) would consist of the instantaneous-reaction term k l (0) Θ( t − t ′ ) only . Accordingly, the time-derivative of k l would be: ˙ k l ( t − t ′ ) = k l δ ( t − t ′ )where k l ≡ k l (0) , so expressions (59 - 60) would coincide with (2).Similarly to introducing the complex compliance, one can define the complex Love numbersas Fourier transforms of (cid:5) k l ( τ ) : Z ∞ ¯ k l ( χ ) e i χτ dχ = (cid:5) k l ( τ ) , (61)the overdot standing for d/dτ . Churkin (1998) suggested to term the time-derivatives (cid:5) k l ( t )as the Love functions . Inversion of (61) trivially yields:¯ k l ( χ ) = Z ∞ ˙ k l ( τ ) e − i χτ dτ = k l (0) + i χ Z ∞ [ k l ( τ ) − k l (0) Θ( τ ) ] e − i χτ dτ , (62)where we integrated only from 0 because the future disturbance contributes nothing to thepresent distortion, so k l ( τ ) vanishes at τ < τ denotes the difference t − t ′ . So τ is reckoned from the present moment t and is directed back into the past.Defining in the standard manner the Fourier components ¯ U l ( χ ) and ¯ W l ( χ ) of functions U l ( t ) and W l ( t ) , we write (59) in the frequency domain:¯ U l ( χ ) = (cid:18) Rr (cid:19) l +1 ¯ k l ( χ ) ¯ W l ( χ ) , (63)where we denote the frequency simply by χ instead of the awkward χ lmpq . To employ (63) in thetidal theory, one has to know the frequency-dependencies ¯ k l ( χ ) . χ ≡ | ω | vs.the positive and negative tidal modes ω It should be remembered that, by relying on formula (63), we place ourselves on thin ice, becausethe similarity of this formula to (50) and (51) is deceptive.In (50) and (51), it was legitimate to limit our expansions of the stress and the strain topositive frequencies χ only. Had we carried out those expansions over both positive and negativefrequencies ω , we would have obtained, instead of (50) and (51), similar expressions2 ¯ u γν ( ω ) = ¯ J ( ω ) ¯ σ γν ( ω ) and ¯ σ γν ( ω ) = 2 ¯ µ ( ω ) ¯ u γν ( ω ) . (64)For positive ω , these would simply coincide with (50) and (51), if we rename ω as χ . Fornegative ω = − χ , the resulting expressions would read as2 ¯ u γν ( − χ ) = ¯ J ( − χ ) ¯ σ γν ( − χ ) and ¯ σ γν ( − χ ) = 2 ¯ µ ( − χ ) ¯ u γν ( − χ ) , (65) Churkin (1998) used functions which he called k l ( t ) and which were, due to a difference in notations, thesame as our (cid:5) k l ( τ ) . χ always stands for a positive quantity. In accordancewith (30), complex conjugation of (65) would then return us to (64).Physically, the negative-frequency components of the stress or strain are nonexistent. Ifbrought into consideration, they are obliged to obey (30) and, thus, should play no role, exceptfor a harmless renormalisation of the Fourier components in (32).When we say that the physically measurable stress σ γν ( t ) is equal to P R e h ¯ σ γν ( χ ) e i χt i ,it is unimportant to us whether the χ -contribution in σ γν ( t ) comes from the term ¯ σ γν ( χ ) e i χt only, or also from the term ¯ σ γν ( − χ ) e i ( − χ ) t . Indeed, the real part of the latter is a clone of thereal part of the former (and it is only the former term that is physical). However, things remainthat simple only for the stress and the strain.As we emphasised in subsection 3.4, the situation with the potentials is drastically different.While the physically measurable potential U ( t ) is still equal to P R e h ¯ U ( χ ) e i χt i , it is now important to distinguish whether the χ -contribution in U ( t ) comes from the term ¯ U γν ( χ ) e i χt or from the term ¯ U ( − χ ) e i ( − χ ) t , or perhaps from both. Although the negative mode − χ wouldbring the same input as the positive mode χ , these inputs will contribute differently into thetidal torque. As can be seen from (285), the secular part of the tidal torque is proportional tosin ǫ l , where ǫ l ≡ ω lmpq ∆ t lmpq , with the time lag ∆ t lmpq being positively defined – see formula(109). Thus the secular part of the tidal torque explicitly contains the sign of the tidal mode ω lmpq .For this reason, as explained in subsection 3.4, a more accurate form of formula (63) shouldbe: ¯ U l ( ω ) = ¯ k l ( ω ) ¯ W l ( ω ) , (66)where ω can be of any sign.If however, we pretend that the potentials depend on the physical frequency χ = | ω | only,i.e., if we always write U ( ω ) as U ( χ ) , then (63) must be written as:¯ U l ( χ ) = ¯ k l ( χ ) ¯ W l ( χ ) , when χ = | ω | for ω > , (67a)and ¯ U l ( χ ) = ¯ k ∗ l ( χ ) ¯ W l ( χ ) , when χ = | ω | for ω < . (67b)Unless we keep this detail in mind, we shall get a wrong sign for the lmpq component of thetorque after the despinning secondary crosses the appropriate commensurability. (We shall, ofcourse, be able to mend this by simply inserting the sign sgn ω lmpq by hand.) While the static Love numbers depend on the static rigidity modulus µ via (3), it is not readilyapparent that the same relation interconnects ¯ k l ( χ ) with ¯ µ ( χ ) , the quantities that are theFourier components of the time-derivatives of k ( t ′ ) and µ ( t ′ ) . Fortunately, the correspondenceprinciple (discussed in Appendix D) tells us that, in many situations, the viscoelastic operationalmoduli ¯ µ ( χ ) or ¯ J ( χ ) obey the same algebraic relations as the elastic parameters µ or J . Thisis why, in these situations, the Fourier or Laplace transform of our viscoelastic equations willmimic (228a - 228b), except that all the functions will acquire overbars: ¯ σ γν = 2 ¯ µ ¯ u γν , etc.15o their solution, too, will be ¯ U l = ¯ k l ¯ W l , with ¯ k l retaining the same functional dependence on ρ , R , and ¯ µ as in (3), except that now µ will have an overbar:¯ k l ( χ ) = 32 ( l −
1) 11 + (2 l + 4 l + 3) ¯ µ ( χ ) l g ρ R = 32 ( l −
1) 11 + A l ¯ µ ( χ ) /µ (68)= 32 ( l −
1) 11 + A l J/ ¯ J ( χ ) = 32 ( l −
1) ¯ J ( χ )¯ J ( χ ) + A l J Here the coefficients A l are defined via the unrelaxed quantities µ = µ (0) = 1 /J = 1 /J (0)in the same manner as the static A l were introduced through the static (relaxed) µ = 1 /J informulae (3).The moral of the story is that, at low frequencies, each ¯ k l depends upon ¯ µ (or upon ¯ J ) inthe same way as its static counterpart k l depends upon the static µ (or upon the static J ).This happens, because at low frequencies we neglect the acceleration term in the equation ofmotion (231b), so this equation still looks like (228b).Representing a complex Love number as¯ k l ( χ ) = R e (cid:2) ¯ k l ( χ ) (cid:3) + i I m (cid:2) ¯ k l ( χ ) (cid:3) = | ¯ k l ( χ ) | e − i ǫ l ( χ ) (69)we can write for the phase lag ǫ l ( χ ) :tan ǫ l ( χ ) ≡ − I m (cid:2) ¯ k l ( χ ) (cid:3) R e (cid:2) ¯ k l ( χ ) (cid:3) (70)or, equivalently: | ¯ k l ( χ ) | sin ǫ l ( χ ) = − I m (cid:2) ¯ k l ( χ ) (cid:3) . (71)The products | ¯ k l ( χ ) | sin ǫ l ( χ ) standing on the left-hand side in (71) emerge also in the Fourierseries for the tidal potential. Therefore it is these products (and not k l /Q ) that should enterthe expansions for forces, torques, and the damping rate. This is the link between the body’srheology and the history of its spin: from ¯ J ( χ ) to ¯ k l ( χ ) to | ¯ k l ( χ ) | sin ǫ ( χ ) , the latter beingemployed in the theory of bodily tides.Through simple algebra, expressions (68) entail: | ¯ k l ( χ ) | sin ǫ l ( χ ) = − I m (cid:2) ¯ k l ( χ ) (cid:3) = 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) . (72)As we know from subsections 3.4 and 4.2, formulae (70 - 72) should be used with care. Sincein reality the potential ¯ U and therefore also ¯ k l are functions not of χ but of ω , then formulae(72) should be equipped with multipliers sgn ω lmpq , when plugged into the expression for the lmpq component of the tidal force or torque. This prescription is equivalent to (67). ¯ k l mpq and ǫ l mpq , or would ¯ k l and ǫ l be enough? In the preceding subsection, the static relation (2) was generalised to evolving settings as U l mpq ( ~r , t ) = (cid:18) Rr (cid:19) l +1 ˆ k l ( t ) W lmpq ( ~R , ~r ∗ , t ′ ) , (73)16here l mpq is a quadruple of integers employed to number a Fourier mode in the Darwin-Kaula expansion (100) of the tide, while U l mpq ( ~r , t ) and W lmpq ( ~R , ~r ∗ , t ′ ) are the harmonicscontaining cos( χ l mpq t − ǫ l mpq ) and cos( χ l mpq t ′ ) correspondingly.One might be tempted to generalise (2) even further to U l mpq ( ~r , t ) = (cid:18) Rr (cid:19) l +1 ˆ k l mpq ( t ) W lmpq ( ~R , ~r ∗ , t ′ ) , with the Love operator (and, consequently, its kernel, the Love function) bearing dependenceupon m , p , and q . Accordingly, (63) would become¯ U lmpq ( χ ) = ¯ k l mpq ( χ ) ¯ W lmpq ( χ ) . (74)Fortunately, insofar as the Correspondence Principle is valid, the functional form of the function¯ k l mpq ( χ ) depends upon l only and, thus, can be written down simply as ¯ k l ( χ l mpq ) . We knowthis from the considerations offered after equations (228a - 228b). There we explained that ¯ k l depends on χ = χ l mpq only via ¯ J ( χ ) , while the functional form of ¯ k l bears no dependence on χ = χ l mpq and, therefore, no dependence on m, p, q .The phase lag is often denoted as ǫ l mpq , a time-honoured tradition established by Kaula(1964). However, as the lag is expressed through ¯ k l via (70), we see that all said above about¯ k l applies to the lag too: while the functional form of the dependency ǫ l mpq ( χ ) may be differentfor different l s , it is invariant under the other three integers, so the notation ǫ l ( χ l mpq ) would bemore adequate.It should be mentioned, though, that for bodies of pronounced non-sphericity coupling be-tween the spherical harmonics furnishes the Love numbers and lags whose expressions throughthe frequency, for a fixed l , have different functional forms for different m, p, q . In these cases,the notations ¯ k l mpq and ǫ l mpq become necessary (Smith 1974; Wahr 1981a,b,c; Dehant 1987a,b).For a slightly non-spherical body, the Love numbers differ from the Love numbers of the sphericalreference body by a term of the order of the flattening, so a small non-sphericity can usually beneglected. For small bodies and small terrestrial planets, the values of A l vary from about unity to dozensto hundreds. For example, A is about 2 for the Earth (Efroimsky 2012), about 20 for Mars(Efroimsky & Lainey 2007), about 80 for the Moon (Efroimsky 2012), and about 200 for Iapetus(Castillo-Rogez et al. 2011). For superearths, the values will be much smaller than unity, though.Insofar as A l J | − J ( χ ) | ≫ , (75)one can approximate (68) with¯ k l ( χ ) = − l − J ( χ ) J ( χ ) + A l J = − J ( χ ) A l J + O (cid:16) | J/ ( A l J ) | (cid:17) , (76)except in the closest vicinity of an lmpq resonance, where the tidal frequency χ lmpq approachesnil, and ¯ J diverges for some rheologies – like, for example, for those of Maxwell or Andrade.Whenever the approximate formula (76) is applicable, we can rewrite (70) astan ǫ ( χ ) ≡ − I m (cid:2) ¯ k l ( χ ) (cid:3) R e (cid:2) ¯ k l ( χ ) (cid:3) ≈ − I m (cid:2) ¯ J ( χ ) (cid:3) R e (cid:2) ¯ J ( χ ) (cid:3) = tan δ ( χ ) , (77)17herefrom we readily deduce that the phase lag ǫ ( χ ) of the tidal frequency χ coincides withthe phase lag of the complex compliance: ǫ ( χ ) ≈ δ ( χ ) , (78)provided χ is not too close to nil (i.e., provided we are not too close to the commensurability).This way, insofar as the condition (71) is fulfilled, the component ¯ U l ( χ ) of the primary’s potentiallags behind the component ¯ W l ( χ ) of the perturbed potential by the same phase angle as thestrain lags behind the stress at frequency χ in a sample of the material. Dependent upon therheology, a vanishing tidal frequency may or may not limit the applicability of (71) and thuscause a considerable difference between ǫ and δ .In other words, the suggested approximation is valid insofar as changes of shape are deter-mined solely by the local material properties, and not by self-gravitation of the object as a whole.Whether this is so or not – depends upon the rheological model. For a Voigt or SAS solid inthe limit of χ → J ( χ ) → J , so the zero-frequency limit of ¯ k l ( χ ) is the static Lovenumber k l ≡ | ¯ k (0) | . In this case, approximation (76 - 78) remains applicable all the way downto χ = 0 . For the Maxwell and Andrade models, however, one obtains, for vanishing frequency:¯ J ( χ ) ∼ / ( ηχ ) , whence ¯ µ ∼ ηχ and ¯ k ( χ ) approaches the hydrodynamical Love number k ( hyd )2 = 3 / A l ≫
1, atall frequencies, because the condition A L ≫ τ − M A − l = µη A − l , andso does the approximation (78). Indeed, at frequencies below this threshold, self-gravitation“beats” the local material properties of the body, and the behaviour of the tidal lag deviatesfrom that of the lag in a sample. This deviation will be indicated more clearly by formula (94)in the next section. The fact that, for some models, the tidal lag ǫ deviates from the materiallag angle δ at the lowest frequencies should be kept in mind when one wants to explore crossing of a resonance.A standard caveat is in order, concerning formulae (76 - 78). Since in reality the potential ¯ U is a function of ω and not χ , our illegitimate use of χ should be compensated by multiplyingthe function ǫ l ( χ lmpq ) with sgn ω lmpq , when the lag shows up in the expression for the tidalforce or torque. Tidal dissipation within a multilayer near-spherical body is studied through expanding the in-volved fields over the spherical harmonics in each layer, setting the boundary conditions on theouter surface, and using the matching conditions on boundaries between layers. This formalismwas developed by Alterman et al (1959). An updated discussion of the method can be found inSabadini & Vermeersen (2004). For a brief review, see Legros et al (2006).Calculation of tidal dissipation in a Jovian planet is an even more formidable task (see Remuset al. 2012a and references therein). However dissipation in a giant planet with a solid core mayturn out to be approachable by analytic means (Remus et al. 2011, 2012b). The acronym
SAS stands for the
Standard Anelastic Solid , which is another name for the Hohenemser-Pragerviscoelastic model. See the Appendix for details. Dissipation at different frequencies
In Efroimsky & Lainey (2007), we considered the generic rheological model Q = ( E χ ) α , (79a)where χ is the tidal frequency and E is a parameter having the dimensions of time. The physicalmeaning of this parameter is elucidated in Ibid. . Under the special choice of α = − Q , this parameter coincides with the time lag ∆ t which, for thisspecial rheology, turns out to be the same at all frequencies.Actual experiments register not the inverse quality factor but the phase lag between thereaction and the action. So the empirical law should rather be written down as1sin δ = ( E χ ) α , (79b)which is equivalent to (79a), provided the Q factor is defined there as Q energy and not as Q work – see subsection 2.2 for details.The applicability realm of the empirical power law (79) is remarkably broad – in terms ofboth the physical constituency of the bodies and their chemical composition. Most intriguing isthe robust universality of the values taken by the index α for very different materials: between0 . . .
14 and 0 . α , also applies to ices. The resultis milestone, taken the physical and chemical differences between ices and silicates. It is agreedupon that in crystalline materials the Andrade regime can find its microscopic origin both inthe dynamics of dislocations (Karato & Spetzler 1990) and in the grain-boundary diffusionalcreep (Gribb & Cooper 1998). As the same behaviour is inherent in metals, silicates, ices, andeven glass-polyester composites (Nechada et al. 2005), it should stem from a single underlyingphenomenon determined by some principles more general than specific material properties. Anattempt to find such a universal mechanism was undertaken by Miguel et al. (2002). See alsothe theoretical considerations offered in Karato & Spetzler (1990).In seismology, the power law (79) became popular in the second part of the XX th century, withthe progress of precise measurements on large seismological basins (Mitchell 1995, Stachnik et al.2004, Shito et al. 2004). Further confirmation of this law came from geodetic experiments thatincluded: (a) satellite laser ranging (SLR) measurements of tidal variations in the J componentof the gravity field of the Earth; (b) space-based observations of tidal variations in the Earth’srotation rate; and (c) space-based measurements of the Chandler Wobble period and damping(Benjamin et al. 2006, Eanes & Bettadpur 1996, Eanes 1995). Not surprisingly, the Andradelaw became a key element in the recent attempt to construct a universal rheological model ofthe Earth’s mantle (Birger 2007). This law also became a component of the non-hydrostatic-equilibrium model for the zonal tides in an inelastic Earth by Defraigne & Smits (1999), amodel that became the basis for the IERS Conventions (Petit & Luzum 2010). While the labexperiments give for α values within 0 . − . . − . χ anelasticitygives way to purely viscoelastic behaviour, so the parameter α becomes close to unity. Forthe Earth’s mantle, the threshold corresponds to the time-scale about a year or slightly longer.Although in Karato & Spetzler (1990) the rheological law is written in terms of 1 /Q , we shallsubstitute it with a law more appropriate to the studies of tides: k l sin ǫ l = ( E χ ) − p , where p = 0 . − . χ > χ and p ∼ χ < χ , (80) χ being the frequency, and χ being the frequency threshold below which viscosity takes overanelasticity.The reason why we write the power scaling law as (80) and not as (79) is that at the lowestfrequencies the geodetic measurements give us actually k l sin ǫ l = − I m (cid:2) ¯ k l ( χ ) (cid:3) and not the lagangle δ in a sample (e.g., Benjamin et al. 2006). For this same reason, we denoted the exponentsin (79) and (80) with different letters, α and p . Below we shall see that these exponents do notalways coincide. Another reason for giving preference to (80) is that not only the sine of the lagbut also the absolute value of the Love number is frequency dependent. Fitting of the LLR data to the power scaling law (79), which was carried out by Williams etal. (2001), has demonstrated that the lunar mantle possesses quite an abnormal value of theexponent: − .
19 . A later reexamination in Williams et al. (2008) rendered a less embar-rassing value, − .
09 , which nevertheless was still negative and thus seemed to contradict ourknowledge about microphysical damping mechanisms in minerals. Thereupon, Williams & Boggs(2009) commented:“
There is a weak dependence of tidal specific dissipation Q on period. The Q increases from ∼ at a month to ∼ at one year. Q for rock is expected to have a weak dependence on tidalperiod, but it is expected to decrease with period rather than increase. The frequency dependenceof Q deserves further attention and should be improved. ”While there always remains a possibility of the raw data being insufficient or of the fittingprocedure being imperfect, the fact is that the negative exponent obtained in Ibid. does not necessarily contradict the scaling law (79) proven for minerals and partial melts. Indeed, theexponent obtained by the LLR Team was not the α from (79) but was the p from (80). Thedistinction is critical due to the difference in frequency-dependence of the seismic and tidal dissi-pation. It turns out that the near-viscous value p ∼ τ M A l (where τ M = η/µ is the Maxwelltime, with η and µ being the lunar mantle’s viscosity and rigidity), the exponent p begins This circumstance was ignored by Defraigne & Smits (1999). Accordingly, if the claims by Karato & Spetzler(1990) are correct, the table of corrections for the tidal variations in the Earth’s rotation in the IERS Conventionsis likely to contain increasing errors for periods of about a year and longer.This detail is missing in the theory of the Chandler wobble of Mars, by Zharkov & Gudkova (2009).
20o decrease with the decrease of the frequency. As the frequency becomes lower, p changes itssign and eventually becomes − χ = 0 . This behaviour follows fromcalculations based on a realistic rheology (see formulae (92 - 94) below), and it goes along wellwith the evident physical fact that the average tidal torque must vanish in a resonance. Insubsection 5.7, comparison of this behaviour with the LLR results will yield us an estimate forthe mean lunar viscosity.
The complex compliance of a Maxwell material contains a term J = J (0) responsible for theelastic part of the deformation and a term − i χη describing the viscosity. Whatever other termsget incorporated into the compliance, these will correspond to other forms of hereditary reaction.The available geophysical data strongly favour a particular extension of the Maxwell approach,the Andrade model (Cottrell & Aytekin 1947, Duval 1976). In modern notations, the model canbe expressed as J ( t − t ′ ) = (cid:2) J + β ( t − t ′ ) α + η − ( t − t ′ ) (cid:3) Θ( t − t ′ ) , (81) α being a dimensionless parameter, β being a dimensional parameter, η denoting the steady-state viscosity, and J standing for the unrelaxed compliance, which is inverse to the unrelaxedrigidity: J ≡ J (0) = 1 /µ (0) = 1 /µ . We see that (81) is the Maxwell model amended with anextra term of a hereditary nature.A simple example illustrating how the model works is rendered by deformation under constantloading. In this case, the anelastic term dominates at short times, the strain thus being a convexfunction of t (the so-called primary or transient creep). As time goes on and the applied loadingis kept constant, the viscous term becomes larger, and the strain becomes almost linear in time– a phenomenon called the secondary creep.Remarkably, for all minerals (including ices) the values of α belong to the interval from 0 . . . β , may be rewritten as β = J τ − α A = µ − τ − α A , (82)the quantity τ A having dimensions of time. This quantity is the timescale associated with theAndrade creep, and it may be termed as the “Andrade time” or the “anelastic time”. It is clearfrom (82) that a short τ A makes the anelasticity more pronounced, while a long τ A makes theanelasticity weak. It is known from Castillo-Rogez et al. (2011) and Castillo-Rogez & Choukroun (2010) thatfor some minerals, within some frequency bands, the Andrade time gets very close to the Maxwelltime: τ A ≈ τ M = ⇒ β ≈ J τ − α M = J − α η − α = µ α − η − α , (83) For example, the principal tidal torque τ lmpq = τ acting on a secondary must vanish when the secondaryis crossing the synchronous orbit. Naturally, this happens because p becomes − χ = 0 . As long as we agree to integrate over t − t ′ ∈ [ 0 , ∞ ) , the terms β ( t − t ′ ) α and η − ( t − t ′ ) can do withoutthe Heaviside step-function Θ( t − t ′ ) . We remind though that the first term, J , does need this multiplier, sothat insertion of (81) into (43) renders the desired J δ ( t − t ′ ) under the integral, after the differentiation in (43)is performed. While the Andrade creep is likely to be caused by “unpinning” of jammed dislocations (Karato & Spetzler1990, Miguel et al 2002), it is not apparently clear if the Andrade time can be identified with the typical time ofunpinning of defects. τ M ≡ ηµ = η J . (84)On general grounds, though, one cannot expect the anelastic timescale τ A and the viscoelastictimescale τ M to coincide in all situations. This is especially so due to the fact that both thesetimes may possess some degree of frequency-dependence. Specifically, there exist indicationsthat in the Earth’s mantle the role of anelasticity (compared to viscoelasticity) undergoes adecrease when the frequencies become lower than 1 / yr – see the miscrophysical model suggestedin subsection 5.2.3 of Karato & Spetzler (1990). It should be remembered, though, that therelation between τ A and τ M may depend also upon the intensity of loading, i.e., upon thedamping mechanisms involved. The microphysical model considered in Ibid. was applicableto strong deformations, with anelastic dissipation being dominated by dislocations unpinning.Accordingly, the dominance of viscosity over anelasticity ( τ A ≪ τ M ) at low frequencies may beregarded proven for strong deformations only. At low stresses, when the grain-boundary diffusionmechanism is dominant, the values of τ A and τ M may remain comparable at low frequencies.The topic needs further research.In terms of the Andrade and Maxwell times, the compliance becomes: J ( t − t ′ ) = J (cid:20) (cid:18) t − t ′ τ A (cid:19) α + t − t ′ τ M (cid:21) Θ( t − t ′ ) . (85)In the frequency domain, compliance (85) will look:¯ J ( χ ) = J + β ( iχ ) − α Γ (1 + α ) − iηχ (86a)= J (cid:2) i χ τ A ) − α Γ (1 + α ) − i ( χ τ M ) − (cid:3) , (86b) χ being the frequency, and Γ denoting the Gamma function. The imaginary and real parts ofthe complex compliance are: I m [ ¯ J ( χ )] = − η χ − χ − α β sin (cid:16) α π (cid:17) Γ( α + 1) (87a)= − J ( χτ M ) − − J ( χτ A ) − α sin (cid:16) α π (cid:17) Γ( α + 1) (87b)and R e [ ¯ J ( χ )] = J + χ − α β cos (cid:16) α π (cid:17) Γ( α + 1) (88a)= J + J ( χτ A ) − α cos (cid:16) α π (cid:17) Γ( α + 1) , (88b)whence we obtain the following dependence of the phase lag upon the frequency:tan δ ( χ ) = − I m (cid:2) ¯ J ( χ ) (cid:3) R e (cid:2) ¯ J ( χ ) (cid:3) = ( η χ ) − + χ − α β sin (cid:16) α π (cid:17) Γ ( α + 1) µ − + χ − α β cos (cid:16) α π (cid:17) Γ ( α + 1) (89a)= z − ζ + z − α sin (cid:16) α π (cid:17) Γ ( α + 1)1 + z − α cos (cid:16) α π (cid:17) Γ ( α + 1) . (89b)22ere z is the dimensionless frequency defined as z ≡ χ τ A = χ τ M ζ , (90)while ζ is a dimensionless parameter of the Andrade model: ζ ≡ τ A τ M . (91) An lmpq term in the expansion for the tidal torque is proportional to the factor k l ( χ ) sin ǫ l ( χ ) = | ¯ k l ( χ lmpq ) | sin ǫ l ( χ lmpq ) . Hence the tidal response of a body is determined by the frequency-dependence of these factors.Combining (72) with (86), and keeping in mind that A l ≫ | ¯ k l ( χ ) | sin ǫ l ( χ ) . Referring the reader to Appendix E.2for details, we present the results, without the sign multiplier. • In the high-frequency band: | ¯ k l ( χ ) | sin ǫ l ( χ ) ≈
32 ( l − A l ( A l + 1) sin (cid:16) απ (cid:17) Γ( α + 1) ζ − α ( τ M χ ) − α , for χ ≫ τ − M . (92)For small bodies and small terrestrial planets (i.e., for A l ≫ χ HI = τ − M ζ α − α . For large terrestrial planets (i.e., for A l ≪ χ HI = τ − A = τ − M ζ − . At high frequencies, anelasticity dominates. So, dependent upon the microphysics of the mantle,the parameter ζ may be of order unity or slightly lower. We say slightly , because we expectboth anelasticity and viscosity to be present near the transitional zone. (A too low ζ wouldeliminate viscosity from the picture completely.) This said, we may assume that the boundary χ HI is comparable to τ − M for both small and large solid objects. This is why in (92) we set theinequality simply as χ ≫ τ − M . • In the intermediate-frequency band: | ¯ k l ( χ ) | sin ǫ l ( χ ) ≈
32 ( l − A l ( A l + 1) ( τ M χ ) − , for τ − M ≫ χ ≫ τ − M ( A l + 1) − . (93)While the consideration in the Appendix E.2 renders τ − M ζ α − α for the upper bound, here weapproximate it with τ − M in understanding that ζ does not differ from unity too much near thetransitional zone. Further advances of rheology may challenge this convenient simplification.23 In the low-frequency band: | ¯ k l ( χ ) | sin ǫ l ( χ ) ≈
32 ( l − A l τ M χ , for τ − M ( A l + 1) − ≫ χ . (94)Scaling laws (92) and (93) mimic, up to constant factors, the frequency-dependencies of | ¯ J ( χ ) | sin δ ( χ ) = − I m [ ¯ J ( χ )] at high and low frequencies, correspondingly, – this can be seenfrom (87).Expression (94) however shows a remarkable phenomenon inherent only in the tidal lagging,and not in the lagging in a sample of material: at frequencies below τ − M ( A l +1) − = µη ( A l +1) − ,the product | ¯ k l ( χ ) | sin ǫ l ( χ ) changes its behaviour and becomes linear in χ .While elsewhere the | ¯ k l ( χ ) | sin ǫ l ( χ ) factor increases with decreasing χ , it changes itsbehaviour drastically on close approach to the zero frequency. Having reached a finite maximumat about χ = τ − M ( A l + 1) − , the said factor begins to scale linearly in χ as χ approaches zero.This way, the factor | ¯ k l ( χ ) | sin ǫ l ( χ ) decreases continuously on close approach to a resonance,becomes nil together with the frequency at the point of resonance. So neither the tidal torquenor the tidal force explodes in resonances. In a somewhat heuristic manner, this change in thefrequency-dependence was pointed out, for l = 2 , in Section 9 of Efroimsky & Williams (2009). Figure 1 shows the absolute value, k ≡ | ¯ k ( χ ) | , as well as the real part, R e (cid:2) ¯ k ( χ ) (cid:3) = k cos ǫ ,and the negative imaginary part, − I m (cid:2) ¯ k ( χ ) (cid:3) = k sin ǫ , of the complex quadrupole Lovenumber. Each of the three quantities is represented by its decadic logarithm as a function ofthe decadic logarithm of the forcing frequency χ (given in Hz). The curves were obtained byinsertion of formulae (87 - 88) into (68). As an example, the case of − I m (cid:2) ¯ k ( χ ) (cid:3) is workedout in Appendix E.2, see formulae (252 - 88).Both in the high- and low-frequency limits, the negative imaginary part of ¯ k ( χ ) , given onFigure 1 by the red curve, approaches zero. Accordingly, over the low- and high-frequency bandsthe real part (the green line) virtually coincides with the absolute value (the blue line).While on the left and on the close right of the peak, dissipation is mainly due to viscosity,friction at higher frequencies is mainly due to anelasticity. This switch corresponds to the changeof the slope of the red curve at high frequencies (for our choice of parameters, at around 10 − Hz). This change of the slope is often called the elbow .Figure 1 was generated for A = 80 . τ M = 3 . × s. The value of A correspondsto the Moon modeled by a homogeneous sphere of rigidity µ = 0 . × Pa. Our choice of thevalue of τ M ≡ η/µ corresponds to a homogeneous Moon with the said value of rigidity and withviscosity set to be η = 3 × Pa s. The reason why we consider an example with such a lowvalue of η will be explained in subsection 5.7. Finally, it was assumed for simplicity that ζ = 1 ,i.e., that τ A = τ M . Although unphysical at low frequencies, this simplification only slightlychanges the shape of the “elbow” and exerts virtually no influence upon the maximum of the redcurve, provided the maximum is located well into the viscosity zone. As ever, we recall that in the expansion for the tidal torque factors (92 - 94) should appear inthe company of multipliers sgn ω . For example, the factor (94) describing dissipation near an lmpq resonance will enter the expansions as | ¯ k l ( χ lmpq ) | sin ǫ l ( χ lmpq ) sgn ω lmpq ≈
32 ( l − A l τ M χ lmpq sgn ω lmpq = 32 ( l − A l τ M ω lmpq . (95)24
11 −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0−3.5−3−2.5−2−1.5−1−0.500.51 lg χ l g ( − k s i n ε ) , l g ( k c o s ε ) , l g ( k ) lg(−k sin ε )lg(k cos ε )lg(k ) Figure 1: Tidal response of a homogeneous spherical Andrade body, set against the decadiclogarithm of the forcing frequency χ (in Hz). The blue curve renders the decadic logarithmof the absolute value of the quadrupole complex Love number, lg k = lg | ¯ k ( χ ) | . The greenand red curves depict the logarithms of the real and the negative imaginary parts of the Lovenumber: lg R e (cid:2) ¯ k ( χ ) (cid:3) = lg ( k cos ǫ ) and lg (cid:8) −I m (cid:2) ¯ k ( χ ) (cid:3)(cid:9) = lg ( − k sin ǫ ) , accordingly.The change in the slope of the red curve (the “elbow”), which takes place to the right of themaximum, corresponds to the switch from viscosity dominance at lower frequencies to anelasticitydominance at higher frequencies. The parameters A and τ M were given values appropriateto a homogeneous Moon with a low viscosity, as described in subsection 5.7. The plots weregenerated for an Andrade body with ζ = 1 at all frequencies. Setting the body Maxwell at lowerfrequencies will only slightly change the shape of the “elbow” and will have virtually no effecton the maximum. 25aturally, the lmpq term of the torque depends on ω lmpq and not just on χ lmpq = | ω lmpq | . Thisterm then goes continuously through zero, and changes its sign as the lmpq resonance is crossed(i.e., as ω lmpq goes through nil and changes its sign).Formula (95) tells us that an lmpq component of the tidal torque goes continuously throughzero as the satellite is traversing the commensurability which corresponds to vanishing of a tidalfrequency χ l mpq . This gets along with the physically evident fact that the principal (i.e., 2200 )term of the tidal torque should vanish as the secondary approaches the synchronous orbit.It is important that a lmpq term of the torque changes its sign and thus creates a chance forentrapment. As the value of an lmpq term of the torque is much lower than that of the principal,2200 term, we see that a perfectly spherical body will never get stuck in a resonance other than2200. (The latter is, of course, the 1 : 1 resonance, i.e., the one in which the principal termof the torque vanishes.) However, the presence of the triaxiality-generated torque is known tocontribute to the probabilities of entrapment into other resonances (provided the eccentricity isnot zero). Typically, in the literature they consider a superposition of the triaxiality-generatedtorque with the principal tidal term. We would point out that the “trap” shape of the lmpq term (95) makes this term relevant for the study of entrapment in the lmpq resonance. In somesituations, one has to take into account also the non-principal terms of the tidal torque. As we mentioned above, fitting of the lunar laser ranging (LLR) data to the power law has resultedin a very small negative exponent p = − .
19 (Williams et al. 2001). Since the measurementsof the lunar damping described in
Ibid. rendered information on the tidal and not seismicdissipation, those results can and should be compared to the scaling law (92 - 94). As the smallnegative exponent was devised from observations over periods of a month to a year, it is naturalto presume that the appropriate frequencies were close to or slightly below the frequency 1 τ M A at which the factor k sin ǫ has its peak:3 × s ≈ . τ M A = ηµ A = 57 η π G ( ρ R ) , (96)as on Figure 1. Hence, were the Moon a uniform viscoelastic body, its viscosity would be only η = 3 × Pa s . (97)For the actual Moon, the estimate means that the lower mantle contains a high percentage ofpartial melt, a fact which goes along well with the model suggested in Weber et al. (2011), andwhich was anticipated yet in Williams et al. (2001) and Williams & Boggs (2009), following anearlier work by Nakamura et al. (1974). Let vector ~r = ( r, λ, φ ) point from the centre of the primary toward a point-like secondaryof mass M sec . Associating the coordinate system with the primary, we reckon the latitude φ from the equator. Setting the coordinate system to corotate, we determine the longitude λ from a fixed meridian. The tidally induced component of the primary’s potential, U , can begenerated either by this secondary itself or by some other secondary of mass M ∗ sec located at ~r ∗ = ( r ∗ , λ ∗ , φ ∗ ) . In either situation, the tidally induced potential U generates a tidal forceand a tidal torque wherewith the secondary of mass M sec acts on the primary.26he scope of this paper is limited to low values of i . When the role of the primary is playedby a planet, the secondary being its satellite, i is the satellite’s inclination. When the role of theprimary is played by the satellite, the planet acting as its secondary, i acquires the meaning ofthe satellite’s obliquity. Similarly, when the planet is regarded as a primary and its host star istreated as its secondary, i is the obliquity of the planet. In all these cases, the smallness of i indicates that the tidal torque acting on the primary can be identified with its polar component,the one orthogonal to the equator of the primary. The other components of the torque will beneglected in this approximation.The polar component of the torque acting on the primary is the negative of the partialderivative of the tidal potential, with respect to the primary’s sidereal angle: T ( ~r ) = − M sec ∂U ( ~r ) ∂θ , (98) θ standing for the primary’s sidereal angle. This formula is convenient when the tidal potential U is expressed through the secondary’s orbital elements and the primary’s sidereal angle. Here and hereafter we are deliberately referring to a primary and a secondary in lieu of a planet and a satellite . The preference stems from our intention to extend the formalism tosetting where a moon is playing the role of a tidally-perturbed primary, the planet being itstide-producing secondary. Similarly, when addressing the rotation of Mercury, we interpret theSun as a secondary that is causing a tide on the effectively primary body, Mercury.
The potential produced at point ~R = ( R , λ , φ ) by a secondary body of mass M ∗ , located at ~r ∗ =( r ∗ , λ ∗ , φ ∗ ) with r ∗ ≥ R , is given by (1). When a tide-raising secondary located at ~r ∗ distortsthe shape of the primary, the potential generated by the primary at some exterior point ~r getschanged. In the linear approximation, its variation is given by (2). Insertion of (1) into (2) entails U ( ~r ) = − G M ∗ sec ∞ X l =2 k l R l + r l + r ∗ l + X m =0 ( l − m )!( l + m )! (2 − δ m ) P l m (sin φ ) P l m (sin φ ∗ ) cos m ( λ − λ ∗ ) . (99)A different expression for the tidal potential was offered by Kaula (1961, 1964), who developeda powerful technique that enabled him to switch from the spherical coordinates to the Keplerelements ( a ∗ , e ∗ , i ∗ , Ω ∗ , ω ∗ , M ∗ ) and ( a, e, i , Ω , ω, M ) of the secondaries located at ~r ∗ and Were the potential written down in the spherical coordinates associated with the primary’s equator andcorotating with the primary, the polar component of the tidal torque could be calculated with aid of the expression T ( ~r ) = M sec ∂U ( ~r ) ∂λ derived, for example, in Williams & Efroimsky (2012). That the expression agrees with (98) can be seen from theformula λ = − θ + Ω + ω + ν + O ( i ) = − θ + Ω + ω + M + 2 e sin M + O ( e ) + O ( i ) ,e , i , ω , Ω , ν and M being the eccentricity, inclination, argument of the pericentre, longitude of the node, trueanomaly, and mean anomaly of the tide-raising secondary. r . Application of this technique to (99) results in U ( ~r ) = − ∞ X l =2 k l (cid:18) Ra (cid:19) l +1 G M ∗ sec a ∗ (cid:18) Ra ∗ (cid:19) l l X m =0 ( l − m )!( l + m )! ( 2 (100) − δ m ) l X p =0 F l mp ( i ∗ ) ∞ X q = −∞ G l pq ( e ∗ ) l X h =0 F l mh ( i ) ∞ X j = −∞ G l hj ( e ) cos (cid:2)(cid:0) v ∗ l mpq − mθ ∗ (cid:1) − ( v l mhj − mθ ) (cid:3) , where v ∗ l mpq ≡ ( l − p ) ω ∗ + ( l − p + q ) M ∗ + m Ω ∗ , (101) v l mhj ≡ ( l − h ) ω + ( l − h + j ) M + m Ω , (102) θ = θ ∗ being the sidereal angle, G lpq ( e ) signifying the eccentricity functions, and F lmp ( i )denoting the inclination functions (Gooding & Wagner 2008).Being equivalent for a planet with an instant response of the shape, (99) and (100) disagreewhen dissipation-caused delays come into play. Kaula’s expression (100), as well as its truncated,Darwin’s version, is capable of accommodating separate phase lags for each mode: U ( ~r ) = − ∞ X l =2 k l (cid:18) Ra (cid:19) l +1 G M ∗ sec a ∗ (cid:18) Ra ∗ (cid:19) l l X m =0 ( l − m )!( l + m )! ( 2 − (103) δ m ) l X p =0 F l mp ( i ∗ ) ∞ X q = −∞ G l pq ( e ∗ ) l X h =0 F l mh ( i ) ∞ X j = −∞ G l hj ( e ) cos (cid:2)(cid:0) v ∗ l mpq − mθ ∗ (cid:1) − ( v l mhj − mθ ) − ǫ l mpq (cid:3) , where ǫ l mpq = h ( l − p ) ˙ ω ∗ + ( l − p + q ) ˙ M ∗ + m ( ˙Ω ∗ − ˙ θ ∗ ) i ∆ t lmpq = ω ∗ lmpq ∆ t lmpq = ± χ ∗ lmpq ∆ t lmpq (104)is the phase lag. The tidal mode ω ∗ lmpq introduced in (104) is ω ∗ l mpq ≡ ( l − p ) ˙ ω ∗ + ( l − p + q ) ˙ M ∗ + m ( ˙Ω ∗ − ˙ θ ∗ ) , (105)while the positively-defined quantity χ ∗ l mpq ≡ | ω ∗ l mpq | = | ( l − p ) ˙ ω ∗ + ( l − p + q ) ˙ M ∗ + m ( ˙Ω ∗ − ˙ θ ∗ ) | (106)is the actual physical l mpq frequency excited by the tide in the primary. The correspond-ing positively-defined time delay ∆ t lmpq = ∆ t l ( χ l mpq ) depends on this physical frequency, thefunctional forms of this dependence being different for different materials. Functions G lhj ( e ) coincide with the Hansen polynomials X ( − l − , ( l − p )( l − p + q ) ( e ) . In Appendix G, we provide atable of the G lhj ( e ) required for expansion of tides up to e , inclusively. While the treatment by Kaula (1964) entails the infinite Fourier series (100), the development by Darwin(1879, 1880) renders its partial sum with | l | , | q | , | j | ≤ . For a simple introduction into Darwin’s method seeFerraz-Mello et al. (2008). Be mindful that in
Ibid. the convention on the notations ~r and ~r ∗ is opposite to ours.
28n neglect of the apsidal and nodal precessions, and also of ˙ M , the above formulae become: ω l mpq = ( l − p + q ) n − m ˙ θ , (107) χ l mpq ≡ | ω l mpq | = | ( l − p + q ) n − m ˙ θ | , (108)and ǫ l mpq ≡ ω l mpq ∆ t l mpq = h ( l − p + q ) n − m ˙ θ i ∆ t l mpq (109a)= χ l mpq ∆ t l ( χ l mpq ) sgn h ( l − p + q ) n − m ˙ θ i , (109b)Formulae (100) and (103) constitute the pivotal result of Kaula’s theory of tides. Importantly,Kaula’s theory imposes no a priori constraint on the form of frequency-dependence of the lags. As explained in Williams & Efroimsky (2012), the empirical model by MacDonald (1964), called
MacDonald torque , tacitly sets an unphysical rheology of the satellite’s material. The rheology isgiven by (79) with α = − Direct differentiation of (103) with respect to − θ will result in the expression T = − ∞ X l =2 (cid:18) Ra (cid:19) l +1 G M ∗ sec M sec a ∗ (cid:18) Ra ∗ (cid:19) l l X m =0 ( l − m )!( l + m )! 2 m l X p =0 F l mp ( i ∗ ) ∞ X q = −∞ G l pq ( e ∗ ) l X h =0 F l mh ( i ) ∞ X j = −∞ G l hj ( e ) k l sin (cid:2) v ∗ l mpq − v l mhj − ǫ l mpq (cid:3) . (110)If the tidally-perturbed and tide-raising secondaries are the same body, then M sec = M ∗ sec , andall the elements coincide with their counterparts with an asterisk. Hence the differences v ∗ l mpq − v l mhj =( l − p + q ) M ∗ − ( l − h + j ) M + m (Ω ∗ − Ω) + l ( ω ∗ − ω ) − p ω ∗ + 2 h ω (111)get simplified to v ∗ l mpq − v l mhj = (2 h − p + q − j ) M ∗ + (2 h − p ) ω ∗ , (112)an expression containing both short-period contributions proportional to the mean anomaly, andlong-period contributions proportional to the argument of the pericentre. For justification of this operation, see Section 6 in Efroimsky & Williams (2009). .1.2 The secular, the purely-short-period, and the mixed-period parts of thetorque Now we see that the terms entering series (110) can be split into three groups:(1) The terms, in which indices ( p , q ) coincide with ( h , j ) , constitute a secular part ofthe tidal torque, because in such terms v l mhj cancel with v ∗ l mpq . This M - and ω -independentpart is furnished by T = ∞ X l =2 G M sec R l + 1 a l + 2 l X m =0 ( l − m )!( l + m )! m l X p =0 F l mp ( i ) ∞ X q = −∞ G l pq ( e ) k l sin ǫ l mpq . (113)(2) The terms with p = h and q = j constitute a purely short-period part of the torque: e T = − ∞ X l =2 G M sec R l +1 a l +2 l X m =0 ( l − m )!( l + m )! m l X p =0 F l mp ( i ) ∞ X q = −∞ ∞ X j = −∞ j = q G l pq ( e ) G l pj ( e ) k l sin [ ( q − j ) M − ǫ l mpq ] . (114)(3) The remaining terms, ones with p = h , make a mixed-period part comprised of both short-and long-period terms: T mixed = − ∞ X l =2 G M sec R l +1 a l +2 l X m =0 ( l − m )!( l + m )! m l X p =0 F l mp ( i ) l X h =0 h = p F l mh ( i ) ∞ X q = −∞ ∞ X j = −∞ G l hq ( e ) G l pj ( e ) k l sin [ (2 h − p + q − j ) M ∗ + (2 h − p ) ω ∗ − ǫ l mpq ] . (115) l = 2 and l = 3 terms in the O ( i ) approximation For l = 2 , index m will take the values 0 , , m = 0 terms enter thepotential, they add nothing to the torque, because differentiation of (103) with respect to − θ furnishes the m multiplier in (110). To examine the remaining terms, we should consider theinclination functions with subscripts ( l mp ) = (220) , (210) , (211) only: F ( i ) = 3 + O ( i ) , F ( i ) = 32 sin i + O ( i ) , F ( i ) = −
32 sin i + O ( i ) , (116)all the other F mp ( i ) being of order O ( i ) or higher. Thence for p = h (i.e., both in the secularand purely-short-period parts) it is sufficient, in the O ( i ) approximation, to keep only the termswith F ( i ) , ignoring those with F ( i ) and F ( i ) . We see that in the O ( i ) approximation • among the l = 2 terms, both in the secular and purely short-period parts,only the terms with ( l mp ) = (220) are relevant.In the case of p = h , i.e., in the mixed-period part, the terms of the leading order in inclinationare: F l mp ( i ) F l mh ( i ) = F ( i ) F ( i ) and F l mp ( i ) F l mh ( i ) = F ( i ) F ( i ) , which happento be equal to one another, and to be of order O ( i ) . This way, in the O ( i ) approximation,30 the mixed-period part of the l = 2 component may be omitted.The inclination functions F lmp = F , F , F , F , F , F , F , F , F , F are of order O ( i ) or higher. The terms containing the squares or cross-products of the thesefunctions may thus be dropped. Specifically, the smallness of the cross-terms tells us that • the mixed-period part of the l = 3 component may be omitted.What remains is the terms containing the squares of functions F ( i ) = −
32 + O ( i ) and F ( i ) = 15 + O ( i ) . (117)In other words, • among the l = 3 terms, both in the secular and purely short-period parts,only the terms with ( l mp ) = (311) and ( l mp ) = (330) are important.All in all, for l = 2 and l = 3 the mixed-period parts of the torque may be neglected, in the O ( i ) approximation. The surviving terms of the secular and the purely short-period parts willbe developed up to e , inclusively. As we just saw, both the secular and short-period parts of the torque may be approximated withthe following degree-2 and degree-3 components: T = T l =2 + T l =3 + O (cid:0) ǫ ( R/a ) (cid:1) = T ( lmp )=(220) + h T ( lmp )=(311) + T ( lmp )=(330) i + O ( ǫ i ) + O (cid:0) ǫ ( R/a ) (cid:1) , (118)and e T = e T l =2 + e T l =3 + O (cid:0) ǫ ( R/a ) (cid:1) = e T ( lmp )=(220) + h e T ( lmp )=(311) + e T ( lmp )=(330) i + O ( ǫ i ) + O (cid:0) ǫ ( R/a ) (cid:1) , (119)were the l = 2 and l = 3 inputs are of the order ( R/a ) and ( R/a ) , accordingly; while the l = 4 , , . . . inputs constitute O ( ǫ ( R/a ) ) .Expressions for T ( lmp )=(220) , T ( lmp )=(311) , and T ( lmp )=(310) are furnished by formulae (283),31285), and (287) in Appendix H. As an example, here we provide one of these components: T ( lmp )=(220) = 32 G M sec R a − (cid:20) e k sin | ǫ − | sgn (cid:16) − n − θ (cid:17) + (cid:18) e − e + 13768 e (cid:19) k sin | ǫ − | sgn (cid:16) n − θ (cid:17) + (cid:18) − e + 638 e − e (cid:19) k sin | ǫ | sgn (cid:16) n − ˙ θ (cid:17) + (cid:18) e − e + 21975256 e (cid:19) k sin | ǫ | sgn (cid:16) n − θ (cid:17) + (cid:18) e − e (cid:19) k sin | ǫ | sgn (cid:16) n − ˙ θ (cid:17) + 7140252304 e k sin | ǫ | sgn (cid:16) n − θ (cid:17) (cid:21) + O ( e ǫ ) + O ( i ǫ ) . (120)Here each term changes its sign on crossing the appropriate resonance. The change of the signtakes place smoothly, as the value of the term goes through zero – this can be seen from formula(95) and from the fact that the tidal mode ω lmpq vanishes in the lmpq resonance.Expressions for e T ( lmp )=(220) , e T ( lmp )=(311) , and e T ( lmp )=(330) are given by formulae (289 - 291)in Appendix I. Although the average of the short-period part of the torque vanishes, this partdoes contribute to dissipation. Oscillating torques contribute also to variations of the surface ofthe tidally-distorted primary, the latter fact being of importance in laser-ranging experiments.Whether the short-period torque may or may not influence also the process of entrapment isworth exploring numerically. The reason why this issue is raised is that the frequencies n ( q − j )of the components of the short-period torque are integers of n and thus are commensurate withthe spin rate ˙ θ near an A/B resonance, A and B being integer. It may be especially interestingto check the role of this torque when q − j = 1 and A/B = N is integer.The hypothetical role of the short-period torque in the entrapment and libration dynamicshas never been discussed so far, as the previous studies employed expressions for the tidal torque,which were obtained through averaging over the period of the secondary’s orbital motion. Consider a tidally-perturbed body caught in a A : B resonance with its tide-raising companion, A and B being integer. Then the spin rate of the body is˙ θ = AB n + ˙ ψ , (121)where the physical-libration angle is ψ = − ψ sin ( ω PL t ) , (122)32 PL being the physical-libration frequency. The oscillating tidal torque exerted on the body iscomprised of the modes ω lmpq = ( l − p + q ) n − m ˙ θ = (cid:18) l − p + q − AB m (cid:19) n − m ˙ ψ . (123)In those lmpq terms for which the combination l − p + q − AB m is not zero, the smallquantity − m ˙ ψ may be neglected. The remaining terms will pertain to the geometric libration.The phase lags will be given by the standard formula ǫ lmpq = ω lmpq ∆ t lmpq .In those lmpq terms for which the combination l − p + q − AB m vanishes, the physical-libration input − m ˙ ψ is the only one left. Accordingly, the multiplier sin h (cid:16) v ∗ lmpq − m θ ∗ (cid:17) − (cid:16) v lmpq − m θ (cid:17) i in the lmpq term of the tidal torque will reduce to sin [ − m ( ψ ∗ − ψ ) ] ≈− m ˙ ψ ∆ t = m ψ ω PL ∆ t cos ω PL t . Here the time lag ∆ t is the one corresponding to thephysical-libration frequency ω PL which may be very different from the usual tidal frequenciesfor nonsynchronous rotation – see Williams & Efroimsky (2012) for a comprehensive discussion. The afore-presented expressions for the secular and purely short-period parts of the tidal torquelook cumbersome when compared to the compact and elegant formulae employed in the literaturehitherto. It will therefore be important to explain why those simplifications are impractical.
Insofar as the quality factor is large and the lag is small (i.e., insofar as sin ǫ can be approximatedwith ǫ ), our expression (282a) assumes a simpler form: ( Q > T l =2 = 32 G M sec R a − k X q = − G q ( e ) ǫ q + O ( e ǫ ) + O ( i ǫ ) + O ( ǫ ) , (124)where the error O ( ǫ ) originates from sin ǫ = ǫ + O ( ǫ ) .The simplification conventionally used in the literature ignores the frequency-dependence ofthe Love number and attributes the overall frequency-dependence to the lag. It also ignores thedifference between the tidal lag ǫ and the lag in the material, δ . This way, the conventionalsimplification makes ǫ obey the scaling law (79b). At this point, most authors also set α = − α arbitrary. From the formula ∆ t lmpq = E − α χ − ( α +1) lmpq (125)derived by Efroimsky & Lainey (2007) in the said approach, we see that the time lags are relatedto the principal-frequency lag ∆ t via:∆ t lmpq = ∆ t χ χ lmpq ! α +1 . (126) The physical-libration input − m ˙ α may be neglected in the expression for ω lmpq even when the magnitudeof the physical libration is comparable to that of the geometric libration (as in the case of Phobos). Let 1sin ǫ = ( E χ ) α , where E is an empirical parameter of the dimensions of time, while ǫ is small enough,so sin ǫ ≈ ǫ . In combination with ǫ lmpq ≡ ω lmpq ∆ t lmpq and χ lmpq = | ω lmpq | , these formulae entail (125). θ ≫ n , the corresponding phase lags are: ǫ lmpq ≡ ∆ t lmpq ω lmpq = − ∆ t χ lmpq χ χ lmpq ! α +1 = − ǫ χ χ lmpq ! α , (127)which helps us to cast the secular part of the torque into the following convenient form: Q > T l =2 = Z (cid:20) − ˙ θ (cid:18) e + 1054 e + O ( e ) (cid:19) + n (cid:18) (cid:18) − α (cid:19) e + (cid:18) − α (cid:19) e + O ( e ) (cid:19)(cid:21) + O ( i /Q ) + O ( Q − ) + O ( αe Q − n/ ˙ θ ) (128a) ≈ Z (cid:20) − ˙ θ (cid:18) e (cid:19) + n (cid:18) (cid:18) − α (cid:19) e (cid:19) (cid:21) , (128b)where the overall factor reads as: Z = 3 G M sec k ∆ t R R a = 3 n M sec k ∆ t ( M prim + M sec ) R a = 3 n M sec k Q ( M prim + M sec ) R a nχ , (129) M prim and M sec being the primary’s and secondary’s masses. Dividing (129) by the primary’sprincipal moment of inertia ξ M primary R , we obtain the contribution that this component ofthe torque brings into the angular deceleration rate ¨ θ :¨ θ = K (cid:26) − (cid:5) θ (cid:20) e + 1054 e + O ( e ) (cid:21) + n (cid:20) (cid:18) − α (cid:19) e + (cid:18) − α (cid:19) e + O ( e ) (cid:21) (cid:27) + O ( i /Q ) + O ( Q − ) + O ( αe Q − n/ ˙ θ ) (130a) ≈ K (cid:20) − (cid:5) θ (cid:18) e (cid:19) + n (cid:18) (cid:18) − α (cid:19) e (cid:19) (cid:21) , (130b)the factor K being given by K ≡ Z ξ M prim R = 3 n M sec k ∆ t ξ M prim ( M prim + M sec ) R a = 3 n M sec k ξ Q M prim ( M prim + M sec ) R a nχ , (131)where ξ is a multiplier emerging in the expression ξ M primary R for the primary’s principalmoment of inertia ( ξ = 2 / For ˙ θ ≫ n , all the modes ω q are negative, so ω q = − χ q . Then, keeping in mind that n/ ˙ θ ≪ q = 1 , like − ∆ t χ χ χ ! α = − ∆ t | n − ˙ θ | | n − ˙ θ || − θ + 3 n | ! α = − ∆ t θ − n ) " α n (cid:5) θ + O ( ( n/ ˙ θ ) ) , and similarly for other q = 0, and then plug the results into (124). This renders us (128). To arrive at the right-hand side of (129), we recalled that χ lmpq ∆ t lmpq = | ǫ lmpq | and that Q − lmpq = | ǫ lmpq | + O ( ǫ ) = | ǫ lmpq | + O ( Q − ) , according to formula (12).
34n the special case of α = − t , Q , and χ instead of ∆ t , Q , and χ standing in formulae (44 - 47) from Williams& Efroimsky (2012). Formula (130b) tells us that the secular part of the tidal torque vanishesfor (cid:5) θ − n = − n e α , (132)which coincides, for α = − occasional or, possibly better to say, exceptional .Formulae (128 - 129) were obtained by insertion of the expressions for the eccentricity func-tions and the phase lags into (124), and by assuming that n ≪ | ˙ θ | . The latter caveat is a crucialelement, not to be overlooked by the users of formulae (128 - 129) and of their corollary (130 -131) for the tidal deceleration rate.The case of α = − without assumingthat n ≪ | (cid:5) θ | . However for α > − n ≪ | (cid:5) θ | remains mandatory, so formulae(128 - 131) become inapplicable when (cid:5) θ reduces to values of about several n .Although formulae (128a) and (130a) contain an absolute error O ( αe Q − n/ ˙ θ ) , this doesnot mean that for (cid:5) θ comparable to n the absolute error becomes O ( αe Q − ) and the relativeone becomes O ( αe ) . In reality, for (cid:5) θ comparable to n , the entire approximation falls apart ,because formulae (126 - 127) were derived from expression (125), which is valid for Q ≫ α = − Thence, in all situations, except for the unrealistic rheology α = − n ≪ | ˙ θ | , but becomes misleading on approach to the physically-interesting resonances. The form in which our approximation (128 - 131) is cast may appear awkward. The formula forthe despinning rate ¨ θ is written as a function of ˙ θ and n , multiplied by the overall factor K .This form would be reasonable, were K a constant. That this is not the case can be seen fromthe presence of the multiplier nχ = n | (cid:5) θ − n | on the right-hand side of (131).Still, when written in this form, our result is easy to juxtapose with an appropriate formulafrom Correia & Laskar (2004, 2009). There, the expression for the despinning rate looks similarto ours, up to an important detail: the overall factor is a constant, because it lacks the saidmultiplier nχ . The multiplier was lost in those two papers, because the quality factor wasintroduced there as 1 / ( n ∆ t ) , see the line after formula (9) in Correia & Laskar (2009). Inreality, the quality factor Q should, of course, be a function of the tidal frequency χ , because Q serves the purpose of describing the tidal damping at this frequency. Had the quality factorbeen taken as 1 / ( χ ∆ t ) , it would render the corrected MacDonald model ( α = − the model is mathematically convenient,because it enables one to write down the secular part of the torque as one expression, avoiding For example, in the case of α > − t lmpq and ǫ lmpq oncrossing the commensurability, i.e., when ω lmpq goes through zero. To be exact, the model is unphysical everywhere except in the closest vicinity of the resonance – see formulae(92 - 94). χ = n . Still,the difference between χ and n in the vicinity of the resonance may alter the probability ofentrapment of Mercury into this rotation mode. The difference between χ and n becomes evenmore considerable near the other resonances of interest. So the probabilities of entrapment intothose resonances must be recalculated.
10 Conclusions
The goal of this paper was to lay the ground for a reliable model of tidal entrapment intospin-orbital resonances. To this end, we approached the tidal theory from the first principlesof solid-state mechanics. Starting from the expression for the material’s compliance in the timedomain, we derived the frequency-dependence of the Fourier components of the tidal torque. Theother torque, one caused by the triaxiality of the rotator, is not a part of this study and will beaddressed elsewhere. • We base our work on the Andrade rheological model, providing arguments in favour of itsapplicability to the Earth’s mantle, and therefore, very likely, to other terrestrial planetsand moons. The model is also likely to apply to the icy moons (Castillo-Rogez et al.2011). We have reformulated the model in terms of a characteristic anelastic timescale τ A (the Andrade time). The ratio of the Andrade time to the viscoelastic Maxwell time, ζ = τ A /τ M , serves as a dimensionless free parameter of the rheological model.The parameters τ A , τ M , ζ cannot be regarded constant, though their values may be changingvery slowly over vast bands of frequency. The shapes of these frequency-dependencies maydepend on the dominating dissipation mechanisms and, thereby, on the magnitude of theload, as different damping mechanisms get activated under different loads.The main question here is whether, in the low-frequency limit, anelasticity becomes muchweaker than viscosity. (That would imply an increase of τ A and ζ as the tidal frequency χ goes down.) The study of ices under weak loads, with friction caused mainly by latticediffusion (Castillo-Rogez et al. 2011, Castillo-Rogez & Choukroun 2010) has not shownsuch a decline of anelasticity. However, Karato & Spetzler (1990) point out that it shouldbe happening in the Earth’s mantle, where the loads are much higher and damping is causedmainly by unpinning of dislocations. According to Ibid. , in the Earth, the decrease of therole of anelasticity happens abruptly as the frequency falls below the threshold χ ∼ − .We then may expect a similar switch in the other terrestrial planets and the Moon, thoughthere the threshold may be different as it is sensitive to the temperature of the mantle.The question, though, remains if this statement is valid also for the small bodies, in whichthe tidal stresses are weaker and dissipation is dominated by lattice diffusion. • By direct calculation, we have derived the frequency dependencies of the factors k l sin ǫ l emerging in the tidal theory. Naturally, the obtained dependencies of these factors uponthe tidal frequency χ lmpq (or, to put it more exactly, upon the tidal mode ω lmpq ) mimicthe frequency-dependence of the imaginary part of the complex compliance. They scale as ∼ χ − α with 0 < α < ∼ χ − at lower frequencies.However in the zero-frequency limit the factors k l sin ǫ l demonstrate a behavior inherentin the tidal lagging and absent in the case of lagging in a sample: in a close vicinity of thezero frequency, these factors (and the appropriate components of the tidal torque) becomelinear in the frequency. This way, k l sin ǫ l first reaches a finite maximum, then decreasescontinuously to nil as the frequency approaches to zero, and then changes its sign. So36he resonances are crossed continuously, with neither the tidal torque nor the tidal forcediverging there. For example, the leading term of the torque vanishes at the synchronousorbit.This continuous traversing of resonances was pointed out in a heuristic manner by Efroimsky& Williams (2009). Now we have derived this result directly from the expression for thecompliance of the material of the rotating body. Our derivation, however, has a problemin it: the frequency, below which the factors k l sin ǫ l and the appropriate componentsof the torque change their frequency-dependence to linear, is implausibly low (lower than10 − Hz, if we take our formulae literally). The reason for this mishap is that in ourformulae we kept using the known value of the Maxwell time τ M all the way down to thezero frequency. Possible changes of the viscosity and, accordingly, of the Maxwell time inthe zero-frequency limit may broaden this region of linear dependence. • We have offered an explanation of the unexpected frequency-dependence of dissipationin the Moon, discovered by LLR. The main point of our explanation is that the LLRmeasures the tidal dissipation whose frequency-dependence is different from that of the seismic dissipation. Specifically, the “wrong” sign of the exponent in the power dissipationlaw may indicate that the frequencies at which tidal friction was observed were below thefrequency at which the lunar k sin ǫ has its peak. Taken the relatively high frequenciesof observation (corresponding to periods of order month to year), this explanation can beaccepted only if the lunar mantle has a low mean viscosity. This may be the case, takenthe presumably high concentration of the partial melt in the low mantle. • We have developed a detailed formalism for the tidal torque, and have singled out itsoscillating component.The studies of entrapment into spin-orbit resonances, performed in the past, took intoaccount neither the afore-mentioned complicated frequency-dependence of the torque inthe vicinity of a resonance, nor the oscillating part of the torque. We have written downa concise and practical expression for the oscillating part, and have raised the questionwhether it may play a role in the entrapment and libration dynamics.
11 Acknowledgments
This paper stems largely from my prior research carried out in collaboration with James G.Williams and discussed on numerous occasions with Sylvio Ferraz Mello, to both of whom I amthankful profoundly. I am grateful to Benoˆıt Noyelles, Julie Castillo-Rogez, Veronique Dehant,Shun-ichiro Karato, Val´ery Lainey, Valeri Makarov, Francis Nimmo, Stan Peale, and Tim VanHoolst for numerous enlightening exchanges and consultations on the theory of tides.I gladly acknowledge the help and inspiration which I obtained from reading the unpublishedpreprint by the late Vladimir Churkin (1998). In Appendix C, I present several key results fromhis preprint.I sincerely appreciate the support from my colleagues at the US Naval Observatory, especiallyfrom John Bangert. 37 ppendix.A Continuum mechanics.A celestial-mechanician’s survival kit
Appendix A offers an extremely short introduction into continuum mechanics. The standardmaterial, which normally occupies large chapters in books, is compressed into several pages.Subsection A.1 presents the necessary terms and definitions. Subsections A.2 explains thebasic concepts employed in the theory of stationary deformation, while subsection A.3 explainsextension of these methods to creep. These subsections also demonstrate the great benefits stem-ming from the isotropy and incompressibility assumptions. Subsection A.4 introduces viscosity,while subsection A.5 offers an example of how elasticity, viscosity get combined with hereditaryreaction, into one expression. Subsection A.6 renders several simple examples.
A.1 Glossary
We start out with a brief guide elucidating the main terms employed in continuum mechanics. • Rheology is the science of deformation and flow. • Elasticity:
This is the most trivial reaction – instantaneous, linear, and fully reversibleafter stressing is turned off. • Anelasticity:
While still linear, this kind of deformation is not necessarily instantaneous,and can demonstrate “memory”, both under loading and when the load is removed. Im-portantly, the term anelasticity always implies reversibility: though with delay, the originalshape is restored. Thus an anelastic material can assume two equilibrium states: one is theunstressed state, the other being the long-term relaxed state. Anelasticity is characterisedby the difference in strain between the two states. It is also characterised by a relaxationtime between these states, and by its inverse – the frequency at which relaxation is most ef-fective. The Hohenemser-Prager model, also called SAS (Standard Anelastic Solid), rendersan example of anelastic behaviour.Anelasticity is an example of but not synonym to hereditary reaction . The latter includesalso those kinds of delayed deformation, which are irreversible. • Inelasticity:
This term embraces all kinds of irreversible deformation, i.e., deformationwhich stays, fully or partially, after the load is removed. • Unelasticity ( = Nonelasticity):
These terms are very broad, in that they embrace any be-haviour which is not elastic. In the literature, these terms are employed both for recoverable(anelastic) and unrecoverable (inelastic) deformations. • Plasticity:
Some authors simply state that plastic deformation is deformation which isirreversible – a very broad and therefore useless definition which makes plasticity soundlike a synonym to inelasticity.More precisely, plastic is a stepwise behaviour: no strain is observed until the stress σ reaches a threshold value σ Y (called yield strength), while a steady flow begins as thestress reaches the said threshold. Plasticity can be either perfect (when deformation isgoing on without any increase in load) or with hardening (when increasingly higher stressesare needed to sustain the flow). It is always irreversible.38n real life, plasticity shows itself in combination with elasticity or/and viscosity. Modelsdescribing these types of behaviour are called elastoplastic, viscoplastic, and viscoelasto-plastic. They are all inelastic, in that they describe unrecoverable changes of shape. • Viscosity:
Another example of inelastic, i.e., irreversible behaviour. A viscous stress isproportional to the time derivative of the viscous strain. • Viscoelasticity:
The term is customarily applied to all situations where both viscous andelastic (but not plastic) reactions are observed. One may then expect that the equationsinterrelating the viscoelastic stress to the strain would contain only the viscosity coefficientsand the elastic moduli. However this is not necessarily true, as some other empiricalconstants may show up. For example, the Andrade model (81) contains an elastic term,a viscous term, and an extra term responsible for a hereditary reaction (the “memory”).Despite the presence of that extra term, the Andrade creep is still regarded viscoelastic. Soit should be understood that viscoelasticity is, generally, more than just viscosity combinedwith elasticity. One might christen such deformations “viscoelastohereditary”, but such aterm would sound awkward. On many occasions, complex viscoelastic models can be illustrated with an infinite set ofviscous and elastic elements. These serve to interpret the hereditary terms as manifestationsof viscosity and elasticity only. While illustrative, these schemes with dashpots and springshave their limitations and should not be taken too literally. In some (not all) situations,the hereditary terms may be interpreted as time-dependent additions to the steady-stateviscosity coefficient, the Andrade model being an example of such situation. • Viscoplasticity:
These are all models wherein both viscosity and plasticity are presentin some combination. In these situations, higher stresses have to be applied to increasethe deformation rate. Just as in the case of viscoelasticity, viscoplastic models may, inprinciple, incorporate extra terms standing for hereditary reaction. • Elastoviscoplasticity ( = Viscoelastoplasticity):
The same as above, though with elasticitypresent. • Hereditary reaction:
While the term is self-explanatory, it would be good to limit its use toeffects other than viscosity. In expression (46) for the stress through strain, the distinctionbetween the viscous and hereditary reactions is clear: while the viscous part of the stress isrendered instantaneously by the delta-function term of the kernel, the hereditary reactionis obtained through integration. In expression (40) for the strain through stress, though,the viscous part shows up, under the integral, in sum with the other delayed terms – see,for example, the Andrade model (81). This is one reason for which we shall use the term hereditary reaction in application to delayed behaviour different from the pure viscosity.Another reason is that viscous flow is always irreversible, while a hereditary reaction maybe either irreversible (inelastic) or reversible (anelastic). • Creep:
Widely accepted is the convention that this term signifies a slow-rate deformationunder loads below the yield strength σ Y .Numerous authors, though, use the oxymoron plastic creep , thereby extending the applica-bility realm of the word creep to any slow deformation.Here we shall understand creep in the former sense, i.e., with no plasticity involved. Sometimes the term elastohereditary is used, but not viscoelastohereditary or elastoviscohereditary .
39t would be important to distinguish between viscoelastic deformations, on the one hand, andviscoplastic (or, properly speaking, viscoelastoplastic) deformations on the other hand. Plasticityshows itself at larger stresses and is, typically, nonlinear. It comes into play when the linearityassertion fails. For most minerals, this happens when the strain approaches the threshold of10 − . Although it is possible that this threshold is transcended in some satellites (for example,in the deeper layers of the Moon), we do not address plasticity in this paper. A.2 Stationary linear deformation of isotropic incompressible media
In the linear approximation, the tensor of elastic stress, ( e ) S , is proportional to the differences indisplacement of the neighbouring elements of the medium. These differences are components ofthe tensor gradient ∇ ⊗ u , where u is the displacement vector.The tensor gradient can be decomposed, in an invariant way, into its symmetric and antisym-metric parts: ∇ ⊗ u = 12 h ( ∇ ⊗ u ) + ( ∇ ⊗ u ) T i + 12 h ( ∇ ⊗ u ) − ( ∇ ⊗ u ) T i . (133)The decomposition being invariant, the two parts should contribute into the stress independently,at least in the linear approximation. However, as well known (Landau & Lifshitz 1986), theantisymmetric part of (133) describes the displacement of the medium as a whole and thusbrings nothing into the stress. This is why the linear dependence is normally written as ( e ) S = B U , (134)where B is a four-dimensional tensor having 3 = 81 components, while the strain tensor U ≡ h ( ∇ ⊗ u ) + ( ∇ ⊗ u ) T i (135)is the symmetric part of the tensor gradient. Its components are related to the displacementvector u through u αβ ≡ (cid:18) ∂u α ∂x β + ∂u β ∂x α (cid:19) .Although the matrix B is comprised of 81 empirical constants, in isotropic materials thedescription can be reduced to two constants only. To see this, recall that the expansion ofthe strain into a part with trace and a traceless part, U = 13 I Sp U + (cid:16) U − I Sp U (cid:17) , isinvariant. Here the trace of U is denoted with Sp U ≡ u αα , summation over repeated indicesbeing implied. The notation I stands for the unity matrix consisting of elements δ γν .In an isotropic medium, the elastic stress must be invariantly expandable into parts propor-tional to the afore-mentioned parts of the strain. The first part of the stress is proportional, withan empirical coefficient 3 K , to the trace part 13 I Sp U of the strain. The second part of the stresswill be proportional, with an empirical coefficient 2 µ , to the traceless part (cid:16) U − I Sp U (cid:17) ofthe strain: ( e ) S = K I Sp U + 2 µ (cid:18) U − I Sp U (cid:19) = − p I + 2 µ (cid:18) U − I Sp U (cid:19) (136a)or, in Cartesian coordinates: σ γν = K δ γν u αα + 2 µ (cid:18) u γν − δ γν u αα (cid:19) = − p δ γν + 2 µ (cid:18) u γν − δ γν u αα (cid:19) , (136b)40here p ≡ −
13 Sp S = − K Sp U (137)is the hydrostatic pressure. Thus the elastic stress gets decomposed, in an invariant way, as: ( e ) S = ( e ) S volumetric + ( e ) S deviatoric , (138)where the volumetric elastic stress is given by ( e ) S volumetric ≡ K I Sp U = I
13 Sp S = − p I , (139)while the deviatoric elastic stress is: ( e ) S deviatoric ≡ µ (cid:18) U − I Sp U (cid:19) . (140)Inverse to (136a - 136b) are the following expressions for the strain tensor: U = 19 K I Sp ( e ) S + 12 µ (cid:18) ( e ) S − I Sp ( e ) S (cid:19) (141a)and u γν = 19 K δ γν ( e ) σ αα + 12 µ (cid:18) ( e ) σ γν − δ γν ( e ) σ αα (cid:19) , (141b)where the term with trace, 19 K I Sp ( e ) S , is called the volumetric strain , while the tracelessterm, 12 µ (cid:18) ( e ) S − I Sp ( e ) S (cid:19) , is named the deviatoric strain . The quantity K is called the bulkmodulus , while µ is called the shear modulus .Expressions (136) trivially entail the following interrelation between traces:Sp ( e ) S = 3 K Sp U or, in terms of components: ( e ) σ αα = 3 K u αα . (142)As demonstrated in many books (e.g., in Landau & Lifshitz 1986), the trace of the strain isequal to the relative variation of the volume, experienced by the material subject to deformation: u αα = ∇ · u = dV ′ − dVdV , where u is the displacement vector. In the no-compressibilityapproximation, the trace of the strain and, according to (142), that of the stress become zero.Then, in the said approximation, the hydrostatic pressure (137) and the volumetric elastic stress(139) become nil, and all we are left with is the deviatoric elastic stress (and, accordingly, thedeviatoric part of the strain). Formulae (136) and (141) get simplified to ( e ) S = 2 µ U , which is the same as ( e ) σ γν = 2 µ u γν , (143)and to 2 U = J ( e ) S , which is the same as 2 u γν = J ( e ) σ γν , (144)the quantity J ≡ /µ being called the compliance of the material.41 .3 Evolving linear deformation of isotropic incompressible isotropicmedia. Hereditary reaction Equations (134 - 144) were written for static deformation, so each of these equations can beassumed to connect the strain and the elastic stress taken at the same instant of time (for astatic deformation their values stay constant anyway).Extension of this machinery is needed when one wants to describe evolving deformation ofmaterials with “memory”. Thence the four-dimensional tensor B becomes a linear operator ˜ B acting on the strain tensor function as a whole. To render the value of the stress at time t , theoperator will “consume” as arguments all the values of strain over the interval t ′ ∈ ( − ∞ , t ] : ( h ) S ( t ) = ˜ B ( t ) U . (145)Thus ˜ B will be an integral operator, with integration going from t ′ = − ∞ through t ′ = t .In the static case, the linearity guaranteed elasticity, i.e., the ability of the body to regainits shape after the loading is turned off: no stress yields no strain. In a more general situationof materials with “memory”, this ability is no longer retained, as the material may demonstrate creep . This is why, in this section, the stress is called hereditary and is denoted with ( h ) S .Just as in the stationary case, we wish the properties of the medium to remain isotropic.As the decomposition of the strain into the trace and traceless parts remains invariant at eachmoment of time, these two parts will, separately, generate the trace and traceless parts of thehereditary stress in an isotropic medium. This means that, in such media, the four-dimensionaltensor operator ˜ B gets reduced to two scalar linear operators ˜ K and ˜ µ : ( h ) S = ( h ) S volumetric + ( h ) S deviatoric = ˜ K I Sp U + 2 ˜ µ (cid:18) U − I Sp U (cid:19) , (146)where both ˜ K and ˜ µ are integral operators acting on the tensor function u γν ( t ′ ) as a whole,i.e., with integration going from t ′ = − ∞ through t ′ = t .If we also assume that, under evolving load, the medium stays incompressible, the trace of thestrain, u αα , will stay zero. An operator generalisation of (142) now reads: σ αα ( t ) = 3 ˜ K ( t ) u αα .Under a reasonable assumption of σ αα being nil in the distant past, this integral operator renders σ αα = 0 at all times. This way, in a medium that is both isotropic and incompressible, we have: U = U deviatoric and, accordingly: ( h ) S = ( h ) S deviatoric . (147)Then the time-dependent analogues to formulae (143) and (144) will be: ( h ) S ( t ) = 2 ˜ µ ( t ) U (148)and 2 U ( t ) = ˆ J ( t ) ( h ) S , (149)where the compliance ˆ J , too, has been promoted to operatorship and crowned with a caret.Formula (148) tells us that in a medium, which is both isotropic and incompressible, relationbetween the stress and strain tensors can be described with one scalar integral operator ˜ µ only,the complementary operator ˆ J being its inverse. (Here the adjective “scalar” does not implymultiplication with a scalar. It means that the operator preserves its functional form under achange of coordinates.)Below we shall bring into the picture also the viscous component of the stress, a compo-nent related to the strain through a four-dimensional tensor whose 3 = 81 components are42ifferential operators. In that case too, the isotropy of the medium will enable us to reduce the81-component tensor operator to two differential operators transforming as scalars. Besides, theincompressibility of the medium makes the viscous stress traceless. Thus it will turn out that, inan isotropic and incompressible medium, the viscous component of the stress can be describedby only one scalar differential operator – much like the elastic and hereditary parts of the stress.(Once again, “scalar” means: indifferent to coordinate transformations.)Eventually, the elastic, hereditary, and viscous deformations will be united under the auspicesof a general viscoelastic formalism. In an isotropic medium, this combined formalism will bereduced to two integro-differential operators only. In a medium which is both isotropic andincompressible, the formalism will be reduced to only one scalar integro-differential operator. A.4 The viscous stress
While the elastic stress ( e ) S is linear in the strain, the viscous stress ( v ) S is linear in the firstderivatives of the components of the velocity with respect to the coordinates: ( v ) S = A ( ∇ ⊗ v ) (150)where A is the so-called viscosity tensor, ∇ ⊗ v is the tensor gradient of the velocity. Thevelocity of a fluid parcel relative to its average position is connected to the displacement vector u through v = d u /dt .The tensor gradient of the velocity can be expanded, in an invariant way, into its antisym-metric and symmetric parts: ∇ ⊗ v = Ω + E , (151)where the antisymmetric part is furnished by the vorticity tensor Ω ≡ h ( ∇ ⊗ v ) − ( ∇ ⊗ v ) T i , (152)while the symmetric part is given by the rate-of-shear tensor E ≡ h ( ∇ ⊗ v ) + ( ∇ ⊗ v ) T i . (153)The latter is obviously related to the strain tensor through E = ∂∂t U . (154)It can be demonstrated (e.g., Landau & Lifshitz 1987) that the antisymmetric vorticity tensordescribes the rotation of the medium as a whole and therefore contributes nothing to thestress. For this reason, the viscous stress can be written as ( v ) S = A E = A ∂∂t U . (155)The matrix A is four-dimensional and contains 3 = 81 components. Just as the matrix B emerging in expression (134) for the elastic stress, the matrix A can be reduced, in an isotropic This is why this tensor’s components coincide with those of the angular velocity ~ω of the body. For example,Ω = 12 (cid:16) ∂v ∂x − ∂v ∂x (cid:17) coincides with ω . Since expansion (151) of the tensor gradient into the vorticity and rate-of-shear tensors is invariant, then sois the conclusion about the irrelevance of the vorticity tensor for the stress picture. E = 13 I ∇ · v + (cid:18) E − I ∇ · v (cid:19) . (156)where the rate-of-expansion tensor I ∇ · v (157)is diagonal and has a trace, while the combination E − I ∇ · v = 12 h ( ∇ ⊗ v ) + ( ∇ ⊗ v ) T i − I ∇ · v (158)is symmetric and traceless. These two parts contribute linearly proportional inputs into thestress. The first input is proportional, with an empirical coefficient 3 ζ , to the rate-of-expansionterm, while the second input into the stress is proportional, with an empirical coefficient 2 η ,to the symmetric traceless combination: ( v ) S = 3 ζ I ∇ · v + 2 η (cid:18) E − I ∇ · v (cid:19) = ζ ∂∂t ( I Sp U ) + 2 η ∂∂t (cid:18) U − I Sp U (cid:19) . (159)Here we recalled that ∇ · v = ∂∂t ∇ · u = ∂∂t u αα = ∂∂t Sp U . Since Sp U is equal to thevolume variation dV ′ − dVdV experienced by the material, we see that the first term in (159) isvolumetric, the second being deviatoric.The quantity η is called the first viscosity or the shear viscosity . The quantity ζ is namedthe second viscosity or the bulk viscosity . A.5 An example of approach to viscoelastic behaviour
In this subsection, we shall consider one possible approach to description of viscoelastic regimes.As we mentioned in subsection A.1, the term viscoelasticity covers not only combinations of elas-ticity and viscosity, but can also include other forms of delayed reaction. So the term viscoelastic is customarily used as a substitution for too long a term viscoelastohereditary .One possible approach would be to assume that the elastic, hereditary, and viscous stressessimply sum up, and that each of them is related the the same strain U : an ( total ) S = ( e ) S + ( h ) S + ( v ) S = (cid:18) B + ˜ B + A ∂∂t (cid:19) U , (160a)or simply ( total ) S = ˆ B U , where ˆ B ≡ B + ˜ B + A ∂∂t , (160b)where the three operators – the integral operator ˜ B , the differential operator A ∂∂t , and theoperator of multiplication by matrix B – comprise an integro-differential operator ˆ B .In an isotropic medium, each of the three matrices, ˜ B , A ∂∂t , and B , includes two terms only.This happens because in such a medium each of the three parts of the stress gets decomposedinvariantly into its deviatoric and volumetric components: The elastic stress becomes: ( e ) S = ( e ) S volumetric + ( e ) S deviatoric = 3 K (cid:18) I Sp U (cid:19) + 2 µ (cid:18) U − I Sp U (cid:19) , (161)44ith K and µ being the bulk elastic modulus and the shear elastic modulus , correspondingly, I standing for the unity matrix, and Sp denoting the trace of a matrix: Sp U ≡ P i U ii .The hereditary stress becomes: ( h ) S = ( h ) S volumetric + ( h ) S deviatoric = 3 ˜ K (cid:18) I Sp U (cid:19) + 2 ˜ µ (cid:18) U − I Sp U (cid:19) , (162)where ˜ K and ˜ µ are the bulk-modulus operator and the shear-modulus operator, accordingly.The viscous stress acquires the form: ( v ) S = ( v ) S volumetric + ( v ) S deviatoric = 3 ζ ∂∂t (cid:18) I Sp U (cid:19) + 2 η ∂∂t (cid:18) U − I Sp U (cid:19) , (163)the quantities ζ and η being termed as the bulk viscosity and the shear viscosity , correspondinglyThe term 13 I Sp U is called the volumetric part of the strain, while U − I Sp U is calledthe deviatoric part. Accordingly, in expressions (161 - 163) for the stresses, the pure-trace termsare called volumetric , the other term being named deviatoric .The total stress, too, can now be split into the total volumetric and the total deviatoric parts: ( total ) S = ( e ) S z }| {(cid:18) ( e ) S volumetric + ( e ) S deviatoric (cid:19) + ( v ) S z }| {(cid:18) ( v ) S volumetric + ( v ) S deviatoric (cid:19) + ( h ) S z }| {(cid:18) ( h ) S volumetric + ( h ) S deviatoric (cid:19) = S volumetric z }| {(cid:18) ( e ) S volumetric + ( v ) S volumetric + ( h ) S volumetric (cid:19) + S deviatoric z }| {(cid:18) ( e ) S deviatoric + ( v ) S deviatoric + ( h ) S deviatoric (cid:19) = (cid:18) K + 3 ˜ K + 3 ζ ∂∂t (cid:19) (cid:18) I Sp U (cid:19) + (cid:18) µ + 2 ˜ µ + 2 η ∂∂t (cid:19) (cid:18) U − I Sp U (cid:19) (164a)= 3 ˆ K (cid:18) I Sp U (cid:19) + 2 ˆ µ (cid:18) U − I Sp U (cid:19) , (164b)where ˆ K ≡ K + ˜ K + ζ ∂∂t and ˆ µ ≡ µ + ˜ µ + η ∂∂t . (165)As expected, a total linear deformation of an isotropic material can be described with two integro-differential operators, one acting on the volumetric strain, another on the deviatoric strain.If an isotropic medium is also incompressible, the relative change of the volume vanishes:Sp U = 0 . Accordingly, the volumetric part of the strain becomes nil, and so do the volumetricparts of the elastic, hereditary, and viscous stresses. For such media, we end up with a simplerelation which includes only deviators: ( total ) S = S deviatoric = ( e ) S deviatoric + ( h ) S deviatoric + ( v ) S deviatoric = 2 µ U + 2 ˜ µ U + 2 η ∂∂t U (166)or simply: ( total ) S = S deviatoric = 2 ˆ µ U , (167)45here U contains only a deviatoric part, whileˆ µ ≡ µ + ˜ µ + η ∂∂t (168)is the total, integro-differential operator, which is mapping the preceding history and the presentrate of change of the strain to the present value of the stress.It should be reiterated that the above approach is based on the assertion that the elastic,viscous, and hereditary stresses sum up, and that all three are related to the same total strain. Asimple example of this approach, called the Kelvin-Voigt model, is rendered below in subsectionA.6.3.A different approach would be to assume that the strain consists of three distinct parts –elastic, hereditary, and viscous – and that these components are related to the same overallstress. A simple example of this treatment, termed the Maxwell model, is set out in subsectionA.6.4. A more complex example of this approach is furnished by the Andrade model presentedin subsection 5.3. Another way of combining elasticity and viscosity (with no hereditary reactioninvolved) is implemented by the Hohenemser-Prager (SAS) model explained in subsection A.6.5below. A.6 Examples of viscoelastic behaviour with no hereditary reaction
A.6.1 Elastic deformation
The truly simplest example of deformation is elastic: ( e ) S = 2 µ U , U = J ( e ) S , (169)where µ and J are the unrelaxed rigidity and compliance: µ = µ (0) , J = J (0) , µ = 1 /J . (170)In the frequency domain, this relation assumes the same form as it would in the time domain:¯ σ γν ( χ ) = 2 µ ¯ u γν ( χ ) , u γν ( χ ) = J ¯ σ γν ( χ ) . (171) A.6.2 Viscous deformation
The next example is that of a purely viscous behaviour: ( v ) S = 2 η ∂∂t U . (172)It is straightforward from (172) and (27) that, in this regime, the Fourier components of thestress are connected to those of the strain through¯ σ γν ( χ ) = 2 ¯ µ ( χ ) ¯ u γν ( χ ) , u γν ( χ ) = ¯ J ( χ ) ¯ σ γν ( χ ) , (173)where the complex rigidity and the complex compliance are given by¯ µ = i η χ , ¯ J = − i η χ . (174) Although we no longer spell it out, the word stress everywhere means: deviatoric stress , as we agreed toconsider the medium incompressible. .6.3 Viscoelastic deformation: a Kelvin-Voigt material The Kelvin-Voigt model, also called the Voigt model, can be represented with a purely viscousdamper and a purely elastic spring connected in parallel. Subject to the same elongation, theseelements have their forces summed up. This illustrates the situation where the total, viscoelasticstress consists of a purely viscous and a purely elastic inputs called into being by the same strain: S = ( ve ) S = ( v ) S + ( e ) S , while U = ( v ) U = ( e ) U . (175)Then the total stress reads: S = (cid:18) µ + 2 η ∂∂t (cid:19) U , (176a)which is often presented in the form of S = 2 µ (cid:18) U + τ V (cid:5) U (cid:19) , (176b)with the so-called Voigt time defined as τ V ≡ η/µ . (177)Comparing (176) with (46), we understand that the kernel of the rigidity operator for the Kelvin-Voigt model can be written down as µ ( t − t ′ ) = µ + η δ ( t − t ′ ) . (178)Suppose the strain is varying in time as u γν ( t ) = σ µ (cid:20) − exp (cid:18) − t − t τ V (cid:19) (cid:21) Θ( t − t ) , (179)so that (cid:5) u γν ( t ) = σ µ τ V exp (cid:18) − t − t τ V (cid:19) Θ( t − t ) . (180)Then insertion of (178) and (180) into ( ?? ) or, equivalently, insertion of (179) into (176a)demonstrates that this strain results from a stress σ γν ( t ) = σ Θ( t − t ) . (181)It would however be a mistake to deduce from this that the compliance function is equal to µ − h − exp (cid:16) − t − t ′ τ V (cid:17) i , even though such a misstatement is sometimes made in the lit-erature. This expression furnishes the compliance function only in the special situation of aHeaviside-step stress (181), but not in the general case. For example, plugging of (178) and (180) into ( ?? ) leads to: σ ( t ) = Z t ′ = tt ′ = − ∞ [ µ + η δ ( t − t ′ ) ] σ µ τ V exp (cid:18) − t ′ − t τ V (cid:19) Θ( t ′ − t ) dt ′ = σ h R t ′ = tt ′ = t exp (cid:16) − t ′ − t τ V (cid:17) dt ′ τ V + exp (cid:16) − t − t τ V (cid:17)i for t ≥ t t < t , which is simply σ Θ( t − t ) .
47s can be easily shown from (27), in the frequency domain model (176) reads as (173), exceptthat the complex rigidity and the complex compliance are now given by¯ µ = µ ( 1 + i χ τ V ) , ¯ J = J i χ τ V . (182)Recall that, for brevity, here and everywhere we write µ and J instead of µ (0) and J (0) .The Kelvin-Voigt material becomes elastic in the low-frequency limit, and viscous in thehigh-frequency limit. A.6.4 Viscoelastic deformation: a Maxwell material
The Maxwell model can be represented with a viscous damper and an elastic spring connectedin series. Experiencing the same force, these elements have their elongations summed up. Thisexample illustrates the situation where the total, viscoelastic strain consists of a purely viscousand a purely elastic contributions generated by the same stress S : U = ( v ) U + ( e ) U , where ( e ) S = 2 µ ( e ) U and ( v ) S = 2 η ∂∂t ( v ) U . (183)Since in the Maxwell regime both contributions to the strain are generated by the same stress S = ( ve ) S = ( v ) S = ( e ) S , (184)formula (183) can be written down as (cid:5) U = 12 µ (cid:5) S + 12 η S (185a)or, in a more conventional form: (cid:5) S + 1 τ M S = 2 µ (cid:5) U , (185b)with the so-called Maxwell time introduced as τ M ≡ η/µ . (186)Although formally the Maxwell time is given by an expression mimicking the definition of theVoigt time, the meaning of these times is different.Comparing (185) with the general expression (43) for the compliance operator, we see that,for the Maxwell model, the compliance operator in the time domain assumes the form: J ( t − t ′ ) = (cid:20) J + ( t − t ′ ) 1 η (cid:21) Θ( t − t ′ ) , (187)where J ≡ /µ . In the frequency domain, (185) can be written down as (173), with the complexrigidity and compliance given by¯ µ ( χ ) = µ i χ τ M i χτ M , ¯ J ( χ ) = J (cid:18) − i χ τ M (cid:19) = J − i χ η . (188)Clearly, such a body becomes elastic in the high-frequency limit, and becomes viscous at lowfrequencies (the latter circumstance making the Maxwell model attractive to seismologists).48 .6.5 Viscoelastic deformation: the Hohenemser-Prager (SAS) model An attempt to combine the Kelvin-Voigt and Maxwell models leads to the Hohenemser-Pragermodel, also known as the Standard Anelastic Solid (SAS): τ M ˙ S + S = 2 µ (cid:16) U + τ V ˙ U (cid:17) , (189)In the limit of τ M → τ V acquiresthe meaning of the Voigt time).A transition from the SAS to Maxwell model, however, can be achieved only through re-definition of parameters. One should set: 2 µ → τ V → ∞ , along with 2 µτ V → η .Then (189) will become (185), with τ M playing the role of the Maxwell time.In the frequency domain, (189) can be put into the form of (173), the complex rigidity andthe complex compliance being expressed through the parameters as¯ µ = µ iτ V χ iτ M χ , ¯ J = J iτ M χ iτ V χ . (190)This entails: tan δ ≡ I m [¯ µ ] / R e [¯ µ ] = ( τ V − τ M ) χ τ V τ M χ , whence it is easy to show that the tangentis related to its maximal value throughtan δ = 2 [tan δ ] max τ χ τ χ , where τ ≡ √ τ M τ V . This is the so-called Debye peak, which is indeed observed in some materials.To prove that the SAS solid is indeed anelastic, one has to make sure that a Heaviside stepstress Θ( t ′ ) entails a strain proportional to 1 − exp( − Γ t ) , and to demonstrate that a predeformedsample subject to stress Θ( − t ′ ) regains it shape as exp( − Γ t ) , where the relaxation constant Γis positive. In subsection C.3 we shall do this for a SAS sphere. B Interconnection between the quality factor and the phaselag
The power P exerted by a tide-raising secondary on its primary can be written as P = − Z ρ ~V · ∇ W d x (191) ρ , ~V , and W signifying the density, velocity, and tidal potential in the small volume d x of the primary. The mass-conservation law ∇ · ( ρ ~V ) + ∂ρ∂t = 0 enables one to shape thedot-product into the form of ρ ~V · ∇ W = ∇ · ( ρ ~V W ) − ρ W ∇ · ~V − ~V W ∇ ρ . (192)Under the realistic assumption of the primary’s incompressibility, the term with ∇ · ~V may beomitted. To get rid of the term with ∇ ρ , one has to accept a much stronger approximation ofthe primary being homogeneous. Then the power will be rendered by P = − Z ∇ · ( ρ ~V W ) d x = − Z ρ W ~V · ~ n dS , (193) ~ n being the outward normal and dS being an element of the surface area of the primary. Thisexpression for the power (pioneered, probably, by Goldreich 1963) enables one to calculate thework through radial displacements only, in neglect of horizontal motion.49enoting the radial elevation with ζ , we can write the power per unit mass, P ≡
P/M , as: P = (cid:18) − ∂W∂r (cid:19) ~V · ~ n = (cid:18) − ∂W∂r (cid:19) dζdt . (194)A harmonic external potential W = W cos( ω lmpq t ) , (195)applied at a point of the primary’s surface, will elevate this point by ζ = h W o g cos( ω lmpq t − ǫ lmpq ) = h W o g cos( ω lmpq t − ω lmpq ∆ t lmpq ) , (196)with g being the surface gravity acceleration, and h denoting the Love number.In formula (196), ω lmpq is one of the modes (105) showing up in the Darwin-Kaula expansion(103). The quantity ǫ lmpq = ω lmpq ∆ t lmpq is the corresponding phase lag, while ∆ t lmpq isthe positively defined time lag at this mode. Although the tidal modes ω lmpq can assume anysign, both the potential W and elevation ζ can be expressed via the positively defined forcingfrequency χ lmpq = | ω lmpq | and the absolute value of the phase lag: W = W cos( χ t ) , (197) ζ = h W o g cos( χ t − | ǫ | ) , (198)subscripts lmpq being dropped here and hereafter for brevity.The vertical velocity of the considered element of the primary’s surface will be dζdt = − h χ W o g sin( χt − | ǫ | ) = − h χ W o g (sin χt cos | ǫ | − cos χt sin | ǫ | ) . (199)Introducing the notation A = h W g ∂W ∂r , we write the power per unit mass as P = (cid:18) − ∂W∂r (cid:19) dζdt = A χ cos( χ t ) sin( χt − | ǫ | ) , (200)and write the work w per unit mass, performed over a time interval ( t , t ) , as: w | tt = Z tt P dt = A Z χtχt cos( χ t ) sin( χt − | ǫ | ) d ( χ t ) = A cos | ǫ | Z χtχt cos z sin z dz − A sin | ǫ | Z χtχt cos z dz = − A χt − | ǫ | ) + 2 χ t sin | ǫ | ] tt . (201)Being cyclic, the first term in (201) renders the elastic energy stored in the body. The secondterm, being linear in time, furnishes the energy damped. This clear interpretation of the twoterms was offered by Stan Peale [2011, personal communication].The work over a time period T = 2 π/χ is equal to the energy dissipated over the period: w | t = Tt =0 = ∆ E cycle = − A π sin | ǫ | . (202)It can be shown that the peak work is obtained over the time span from π to | ǫ | and assumesthe value E ( work ) peak = A h cos | ǫ | − sin | ǫ | (cid:16) π − | ǫ | (cid:17) i , (203)50hence the appropriate quality factor is given by: Q − work = − ∆ E cycle π E ( work ) peak = tan | ǫ | − (cid:16) π − | ǫ | (cid:17) tan | ǫ | . (204)To calculate the peak energy stored in the body, we would note that the first term in (201)is maximal when taken over the span from χ t = π/ | ǫ | / χ t = 3 π/ | ǫ | / E ( energy ) peak = A , (205)and the corresponding quality factor is: Q − energy = − ∆ E cycle π E ( energy ) peak = sin | ǫ | . (206)Goldreich (1963) suggested to employ the span χ t = (0 , π/
4) . The absolute value of theresulting power, denoted in
Ibid. as E ∗ , is equal to E ∗ = A | ǫ | (207)and is not the peak value of the energy stored nor of the work performed. Goldreich (1963)however employed it to define a quality factor, which we shall term Q Goldreich . This factor isintroduced via Q − Goldreich = − ∆ E cycle π E ∗ = tan | ǫ | . (208)In our opinion, the quality factor Q energy defined through (206) is preferable, because the ex-pansion of tides contains terms proportional to k l ( χ lmpq ) sin ǫ l ( χ lmpq ) . Since the long-establishedtradition suggests to substitute sin ǫ with 1 /Q , it is advisable to define the Q exactly as (206),and also to call it Q l , to distinguish it from the seismic quality factor (Efroimsky 2012). C Tidal response of a homogeneous viscoelastic sphere(Churkin 1998)
This section presents some results from the unpublished preprint by Churkin (1998). We tookthe liberty of upgrading the notations and correcting some minor oversights. C.1 A homogeneous Kelvin-Voigt spherical body
In combination with the Correspondence Principle, the formulae from subsection A.6.3 furnishthe following expression for the complex Love numbers of a Kelvin-Voigt body:¯ k l ( χ ) = 32( l −
1) 11 + A l ( 1 + τ V i χ ) , (209) Churkin (1998) employed the notation k l ( τ ) for what we call (cid:5) k l ( τ ) . Our notations are more convenient inthat they amplify the close analogy between the Love functions and the compliance function.
51t then can be demonstrated, with aid of (61), that the time-derivative of the corresponding Lovefunction is (cid:5) k l ( τ ) = l −
1) 1 A l τ V exp( − τ ζ l ) Θ( τ ) for τ V > l −
1) 11 + A l δ ( τ ) for τ V = 0 , (210)while the Love function itself has the form of k l ( τ ) = 32( l −
1) 1 A l ζ l τ V h − exp( − τ ζ l ) i Θ( τ ) (211)= 32( l −
1) 11 + A l h − exp( − τ ζ l ) i Θ( τ ) , where ζ l ≡ A l A l τ V . (212)Formulae (210) may look confusing, in that exp( − τ ζ l ) simply vanishes in the elastic limit, i.e.,when τ V → ζ l → ∞ . We however should not be misled by this mathematical artefactstemming from the nonanaliticity of the exponent function. Instead, we should keep in mind thata physical meaning is attributed not to the Love functions or their derivatives but to the resultsof the Love operator’s action on realistic disturbances. For example, a Heaviside step potential W l ( ~R , ~r ∗ , t ′ ) = W Θ( t ′ ) (213)applied to a homogeneous Kelvin-Voigt spherical body will furnish, through relation (60), thefollowing response of the potential: U l ( ~r , t ) = (cid:18) Rr (cid:19) l +1 Z t ′ = tt ′ = −∞ (cid:5) k l ( t − t ′ ) W Θ( t ′ ) dt ′ = (cid:18) Rr (cid:19) l +1 Z t ′ = tt ′ = 0 (cid:5) k l ( t − t ′ ) W dt ′ (214)= W (cid:18) Rr (cid:19) l +1 Z τ = tτ = 0 (cid:5) k l ( τ ) dτ = 32( l −
1) 1 − exp( − t ζ l )1 + A l (cid:18) Rr (cid:19) l +1 W .
In the elastic limit, this becomes: τ V → ⇒ ζ l → ∞ = ⇒ U l ( ~r , t ) → l −
1) 11 + A l (cid:18) Rr (cid:19) l +1 W , (215)which reproduces the case described by the static Love number k l = 32 11 + A l .An alternative way of getting (215) would be to employ formulae (211) and (59a). C.2 A homogeneous Maxwell spherical body
Using the formulae presented in the Appendix A.6.4, and relying upon the CorrespondencePrinciple, we write down the complex Love numbers for a Maxwell material as¯ k l ( χ ) = 32( l −
1) 11 + A l τ M i χ τ M i χ = 32( l −
1) 11 + A l (cid:20) A l A l ) τ M i χ (cid:21) , (216)52hich corresponds, via (61), to (cid:5) k l ( τ ) = 32( l − δ ( τ ) + A l γ l exp( − τ γ l ) Θ( τ )1 + A l (217)and k l ( τ ) = 32( l −
1) 1 + A l h − exp( − τ γ l ) i A l Θ( τ ) , (218)where γ l ≡
1( 1 + A l ) τ M . (219)A Heaviside step potential W l ( ~R , ~r ∗ , t ′ ) = W Θ( t ′ ) (220)will, according to formula (60), render the following response: U l ( ~r , t ) = (cid:18) Rr (cid:19) l +1 Z t ′ = tt ′ = −∞ (cid:5) k l ( t − t ′ ) W Θ( t ′ ) dt ′ = (cid:18) Rr (cid:19) l +1 Z t ′ = tt ′ = 0 (cid:5) k l ( t − t ′ ) W dt ′ (221)= W (cid:18) Rr (cid:19) l +1 Z τ = tτ = 0 (cid:5) k l ( τ ) dτ = 32( l −
1) 1 + A l h − exp( − t γ l ) i A l (cid:18) Rr (cid:19) l +1 W Θ( t ) . In the elastic limit, we obtain: τ M → ∞ = ⇒ γ l → ⇒ U l ( ~r , t ) → l −
1) 11 + A l (cid:18) Rr (cid:19) l +1 W , (222)which corresponds to the situation described by the static Love number k l = 32( l −
1) 11 + A l . C.3 A homogeneous Hohenemser-Prager (SAS) spherical body
The Correspondence Principle, along with the formulae from subsection A.6.5, yields the followingexpression for the complex Love numbers of a Hohenemser-Prager (SAS) spherical body:¯ k l ( χ ) = 32 ( l −
1) 11 + A l i χ τ V i χ τ M . (223)Combined with (61), this entails: (cid:5) k l ( τ ) = 32 ( l −
1) 11 + A l τ V τ M " δ ( τ ) + A l τ M τ V − τ M τ M + A l τ V exp − A l τ M + A l τ V τ ! (224)and k l ( τ ) = 32 ( l −
1) 1 − A l τ M τ V − τ M A l (cid:20) − exp (cid:18) − A l τ M + A l τ V τ (cid:19) (cid:21) A l τ V τ M Θ( τ ) (225)53 Heaviside step potential W l ( ~R , ~r ∗ , t ′ ) = W Θ( t ′ ) (226)applied to a SAS spherical body will then result in the following variation of its potential: U l ( ~r , t ) = (cid:18) Rr (cid:19) l +1 Z t ′ = tt ′ = −∞ (cid:5) k l ( t − t ′ ) W Θ( t ′ ) dt ′ = (cid:18) Rr (cid:19) l +1 Z t ′ = tt ′ = 0 (cid:5) k l ( t − t ′ ) W dt ′ = W (cid:18) Rr (cid:19) l +1 Z τ = tτ = 0 (cid:5) k l ( τ ) dτ = 32( l −
11 + A l τ V τ M + A l A l τ V − τ M τ M + A l τ V " − exp − A l τ M + A l τ V t ! Rr (cid:19) l +1 W Θ( t ) . (227)Within this model, the elastic limit is achieved by setting τ M = τ V , whence we obtain thecase described by the static Love number k l = 32 11 + A l . Interestingly, the elastic regime isachieved even when these times are not zero. Their being equal to one another turns out to besufficient.Repeating the above calculation for tidal disturbance W Θ( − t ′ ) , we shall see that, after thetidal perturbation is removed, a tidally prestressed sphere regains its shape, the stress relaxingat a rate proportional to exp (cid:18) − A l τ M + A l τ V t (cid:19) . D The correspondence principle(elastic-viscoelastic analogy)
D.1 The correspondence principle, for nonrotating bodies
While the static Love numbers depend on the static rigidity µ through (3), it is not immediatelyclear if a similar formula interconnects also ¯ k l ( χ ) with ¯ µ ( χ ) . To understand why and whenthe relation should hold, recall that formulae (3) originate from the solution of a boundary-valueproblem for a system incorporating two equations: σ βν = 2 µ u βν , (228a)0 = ∂σ βν ∂x ν − ∂p∂x β − ρ ∂ ( W + U ) ∂x β , (228b)the latter being simply the equation of equilibrium written for a static viscoelastic medium,in neglect of compressibility and heat conductivity. The notations σ βν and u βν stand for the deviatoric stress and strain, p ≡ −
13 Sp S is the pressure (set to be nil in incompressible media),while W and U are the perturbing and perturbed potentials. By solving the system, one arrivesat the static relation U l = k l W l , with the customary static Love numbers k l expressed via ρ , R , and µ by (3).Now let us write equation like (228a - 228b) for time-dependent deformation of a nonrotating body: S = 2 ˆ µ U , (229a) ρ ¨u = ∇ S − ∇ p − ∇ ( W + U ) (229b)54r, in terms of components: σ βν = 2 ˆ µ u βν , (230a) ρ ¨ u β = ∂σ βν ∂x ν − ∂p∂x β − ρ ∂ ( W + U ) ∂x β . (230b)In the frequency domain, this will look:¯ σ βν ( χ ) = 2 ¯ µ ( χ ) ¯ u βν ( χ ) , (231a) ρ χ ¯ u βν ( χ ) = ∂ ¯ σ βν ( χ ) ∂x ν − ∂ ¯ p ( χ ) ∂x β − ρ ∂ (cid:2) ¯ W ( χ ) + ¯ U ( χ ) (cid:3) ∂x β , (231b)where a bar denotes a spectral component for all functions except µ – recall that ¯ µ is a spectralcomponent not of the kernel µ ( τ ) but of its time-derivative ˙ µ ( τ ) .Unless the frequencies are extremely high, we can neglect the body-fixed acceleration term χ ¯ u βν ( χ ) in the second equation, in which case our system of equations for the spectral com-ponents will mimic (228). Thus we arrive at the so-called correspondence principle (also knownas the elastic-viscoelastic analogy ), which maps a solution of a linear viscoelastic boundary-valueproblem to a solution of a corresponding elastic problem with the same initial and boundaryconditions. As a result, the algebraic equations for the Fourier (or Laplace) components of thestrain and stress in the viscoelastic case mimic the equations connecting the strain and stress inthe appropriate elastic problem. So the viscoelastic operational moduli ¯ µ ( χ ) or ¯ J ( χ ) obey thesame algebraic relations as the elastic parameters µ or J .In the literature, there is no consensus on the authorship of this principle. For example,Haddad (1995) mistakenly attributes it to several authors who published in the 1950s and 1960s.In reality, the principle was pioneered almost a century earlier by Darwin (1879), for isotropicincompressible media. The principle was extended to more general types of media by Biot (1954,1958), who also pointed out some limitations of this principle. D.2 The correspondence principle, for rotating bodies
Consider a body of mass M prim , which is spinning at a rate ~ω and is also performing someorbital motion (for example, is orbiting, with its partner of mass M sec , around their mutualcentre of mass). Relative to some inertial coordinate system, the centre of mass of the body islocated at ~x CM , while a small parcel of its material is positioned at ~x . Relative to the centre ofmass of the body, the parcel is located at ~r = ~x − ~x CM . The body being deformable, we candecompose ~r into its average value, ~r , and an instantaneous displacement ~u : ~x = ~x CM + ~r~r = ~r + ~u = ⇒ ~x = ~x CM + ~r + ~u . (232)Denote with D/Dt the time-derivative in the inertial frame. The symbol d/dt and its synonym,overdot, will be reserved for the time-derivative in the body frame, so d ~r /dt = 0 . Then D ~r Dt = d ~r dt + ~ω × ~r and D ~r Dt = d ~r dt + 2 ~ω × d ~r dt + ~ω × ( ~ω × ~r ) + ˙ ~ω × ~r . (233)Together, the above formulae result in D ~x Dt = D ~x CM Dt + D ~r Dt = D ~x CM Dt + d ~r dt + 2 ~ω × d ~r dt + ~ω × ( ~ω × ~r ) + ˙ ~ω × ~r = D ~x CM Dt + d ~u dt + 2 ~ω × d ~u dt + ~ω × ( ~ω × ~r ) + ˙ ~ω × ~r . (234)55he equation of motion for a small parcel of the body’s material will read as ρ D ~x Dt = ∇ S − ∇ p + ~F self + ~F ext , (235)where ~F ext is the exterior gravity force per unit volume , while ~F self is the “interior” gravityforce per unit volume , i.e., the self-force wherewith the rest of the body is acting upon the selectedparcel of medium. Insertion of (234) in (235) furnishes: ρ (cid:20) D ~x CM Dt + (cid:5) (cid:5) ~u + 2 ~ω × (cid:5) ~u + ~ω × ( ~ω × ~r ) + ˙ ~ω × ~r (cid:21) = ∇ S − ∇ p + ~F self + ~F ext . (236)At the same time, for the primary body as a whole, we can write: M prim D ~x CM Dt = Z V ~F ext d ~r , (237)the integration being carried out over the volume V of the primary. (Recall that ~F ext is a forceper unit volume.) Combined together, the above two equations will result in ρ (cid:20) (cid:5) (cid:5) ~u + 2 ~ω × (cid:5) ~u + ~ω × ( ~ω × ~r ) + ˙ ~ω × ~r (cid:21) = ∇ S − ∇ p + ~F self + ~F ext − ρM prim Z V ~F ext d ~r . (238)For a spherically-symmetrical (not necessarily radially-homogeneous) body, the integral on theright-hand side clearly removes the Newtonian part of the force, leaving the harmonics intact: ~F ext − ρM prim Z V ~F ext d ~r = ρ ∞ X l =2 ∇ W l , (239)where the harmonics are given by W l ( ~r , ~r ∗ ) = − G M sec r ∗ (cid:16) rr ∗ (cid:17) l P l (cos γ ) , (240) ~r ∗ being the vector pointing from the centre of mass of the primary to that of the secondary,and γ being the angular separation between ~r and ~r ∗ , subtended at the centre of mass of theprimary.In reality, a tiny extra force F , the tidal force per unit volume, is left over due to the bodybeing slightly distorted: ~F ext − ρM prim Z V ~F ext d ~r = ρ ∞ X l =2 ∇ W l + F . (241)Here F is the density multiplied by the average tidal acceleration experienced by the body as awhole. In neglect of F , we arrive at ρ (cid:20) (cid:5) (cid:5) ~u + 2 ~ω × (cid:5) ~u + ~ω × ( ~ω × ~r ) + ˙ ~ω × ~r (cid:21) = ∇ S − ∇ p − ρ ∞ X l =2 ∇ ( U l + W l ) . (242)Here, to each disturbing term of the exterior potential, W l , corresponds a term U l of the self-potential, the self-force thus being expanded into ~F self = − P ∞ l =2 ∇ U l .Equation (242) could as well have been derived in the body frame, where it would haveassumed the same form.Denoting the tidal frequency with χ , we see that the terms on the left-hand side have theorder of ρ χ u , ρ ω χ u , ρ ω r , and ρ ˙ χ ω r , correspondingly. In realistic situations, the firsttwo terms, thus, can be neglected, and we end up with0 = ∇ S − ∇ p − ρ ∞ X l =2 ∇ ( U l + W l ) − ρ ~ω × ( ~ω × ~r ) − ρ ˙ ~ω × ~r , (243)the term − ∇ p vanishing in an incompressible media.56 .3 The centripetal term and the zero-degree Love number The centripetal term in (243) can be split into a purely radial part and a part that can beincorporated into the W term of the tide-raising potential, as was suggested by Love (1909,1911). Introducing the colatitude φ ′ through cos φ ′ = ~ω | ~ω | · ~r | ~r | , we can write down theevident equality ~ω × ( ~ω × ~r ) = ~ω ( ~ω · ~r ) − ~r ~ω = ∇ (cid:20)
12 ( ~ω · ~r ) − ~ω ~r (cid:21) = ∇ (cid:20) ~ω ~r (cid:0) cos φ ′ − (cid:1) (cid:21) . The definition P (cos φ ′ ) = 12 (3 cos φ ′ −
1) easily renders: cos φ ′ = 23 P (cos φ ′ ) + 13 ,whence: ~ω × ( ~ω × ~r ) = ∇ (cid:20) ~ω ~r [ P (cos φ ′ ) − (cid:21) . (244)We see that the centripetal force splits into a second-harmonic and purely-radial parts: − ρ ~ω × ( ~ω × ~r ) = − ∇ h ρ ~ω ~r P (cos φ ′ ) i + ∇ h ρ ~ω ~r i , (245)where we assume the body to be homogeneous. The second-harmonic part can be incorporatedinto the external potential. The response to this part will be proportional to the degree-2 Lovenumber k .The purely radial part of the centripetal potential generates a radial deformation. Thispart of the potential is often ignored, the associated deformation being tacitly included into theequilibrium shape of the body. Compared to the main terms of the equation of motion, thisradial term is of the order of 10 − for the Earth, and is smaller for most other bodies. As therotation variations of the Earth are of the order of 10 − , this term leads to a tiny change in thegeopotential and to an associated displacement of the order of a micrometer. However, for other rotators the situation may be different. For example, in Phobos, whoselibration magnitude is large (about 1 degree), the radial term may cause an equipotential-surfacevariation of about 10 cm. This magnitude is large enough to be observed by future missionsand should be studied in more detail. The emergence of the purely radial deformation givesbirth to the zero-degree Love number (Dahlen 1976, Matsuyama & Bills 2010). Using Dahlen’sresults, Yoder (1982, eqns 21 - 22) demonstrated that the contribution of the radial part of thecentripetal potential to the change in mean motion of Phobos is about 3%, which is smallerthan the uncertainty in our knowledge of Phobos’ k /Q . It should be mentioned, however, thatthe calculations by Dahlen (1976) and Matsuyama & Bills (2010) were performed for steady (orslowly changing) rotation, and not for libration. This means that Yoder’s application of Dahlen’sresult to Phobos requires extra justification.What is important for us here is that the radial term does not interfere with the calculationof the Love number. Being independent of the longitude, this term generates no tidal torqueeither, provided the obliquity is neglected. D.4 The toroidal term
The inertial term − ρ ˙ ~ω × ~r in the equation of motion (243) can be cast into the form − ρ ˙ ~ω × ~r = ρ ~r × ∇ ( ˙ ~ω · ~r ) , (246) Tim Van Hoolst, private communication. Tim Van Hoolst, private communication. h l ) or the potential ( k l ). As this deformation yields no change in thegravitational potential of the tidally-perturbed body, there is no tidal torque associated withthis deformation. Being divergence-free, this deformation entails no contraction or expansioneither, i.e., it is purely shear. Still, this deformation contributes to dissipation. Besides, sincethe toroidal forcing results in the toroidal deformation, it can, in principle, be associated with a“toroidal” Love number.To estimate the dissipation caused by the toroidal rotational force, Yoder (1982) introducedan equivalent effective torque. He pointed out that this force becomes important when themagnitude of the physical libration is comparable to that of the optical libration. According to Ibid. , the toroidal force contributes to the change of the mean motion of Phobos about 1.6%,which is less than the input from the purely radial part.
E The Andrade and Maxwell models at different frequen-cies
E.1 Response of a sample obeying the Andrade model
Within the Andrade model, the tangent of the phase lag demonstrates the so-called “elbowdependence”. At high frequencies, the tangent of the lag obeys a power law with an exponentequal to − α , where 0 < α < − (1 − α ) . This model fits well the behaviourof ices, metals, silicate rocks, and many other materials.However the applicability of the Andrade law may depend upon the intensity of the loadand, accordingly, upon the damping mechanisms involved. Situations are known, when, at lowfrequencies, anelasticity becomes much less efficient than viscosity. In these cases, the Andrademodel approaches, at low frequencies, the Maxwell model. E.1.1 The high-frequency band
At high frequencies, expression (89b) gets simplified. In the numerator, the term with z − α dominates: z − α ≫ z − ζ , which is equivalent to z ≫ ζ − α . In the denominator, the constantterm dominates: 1 ≫ z − α , or simply: z ≫ z ≫ ζ − α or z ≫ α − term in (85) is large enough. In other words, the Andrade timescale τ A should besmaller (or, at least, not much higher) than the viscoelastic time τ M . Accordingly, at highfrequencies, ζ is smaller (or, at least, not much higher) than unity. Hence, within the high-frequency band, either the condition z ≫ z ≫ ζ − α or the two conditionsare about equivalent. This, along with (90) and (91) enables us to write:tan δ ≈ ( χ τ A ) − α sin (cid:16) α π (cid:17) Γ( α + 1) for χ ≫ τ − A = τ − M ζ − . (247)The tangent being small, the expression for sin δ looks identical:sin δ ≈ ( χ τ A ) − α sin (cid:16) α π (cid:17) Γ( α + 1) for χ ≫ τ − A = τ − M ζ − . (248)58 .1.2 The intermediate region In the intermediate region, the behaviour of the phase lag δ depends upon the frequency-dependence of ζ . For example, if there happens to exist an interval of frequencies over whichthe conditions 1 ≫ z ≫ ζ − α are obeyed, then over this interval we shall have: 1 ≪ z − α and z − α ≫ ζ z − . Applying these inequalities to (89b), we see that over such an interval offrequencies tan δ will behave as z − α tan (cid:16) α π (cid:17) . E.1.3 The low-frequency band
At low frequencies, the term z − ζ becomes leading in the numerator of (89b): z − α ≪ z − ζ ,which requires z ≪ ζ − α . In the denominator, the term with z − α becomes the largest: 1 ≪ z − α , whence z ≪ ζ is larger (at least, not much smaller) thanunity. Thence the condition z ≪ z ≪ ζ − α . Thus we state:tan δ ≈ ( χ τ A ) − (1 − α ) ζ cos (cid:16) α π (cid:17) Γ( α + 1) for χ ≪ τ − A = τ − M ζ − . (249)The appropriate expression for sin δ will be:sin δ ≈ − O (cid:0) ( χ τ A ) − α ) ζ − (cid:1) for χ ≪ τ − A = τ − M ζ − , (250)It would be important to emphasise that the threshold τ − A = τ − M ζ − standing in (247) and(248) is different from the threshold τ − A = τ − M ζ − showing up in (249) and (250), even thoughthese two thresholds are given by the same expression. The reason for this is that the timescales τ A and τ M are not fixed constants. While the Maxwell time is likely to be a very slow functionof the frequency, the Andrade time may undergo a faster change over the transitional region: τ A must be larger than τ M at low frequencies (so anelasticity yields to viscosity), and must becomeshorter than or of the order of τ M at high frequencies (so anelasticity becomes stronger). Thisway, the threshold τ − A standing in (249 - 250) is lower than the threshold τ − A standing in (247- 248). The gap between these thresholds is the region intermediate between the two pronouncedpower laws (247) and (249). E.1.4 The low-frequency band: a special case, the Maxwell model
Suppose that, below some threshold χ , anelasticity quickly becomes much less efficient thanviscosity. This would imply a steep increase of ζ (equivalently, of τ A ) at low frequencies. Then,in (89b), we shall have: 1 ≫ z − α and z − α ≪ ζ z − . This means that, for frequencies below χ ,the tangent will behave astan δ ≈ z − ζ = ( χ τ M ) − for χ ≪ χ . (251)the well-known viscous scaling law for the lag.The study of ices and minerals under weak loads (Castillo-Rogez et al. 2011, Castillo-Rogez2011) has not shown such an abrupt vanishing of anelasticity. However, Karato & Spetzler (1990)point out that this should be happening in the Earth’s mantle, where the loads are much higherand anelasticity is caused by unpinning of dislocations.59 .2 The behaviour of | k l ( χ ) | sin ǫ l ( χ ) = − I m (cid:2) ¯ k l ( χ ) (cid:3) within the Andrade and Maxwell models As we explained in subsection 4.1, products k l ( χ l mpq ) sin ǫ l ( χ l mpq ) enter the l mpq term of theDarwin-Kaula series for the tidal potential, force, and torque. Hence the importance to knowthe behaviour of these products as functions of the tidal frequency χ l mpq . E.2.1 Prefatory algebra
It ensues from (68) that¯ k l ( χ ) = 32 ( l − (cid:0) R e (cid:2) ¯ J ( χ ) (cid:3) (cid:1) + (cid:0) I m (cid:2) ¯ J ( χ ) (cid:3) (cid:1) + A l J R e (cid:2) ¯ J ( χ ) (cid:3) + i 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) , (252)whence | ¯ k l ( χ ) | sin ǫ l ( χ ) = − I m (cid:2) ¯ k l ( χ ) (cid:3) = 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) , (253) J = J (0) ≡ /µ = 1 /µ (0) being the unrelaxed compliance (the inverse of the unrelaxed shearmodulus µ ). For an Andrade material, the compliance ¯ J in the frequency domain is renderedby (86). Its imaginary and real parts are given by (87 - 88). It is then easier to rewrite (253) as | ¯ k l ( χ ) | sin ǫ l ( χ ) =3 A l l − ζ z − + z − α sin (cid:16) α π (cid:17) Γ( α + 1) h A l + 1 + z − α cos (cid:16) α π (cid:17) Γ( α + 1) i + h ζ z − + z − α sin (cid:16) α π (cid:17) Γ(1 + α ) i , (254)where z ≡ χ τ A = χ τ M ζ (255)and ζ ≡ τ A τ M . (256)For β → τ A → ∞ , (254) coincides with the appropriate expression for a sphericalMaxwell body. E.2.2 The high-frequency band
Within the upper band, the term with z − α dominates the numerator, while A l dominates thedenominator. The domination of z − α in the numerator requires that z ≫ ζ − α , which is thesame as χ ≫ τ − M ζ α − α . The domination of A l in the denominator requires: z ≫ A − /α , whichis the same as χ ≫ τ − M ζ − A − /α . It also demands that ζ z − ≪ A l , which is: χ ≫ τ − M A − l .For realistic values of A l (say, 10 ) and α (say, 0.25), we have: A − /α ∼ − . At highfrequencies, anelasticity beats viscosity, so ζ is less than unity (or, at least, is not much largerthan unity). On these grounds, the requirement χ ≫ τ − M ζ α − α is the strongest here. Itsfulfilment guarantees that of both χ ≫ τ − M ζ − A − /α and χ ≫ τ M /A l . Thus we have: | ¯ k l ( χ ) | sin ǫ l ( χ ) ≈
32 ( l −
1) 1 A l sin (cid:16) α π (cid:17) Γ( α + 1) ζ − α ( τ M χ ) − α for χ ≫ τ − M ζ α − α . (257a)60 .2.3 The intermediate band Within the intermediate band, the term ζ z − takes over in the numerator, while A l still dom-inates in the denominator. The domination of ζ z − α in the numerator implies that z ≪ ζ − α ,which is equivalent to χ ≫ τ − M ζ α − α . The domination of A l in the denominator requires χ ≫ τ − M ζ − A − /αl and χ ≫ τ − M A − l , as we just saw above.As we are considering the band where viscosity takes over anelasticity, we may expect thathere ζ is about or, likely, larger than unity. Taken the large value of A l , we see that the condition χ ≫ τ − M A − l is stronger. Its fulfilment guarantees the fulfilment of χ ≫ τ − M ζ − A − /αl . Thisway, we obtain: | ¯ k l ( χ ) | sin ǫ l ( χ ) ≈
32 ( l −
1) 1 A l ( τ M χ ) − for τ − M ζ α − α ≫ χ ≫ τ − M A − l . (257b) E.2.4 The low-frequency band
For frequencies lower than τ − M A − l the Andrade model renders the same frequency-dependencyas that given (at frequencies below τ − M ) by the Maxwell model: | ¯ k l ( χ ) | sin ǫ l ( χ ) ≈
32 ( l − A l τ M χ for τ − M A − l ≫ χ . (257c) E.2.5 Interpretation
Formulae (257a) and (257b) render a frequency-dependence mimicking that of | − J ( χ ) | sin δ ( χ )= − I m (cid:2) ¯ J ( χ ) (cid:3) in the high- and low-frequency bands. This can be seen from comparing (257a)and (257b) with (248).In contrast, (257c) reveals a peculiar feature inherent in the tidal lagging, and absent in thelagging in a sample.For terrestrial bodies, the condition τ − M A − l ≫ χ puts the values of χ below 10 − Hz , giveor take several orders of magnitude. Hence | ¯ k l ( χ ) | sin ǫ l ( χ ) follows the linear scaling law (257c)only in an extremely close vicinity of the commensurability where the frequency χ vanishes.Nonetheless it is very important that | ¯ k ( χ ) | sin ǫ ( χ ) first reaches a finite maximum and thendecreases continuously and vanishes, as the frequency goes to zero. This confirms that neitherthe tidal torque nor the tidal force becomes infinite in resonances. F The behaviour of k l ( χ ) ≡ | ¯ k l ( χ ) | in thelimit of vanishing tidal frequency χ ,within the Andrade and Maxwell models From (68), we obtain: | ¯ k l ( χ ) | = (cid:18) l − (cid:19) | J ( χ ) || A l J + J ( χ ) | = (cid:18) l − (cid:19) (cid:16) R e h J ( χ ) i (cid:17) + (cid:16) I m h J ( χ ) i (cid:17) (cid:16) R e h J ( χ ) i + A l J (cid:17) + (cid:16) I m h J ( χ ) i (cid:17) . (258)Bringing in expressions for the imaginary and real parts of the compliance, and introducingnotations E ≡ η − , B ≡ β sin (cid:16) α π (cid:17) Γ(1 + α ) , D ≡ β cos (cid:16) α π (cid:17) Γ(1 + α ) , (259)61e can write: | k l ( χ ) | = (cid:18) l − (cid:19) (cid:20) − A l J DE χ − α − A l J + A l J E χ + O ( χ − α ) (cid:21) (260)and | k l ( χ ) | − = (cid:18) l − (cid:19) (cid:20) A l J DE χ − α + 2 A l J + A l J E χ + O ( χ − α ) (cid:21) , (261)whence | k l ( χ ) | = 32( l − (cid:20) − A l J DE χ − α − A l J J + A l J/ E χ + O ( χ − α ) (cid:21) (262)and | k l ( χ ) | − = 2( l − (cid:20) A l J DE χ − α + A l J J + A l J/ E χ + O ( χ − α ) (cid:21) , (263)the expansions being valid for χ J/E ≪ A l , i.e., for χ τ M ≪ A l .Rewriting (68) as¯ k l ( χ ) = 32( l − (cid:16) R e h J ( χ ) i + i I m h J ( χ ) i(cid:17) (cid:16) R e h J ( χ ) i + A l J − i I m h J ( χ ) i(cid:17)(cid:16) R e h J ( χ ) i + A l J (cid:17) + (cid:16) I m h J ( χ ) i(cid:17) , (264)we extract its real part: R e (cid:2) ¯ k l ( χ ) (cid:3) = 32( l − (cid:16) R e h J ( χ ) i(cid:17) + (cid:16) I m h J ( χ ) i(cid:17) + A l J R e h J ( χ ) i(cid:16) R e h J ( χ ) i + A l J (cid:17) + (cid:16) I m h J ( χ ) i(cid:17) = 32( l − − A l J R e h J ( χ ) i + A l J (cid:16) R e h J ( χ ) i + A l J (cid:17) + (cid:16) I m h J ( χ ) i(cid:17) . (265)Insertion of the expressions for R e h J ( χ ) i and I m h J ( χ ) i into the latter formula entails: R e (cid:2) ¯ k l ( χ ) (cid:3) = 32( l − (cid:20) − A l J DE χ − α − A l J J + A l JE χ + O ( χ − α ) (cid:21) . (266)Expressions (263) and (266) enable us to write down the cosine of the shape lag:cos ǫ l = R e [ k l ( χ )] | k l ( χ ) | = 1 − (cid:18) A l JE (cid:19) χ + O ( χ − α ) = 1 − A l ( τ M χ ) + O ( χ − α ) . (267)Comparing this expression with (262), we see that, for the Andrade ( α = 0) model, theevolution of k l ( χ ) ≡ | ¯ k ( χ ) | in the limit of small χ , unfortunately, cannot be approximated witha convenient expression k l ( χ ) ≈ k l (0) cos ǫ ( χ ) , which is valid for simpler models (like the oneof Kelvin-Voigt or SAS).However, for the Maxwell model ( β = 0) expression (262) becomes: | k l ( χ ) | = 32( l − (cid:20) − A l JE ( J + A l J/ χ + O ( χ ) (cid:21) = 32( l −
1) [1 − A l (1 + A l /
2) ( τ M χ ) + O ( χ )] , (268)62hich can be written as | k l ( χ ) | ≈ l − " − (cid:18) A l JE (cid:19) χ + O ( χ ) = 32( l − (cid:20) − A l ( τ M χ ) + O ( χ ) (cid:21) , (269)insofar as A l ≫ A l is small), the following convenient approximation isvalid, provided the Maxwell model is employed: k l ( χ ) ≈ k l (0) cos ǫ ( χ ) , for χ τ M ≪ A l where A l ≫ . (270) G The eccentricity functions
In our development, we take into account the expansions for G lpq ( e ) over the powers of eccen-tricity, keeping the terms up to e , inclusively. The table of the eccentricity functions presentedin the book by Kaula (1966) is not sufficient for our purposes, because some of the G lpq ( e )functions in that table are given with lower precision. For example, the e term is missing in theapproximation for G ( e ) . Besides, that table omits several functions which are of order e .So here we provide a more comprehensive table. The table is based on the information borrowedfrom Cayley (1861) who tabulated various expansions employed in astronomy. Among those,were series (cid:16) ra (cid:17) − ( l +1) (cid:20) [ cos ] i cos[ sin ] i sin (cid:21) jν = ∞ X i = − ∞ (cid:20) cossin (cid:21) i M , (271) ν and M signifying the true and mean anomalies, while [ cos ] i and [ sin ] i denoting the co-efficients tabulated by Cayley. These coefficients are polynomials in the eccentricity. Cayley’sinteger indices i , j are connected with Kaula’s integers l , p , q via l − p = j , l − p + q = i . (272)With the latter equalities kept in mind, the eccentricity functions, for i ≥ G lpq ( e ) = [ cos ] i + [ sin ] i , for i ≥ . (273)To obtain the eccentricity functions for i < − i = [ cos ] i ,while [ sin ] i = − [ sin ] i . It is then possible to demonstrate that G lpq ( e ) = [ cos ] i − [ sin ] i , for i < . (274)63hen the following expressions, for l = 2 , ensue from Cayley’s tables: G − ( e ) = G − ( e ) = G − ( e ) = G − ( e ) = 0 , (275a) G − ( e ) = 15625129024 e , (275b) G − ( e ) = 445 e , (275c) G − ( e ) = 811280 e + 812048 e , (275d) G − ( e ) = 124 e + 7240 e , (275e) G − ( e ) = 148 e + 11768 e + 31330720 e , (275f) G − ( e ) = 0 , (275g) G − ( e ) = − e + 116 e − e − e , (275h) G ( e ) = 1 − e + 1316 e − e , (275i) G ( e ) = 72 e − e + 489128 e − e (275j) G ( e ) = 172 e − e + 60148 e , (275k) G ( e ) = 84548 e − e + 2082256144 e , (275l) G ( e ) = 53316 e − e , (275m) G ( e ) = 2283473840 e − e , (275n) G ( e ) = 73369720 e , (275o) G ( e ) = 1214427371680 e , (275p)the other values of q generating polynomials G q ( e ) whose leading terms are of order e andhigher.Since in our study we intend to employ the squares of these functions, with terms up to e only, then we may completely omit the eccentricity functions with | q | ≥ G − ( e ) = 12304 e + O ( e ) , (276a) G − ( e ) = 0 , (276b) G − ( e ) = 14 e − e + 13768 e + O ( e ) , (276c) G ( e ) = 1 − e + 638 e − e + O ( e ) , (276d) G ( e ) = 494 e − e + 21975256 e + O ( e ) , (276e) G ( e ) = 2894 e − e + O ( e ) , (276f) G ( e ) = 7140252304 e + O ( e ) , (276g)the squares of the others being of the order of e or higher.Be mindful that, for l = 2 we considered only the functions with p = 0 . This is dictatedby the fact that the inclination functions F lmp = F p are of order i (and, accordingly, theirsquares and cross-products are of order i ) for all the values of p except zero.For l = 3 , the situation changes. The inclination functions F lmp = F , F , F , F ,F , F , F , F , F , F are of the order O ( i ) or higher. The terms containing thesquares or cross-products of the these functions may thus be omitted. (Specifically, by neglectingthe cross-terms we get rid of the mixed-period part of the l = 3 component.) What is left is theterms with lmp = 311 and lmp = 330 . These terms contain the squares of functions F ( i ) = −
32 + O ( i ) and F ( i ) = 15 + O ( i ) , (277)accordingly. From here, we see that, for l = 3 , we shall need to employ the eccentricity functions G lpq ( e ) = G q ( e ) and G lpq ( e ) = G q ( e ) .The following expressions, for l = 3 and p = 0 , ensue from Cayley’s tables: G − ( e ) = G − ( e ) = G − ( e ) = G − ( e ) = 0 , (278a) G − ( e ) = 8315 e , (278b) G − ( e ) = 815120 e , (278c) G − ( e ) = 1120 e + 131440 e , (278d) G − ( e ) = 1384 e + 1384 e , (278e)65 − ( e ) = 0 , (278f) G − ( e ) = 18 e + 148 e + 553072 e , (278g) G − ( e ) = − e + 54 e − e + 23288 e , (278h) G ( e ) = 1 − e + 42364 e − e , (278i) G ( e ) = 5 e − e + 60724 e − e (278j) G ( e ) = 1278 e − e + 2438053072 e , (278k) G ( e ) = 1634 e − e + 10895 e , (278l) G ( e ) = 35413384 e − e , (278m) G ( e ) = 23029120 e − e , (278n) G ( e ) = 3850951024 e , (278o) G ( e ) = 4437763 e , (278p)the other values of q generating polynomials G q ( e ) and G q ( e ) , whose leading terms are oforder e and higher.The squares of some these functions, will read, up to e terms inclusively, as: G − ( e ) = 0 , (279a) G − ( e ) = 164 e + 1192 e + O ( e ) , (279b) G − ( e ) = e − e + 8948 e + O ( e ) , (279c) G ( e ) = 1 − e + 157532 e − e + O ( e ) , (279d) G ( e ) = 25 e − e + 884312 e + O ( e ) , (279e) G ( e ) = 1612964 e − e + O ( e ) , (279f) G ( e ) = 2656916 e + O ( e ) , (279g)the squares of the others being of the order of e or higher.66inally, we write down the expressions for the eccentricity functions with l = 3 and p = 1 : G − ( e ) = G − ( e ) = 0 , (280a) G − ( e ) = 163372240 e , (280b) G − ( e ) = 482039216 e , (280c) G − ( e ) = 899240 e + 2441480 e , (280d) G − ( e ) = 343128 e + 2819640 e , (280e) G − ( e ) = 2312 e + 8924 e + 5663960 e , (280f) G − ( e ) = 118 e + 4916 e + 156653072 e , (280g) G − ( e ) = e + 52 e + 358 e + 10516 e , (280h) G ( e ) = 1 + 2 e + 23964 e + 3323576 e , (280i) G ( e ) = 3 e + 114 e + 24548 e + 46364 e (280j) G ( e ) = 538 e + 3916 e + 70411024 e , (280k) G ( e ) = 776 e − e + 4751480 e , (280l) G ( e ) = 2955128 e − e , (280m) G ( e ) = 316780 e − e , (280n) G ( e ) = 302463746080 e , (280o) G ( e ) = 1783311680 e , (280p)67nd the squares: G − ( e ) = 529144 e + O ( e ) , (281a) G − ( e ) = 12164 e + 53964 e + O ( e ) , (281b) G − ( e ) = e + 5 e + 15 e + O ( e ) , (281c) G ( e ) = 1 + 4 e + 36732 e + 7625288 e + O ( e ) , (281d) G ( e ) = 9 e + 332 e + 61116 e + O ( e ) , (281e) G ( e ) = 280964 e + 206764 e + O ( e ) , (281f) G ( e ) = 592936 e + O ( e ) , (281g)the squares of the other functions from this set being of the order e or higher. H The l = 2 and l = 3 terms of the secular part of thetorque H.1 The l = 2 terms of the secular torque Extracting the l = 2 input from (113), we recall that only the ( lmpq ) = (220 q ) terms matter.Out of these terms, we need only the ones up to e . These are the terms with | q | ≤ T l =2 = T ( lmp )=(220) + O ( i ǫ ) = 32 G M sec R a − X q = − G q ( e ) k sin ǫ q + O ( e ǫ ) + O ( i ǫ ) (282a)= 32 G M sec R a − (cid:20) e k sin ǫ − + (cid:18) e − e + 13768 e (cid:19) k sin ǫ − + (cid:18) − e + 638 e − e (cid:19) k sin ǫ + (cid:18) e − e + 21975256 e (cid:19) k sin ǫ + (cid:18) e − e (cid:19) k sin ǫ + 7140252304 e k sin ǫ (cid:21) + O ( e ǫ ) + O ( i ǫ ) , (282b)where the absolute error O ( e ǫ ) has emerged because of our neglect of terms with | q | ≥ O ( i ǫ ) came into being after the truncation of terms with p ≥ k l sin ǫ lmpq will be rewritten as:68 l sin | ǫ lmpq | sgn h ( l − p + q ) n − m ˙ θ i . This will render: T l =2 = 32 G M sec R a − (cid:20) e k sin | ǫ − | sgn (cid:16) − n − θ (cid:17) + (cid:18) e − e + 13768 e (cid:19) k sin | ǫ − | sgn (cid:16) n − θ (cid:17) + (cid:18) − e + 638 e − e (cid:19) k sin | ǫ | sgn (cid:16) n − ˙ θ (cid:17) + (cid:18) e − e + 21975256 e (cid:19) k sin | ǫ | sgn (cid:16) n − θ (cid:17) + (cid:18) e − e (cid:19) k sin | ǫ | sgn (cid:16) n − ˙ θ (cid:17) + 7140252304 e k sin | ǫ | sgn (cid:16) n − θ (cid:17) (cid:21) + O ( e ǫ ) + O ( i ǫ ) , (283) H.2 The l = 3 , m = 1 terms of the secular torque Getting the l = 3 , m = 1 input from (113) and leaving in it only the terms up to e , we obtain,with aid of formulae (276) from Appendix G, the following expression: T ( lmp )=(311) = 38 G M sec R a − X q = − G q ( e ) k sin ǫ q + O ( e ǫ ) (284a)= 38 G M sec R a − (cid:20) e k sin ǫ − + (cid:18) e + 53964 e (cid:19) k sin ǫ − + (cid:0) e + 5 e + 15 e (cid:1) k sin ǫ − + (cid:18) e + 36732 e + 7625288 e (cid:19) k sin ǫ + (cid:18) e + 332 e + 61116 e (cid:19) k sin ǫ + (cid:18) e + 206764 e (cid:19) k sin ǫ + 592936 e k sin ǫ (cid:21) + O ( e ǫ ) . (284b)69ith the signs depicted explicitly, this will look: T ( lmp )=(311) = 38 G M sec R a − (cid:20) e k sin | ǫ − | sgn (cid:16) − n − ˙ θ (cid:17) + (cid:18) e + 53964 e (cid:19) k sin | ǫ − | sgn (cid:16) − n − ˙ θ (cid:17) + (cid:0) e + 5 e + 15 e (cid:1) k sin | ǫ − | sgn (cid:16) − ˙ θ (cid:17) + (cid:18) e + 36732 e + 7625288 e (cid:19) k sin | ǫ | sgn (cid:16) n − ˙ θ (cid:17) + (cid:18) e + 332 e + 61116 e (cid:19) k sin | ǫ | sgn (cid:16) n − ˙ θ (cid:17) + (cid:18) e + 206764 e (cid:19) k sin | ǫ | sgn (cid:16) n − ˙ θ (cid:17) + 592936 e k sin | ǫ | sgn (cid:16) n − ˙ θ (cid:17) (cid:21) + O ( e ǫ ) . (285) H.3 The l = 3 , m = 3 terms of the secular torque The second relevant group of terms with l = 3 will read: T ( lmp )=(330) = 158 G M sec R a − X q = − G q ( e ) k sin ǫ q + O ( e ǫ ) (286a)= 158 G M sec R a − (cid:20) (cid:18) e + 1192 e (cid:19) k sin ǫ − + (cid:18) e − e + 8948 e (cid:19) k sin ǫ − + (cid:18) − e + 157532 e − e (cid:19) k sin ǫ + (cid:18) e − e + 884312 e (cid:19) k sin ǫ + (cid:18) e − e (cid:19) k sin ǫ + 2656916 e k sin ǫ (cid:21) + O ( e ǫ ) (286b)70r, with the signs shown explicitly: T ( lmp )=(330) = 158 G M sec R a − (cid:20) (cid:18) e + 1192 e (cid:19) k sin | ǫ − | sgn (cid:16) − n − ˙ θ (cid:17) + (cid:18) e − e + 8948 e (cid:19) k sin | ǫ − | sgn (cid:16) − ˙ θ (cid:17) + (cid:18) − e + 157532 e − e (cid:19) k sin | ǫ | sgn (cid:16) n − ˙ θ (cid:17) + (cid:18) e − e + 884312 e (cid:19) k sin | ǫ | sgn (cid:16) n − ˙ θ (cid:17) + (cid:18) e − e (cid:19) k sin | ǫ | sgn (cid:16) n − ˙ θ (cid:17) + 2656916 e k sin | ǫ | sgn (cid:16) n − ˙ θ (cid:17) (cid:21) + O ( e ǫ ) . (287) I The l = 2 and l = 3 terms of the short-period part ofthe torque The short-period part of the torque may be approximated with terms of degrees 2 and 3: e T = e T l =2 + e T l =3 + O (cid:0) ǫ ( R/a ) (cid:1) = e T ( lmp )=(220) + h e T ( lmp )=(311) + e T ( lmp )=(330) i + O ( ǫ i ) + O (cid:0) ǫ ( R/a ) (cid:1) , (288)where e T ( lmp )=(220) = 3 G M sec R a − X q = − X j = − j < q G q ( e ) G j ( e ) n cos [ M ( q − j ) ] k sin ǫ q − sin [ M ( q − j ) ] k cos ǫ q o + O ( i ǫ ) + O ( e ǫ ) , (289a)= − G M sec R a − X q = − X j = − j < q G q ( e ) G j ( e ) k sin [ M ( q − j )] + O ( i ǫ ) + O ( eǫ ) , (289b)71 T ( lmp )=(311) = 34 G M sec R a − X q = − X j = − j < q G q ( e ) G j ( e ) n cos [ M ( q − j ) ] k sin ǫ q − sin [ M ( q − j ) ] k cos ǫ q o + O ( i ǫ ) + O ( e ǫ ) , (290a)= − G M sec R a − X q = − X j = − j < q G q ( e ) G j ( e ) k sin [ M ( q − j )] + O ( i ǫ ) + O ( eǫ ) , (290b) e T ( lmp )=(330) = 154 G M sec R a − X q = − X j = − j < q G q ( e ) G j ( e ) cos [ M ( q − j ) ] k sin ǫ q − sin [ M ( q − j ) ] k cos ǫ q o + O ( i ǫ ) + O ( e ǫ ) , (291a)= − G M sec R a − X q = − X j = − j < q G q ( e ) G j ( e ) k sin [ M ( q − j )] + O ( i ǫ ) + O ( eǫ ) , (291b)the expressions for the eccentricity functions being provided in Appendix G. The overall nu-merical factors in (289 - 291) are twice the numerical factors in (114), because in (289 - 291)we have j < q and not j = q . The right-hand sides of (289 - 291) contain O ( eǫ ) in-stead of O ( e ǫ ) , because at the final step we approximated cos [ M ( q − j ) ] k l sin ǫ lmpq − sin [ M ( q − j ) ] k l cos ǫ lmpq simply with − sin [ M ( q − j ) ] k l . Doing so, we replaced thecosine with unity, because the entire Darwin-Kaula formalism is a linear approximation in thelags. We also neglected k l sin ǫ lmpq and kept only the leading term with k l . This neglect wouldbe illegitimate in the secular part of the torque, but is probably acceptable in the purely short-period part, because the latter part has a zero average and therefore should be regarded as asmall correction even in its leading order. The latter circumstance also will justify approximationof k l = k l ( χ ) with k l (0) in (289 - 291). References [1] Alterman, Z.; Jarosch, H.; and Pekeris, C. 1959. “Oscillations of the Earth.”
Proceedings ofthe Royal Society of London, Series A , Vol. , pp. 80 - 95.[2] Andrade, E. N. da C. 1910. “On the Viscous Flow in Metals, and Allied Phenomena.”
Pro-ceedings of the Royal Society of London. Series A.
Vol. , pp. 1 - 12[3] Benjamin, D.; Wahr, J. ; Ray, R. D.; Egbert, G. D.; and Desai, S. D. 2006. “Constraintson mantle anelasticity from geodetic observations, and implications for the J anomaly.” Geophysical Journal International , Vol. , pp. 3 - 16724] Bills, B. G.; Neumann, G. A.; Smith, D.E.; and Zuber, M.T. 2005. “Improved estimate oftidal dissipation within Mars from MOLA observations of the shadow of Phobos.”
Journal ofGeophysical Research , Vol. , pp. 2376 - 2406. doi:10.1029/2004JE002376, 2005[5] Biot, M. A. 1954. “Theory of Stress-Strain Relaxation in Anisotropic Viscoelasticity andRelaxation Phenomena.”
Journal of Applied Physics , Vol. Linear thermodynamics and the mechanics of solids.
Izvestiya. Physics of the Solid Earth.
Vol. , pp. 635 - 641[8] Castillo-Rogez, J. 2009. “New Approach to Icy Satellite Tidal Response Modeling.” AmericanAstronomical Society, DPS meeting 41, 61.07.[9] Castillo-Rogez, J. C.; Efroimsky, M., and Lainey, V. 2011. “The tidal history of Iapetus.Dissipative spin dynamics in the light of a refined geophysical model”. Journal of GeophysicalResearch – Planets , Vol. , p. E09008doi:10.1029/2010JE003664[10] Castillo-Rogez, J. C., and Choukroun, M. 2010. “Mars’ Low Dissipation Factor at 11-h.Interpretation from an Anelasticity-Based Dissipation Model.” American Astronomical Society,DPS Meeting No 42, Abstract 51.02.
Bulletin of the American Astronomical Society , Vol. ,p. 1069[11] Churkin, V. A. 1998. “The Love numbers for the models of inelastic Earth.” Preprint No121. Institute of Applied Astronomy. St.Petersburg, Russia. /in Russian/[12] Correia, A. C. M., and Laskar, J. 2009. “Mercury’s capture into the 3 / Icarus , Vol. , pp. 1 - 11[13] Correia, A. C. M., and Laskar, J. 2004. “Mercury’s capture into the 3 / Nature , Vol. , pp. 848 - 850[14] Cottrell, A. H., and Aytekin, V. 1947. “Andrade’s creep law and the flow of zinc crystalls.”
Nature , Vol. , pp. 328 - 329[15] Dahlen, F. A. 1976. “The passive influence of the oceans upon rotation of the Earth.”
Geophys. Journal of the Royal Astronomical Society.
Vol. , pp. 363 - 406[16] Darwin, G. H. 1879. “On the precession of a viscous spheroid and on the remote history ofthe Earth.” Philosophical Transactions of the Royal Society of London , Vol.
Philosophical Transactions of the Royal Society ofLondon , Vol.
Geophysical J. International , Vol. , pp. 563 - 5727319] Dehant V. 1987a. “Tidal parameters for an inelastic Earth.”
Physics of the Earth and Plan-etary Interiors , Vol. , pp. 97 - 116[20] Dehant V. 1987b. “Integration of the gravitational motion equations for an elliptical uni-formly rotating Earth with an inelastic mantle.” Physics of the Earth and Planetary Interiors ,Vol. , pp. 242 - 258[21] Duval, P. 1976. “Temporary or permanent creep laws of polycrystalline ice for different stressconditions.” Annales de Geophysique , Vol. , pp. 335 - 350[22] Eanes, R. J. 1995. A study of temporal variations in Earth’s gravitational field usingLAGEOS-1 laser ranging observations . PhD thesis, University of Texas at Austin[23] Eanes, R. J., and Bettadpur, S. V. 1996. “Temporal variability of Earth’s gravitational fieldfrom laser ranging.” In: Rapp, R. H., Cazenave, A. A., and Nerem, R. S. (Eds.)
Global gravityfield and its variations. Proceedings of the International Association of Geodesy SymposiumNo 116 held in Boulder CO in July 1995.
IAG Symposium Series. Springer 1997ISBN: 978-3-540-60882-0[24] Efroimsky, M., and V. Lainey. 2007. “The Physics of Bodily Tides in Terrestrial Planets, andthe Appropriate Scales of Dynamical Evolution.”
Journal of Geophysical Research – Planets ,Vol. , p. E12003. doi:10.1029/2007JE002908[25] Efroimsky, M., and Williams, J. G. 2009. “Tidal torques. A critical review of some tech-niques.”
Celestial mechanics and Dynamical Astronomy,
Vol. , pp. 257 - 289arXiv:0803.3299[26] Efroimsky, M. 2012. “Tidal dissipation compared to seismic dissipation: in small bodies,earths, and superearths.” the Astrophysical Journal , Vol. , No 2, p. 150doi:10.1088/0004-637X/746/2/150arXiv:1105.3936[27] Ferraz-Mello, S.; Rodr´ıguez, A.; and Hussmann, H. 2008. “Tidal friction in close-in satellitesand exoplanets: The Darwin theory re-visited.”
Celestial mechanics and Dynamical Astron-omy,
Vol. , pp. 171 - 201[28] Fontaine, F. R.; Ildefonse, B.; and Bagdassarov, N. 2005. “Temperature dependence of shearwave attenuation in partially molten gabbronorite at seismic frequencies.”
Geophysical JournalInternational , Vol. , pp. 1025 - 1038[29] Goldreich, P. 1963. “On the eccentricity of the satellite orbits in the Solar System.”
MonthlyNotices of the Royal Astronomical Society of London , Vol. , pp. 257 - 268[30] Gooding, R.H., and Wagner, C.A. 2008. “On the inclination functions and a rapid stableprocedure for their evaluation together with derivatives.”
Celestial Mechanics and DynamicalAstronomy , Vol. , pp. 247 - 272[31] Gribb, T.T., and Cooper, R.F. 1998. “Low-frequency shear attenuation in polycrystallineolivine: Grain boundary diffusion and the physical significance of the Andrade model forviscoelastic rheology.” Journal of Geophysical Research – Solid Earth, Vol. , pp. 27267 -27279[32] Haddad, Y. M. 1995.
Viscoelasticity of Engineering Materials.
Chapman and Hall, LondonUK, p. 279 7433] Hut, P. 1981. “Tidal evolution in close binary systems.”
Astronomy and Astrophysics , Vol. , pp. 126 - 140[34] Karato, S.-i. 2008. Deformation of Earth Materials. An Introduction to the Rheology of SolidEarth . Cambridge University Press, UK.[35] Karato, S.-i., and Spetzler, H. A. 1990. “Defect Microdynamics in Minerals and Solid-StateMechanisms of Seismic Wave Attenuation and Velocity Dispersion in the Mantle.”
Reviews ofGeophysics , Vol. , pp. 399 - 423[36] Kaula, W. M. 1961. “Analysis of gravitational and geometric aspects of geodetic utilisationof satellites.” The Geophysical Journal , Vol. , pp. 104 - 133[37] Kaula, W. M. 1964. “Tidal Dissipation by Solid Friction and the Resulting Orbital Evolu-tion.” Reviews of Geophysics , Vol. , pp. 661 - 684[38] Kaula, W. M. 1966. Theory of Satellite Geodesy: Applications of Satellites to Geodesy.
Blaisdell Publishing Co, Waltham MA. (Re-published in 2006 by Dover. ISBN: 0486414655.)[39] Landau, L., and Lifshitz, E. M. 1986.
The Theory of Elasticity.
Pergamon Press, Oxford1986.[40] Landau, L., and Lifshitz, E. M. 1987.
Fluid Mechanics.
Pergamon Press, Oxford 1987.[41] Legros, H.; Greff, M.; and Tokieda, T. 2006 “Physics inside the Earth. Deformation andRotation.”
Lecture Notes in Physics , Vol. , pp. 23 - 66. Springer, Heidelberg[42] Love, A. E. H. 1909. “The Yielding of the Earth to Disturbing Forces.”
Proceedings of theRoyal Society of London. Series A,
Vol. , pp. 73 - 88[43] Love, A. E. H. 1911. Some problems of geodynamics.
Cambridge University Press, London.Reprinted by: Dover, NY 1967.[44] MacDonald, G. J. F. 1964. “Tidal Friction.”
Reviews of Geophysics.
Vol. , pp. 467 - 541[45] Matsuyama, I., and Bills, B. G. 2010. “Global contraction of planetary bodies due to de-spinning. Application to Mercury and Iapetus.” Icarus , Vol. , pp. 271 - 279[46] McCarthy, C.; Goldsby, D. L.; and Cooper, R. F. 2007. “Transient and Steady-State CreepResponses of Ice-I/Magnesium Sulfate Hydrate Eutectic Aggregates.” , held on 12 - 16 March 2007 in League City, TX. LPIContribution No 1338, p. 2429[47] Mignard, F. 1979. “The Evolution of the Lunar Orbit Revisited. I.”
The Moon and thePlanets.
Vol. , pp. 301 - 315.[48] Mignard, F. 1980. “The Evolution of the Lunar Orbit Revisited. II.” The Moon and thePlanets.
Vol. , pp. 185 - 201[49] Miguel, M.-C.; Vespignani, A.; Zaiser, M.; and Zapperi, S. 2002. “Dislocation Jamming andAndrade Creep.” Physical Review Letters , Vol. , pp. 165501 - 1655[50] Mitchell, B. J. 1995. “Anelastic structure and evolution of the continental crust and uppermantle from seismic surface wave attenuation.” Reviews of Geophysics , Vol. , No 4, pp. 441- 462. 7551] Munk, W. H., and MacDonald, G. J. F. 1960. The rotation of the earth; a geophysicaldiscussion.
Cambridge University Press, 323 pages.[52] Nakamura, Y.; Latham, G.; Lammlein, D.; Ewing, M.; Duennebier, F.; and Dorman, J.1974. “Deep lunar interior inferred from recent seismic data.”
Geophysical Research Letters ,Vol. , pp. 137 - 140[53] Nechada, H.; Helmstetterb, A.; El Guerjoumaa, R.; and Sornette, D. 2005. “Andrade andcritical time-to-failure laws in fiber-matrix composites. Experiments and model.” Journal ofthe Mechanics and Physics of solids , Vol. Geophysical Research Letters , Vol. , p. L04202doi:10.1029/2009GL041465[56] Remus, F.; Mathis, S.; and Zahn, J.-P. 2012a. “The Equilibrium Tide in Stars and GiantPlanets. I - The Coplanar Case.” Submitted to: Astronomy & Astrophysics [57] Remus, F.; Mathis, S.; Zahn, J.-P.; and Lainey, V. 2012b. “Anelastic tidal dissipation inmulti-layers planets.” Submitted to:
Astronomy & Astrophysics [58] Remus, F.; Mathis, S.; Zahn, J.-P.; and Lainey, V. 2011. “The elasto-viscous equilibriumtide in exoplanetary systems.” EPSC-DPS Joint Meeting 2011, Abstract 1372.[59] Rodr´ıguez; A., Ferraz Mello, S.; and Hussmann, H. 2008. “Tidal friction in close-in planets.”In: Y.S.Sun, S.Ferraz-Mello and J.L.Zhou (Eds.)
Exoplanets: Detection, Formation and Dy-namics. Proceedings of the IAU Symposium No 249, pp. 179 - 186doi:10.1017/S174392130801658X[60] Sabadini, R., and Vermeersen, B. 2004.
Global Dynamics of the Earth: Applications ofNormal Mode Relaxation Theory to Solid-Earth Geophysics.
Kluwer, Dordrecht 2004[61] Shito, A.; Karato, S.-i.; and Park, J. 2004. “Frequency dependence of Q in Earth’s uppermantle, inferred from continuous spectra of body wave.” Geophysical Research Letters , Vol. , No 12, p. L12603, doi:10.1029/2004GL019582[62] Singer, S. F. 1968. “The Origin of the Moon and Geophysical Consequences.” The Geophys-ical Journal of the Royal Astronomical Society , Vol. , pp. 205 - 226[63] Smith, M. 1974. “The scalar equations of infinitesimal elastic-gravitational motion for arotating, slightly elliptical Earth.” The Geophysical Journal of the Royal Astronomical Society ,Vol. , pp. 491 - 526[64] Stachnik, J. C.; Abers, G. A.; and Christensen, D. H. 2004. “Seismic attenuation and mantlewedge temperatures in the Alaska subduction zone.” Journal of Geophysical Research – SolidEarth , Vol. , No B10, p. B10304, doi:10.1029/2004JB003018[65] Tan, B. H.; Jackson, I.; and Fitz Gerald J. D. 1997. “Shear wave dispersion and attenuationin fine-grained synthetic olivine aggregates: preliminary results.”
Geophysical Research Letters ,Vol. , No 9, pp. 1055 - 1058, doi:10.1029/97GL008607666] Taylor, P. A., and Margot, J.-L. 2010. “Tidal evolution of close binary asteroid systems.” Celestial Mechanics and Dynamical Astronomy , Vol. , pp. 315 - 338[67] Wahr, J.M. 1981a. “A normal mode expansion for the forced response of a rotating Earth.”
The Geophysical Journal of the Royal Astronomical Society , Vol. , pp. 651 - 675[68] Wahr, J.M. 1981b. “Body tides on an elliptical, rotating, elastic and oceanless Earth.” TheGeophysical Journal of the Royal Astronomical Society , Vol. , pp. 677 - 703[69] Wahr, J.M. 1981c. “The forced nutations of an elliptical, rotating, elastic and oceanlessEarth.” The Geophysical Journal of the Royal Astronomical Society , Vol. , pp. 705 - 727[70] Weertman, J., and Weertman, J. R. 1975. “High Temperature Creep of Rock and MantleViscosity.” Annual Review of Earth and Planetary Sciences , Vol. , pp. 293 - 315[71] Weber, R. C.; Lin, Pei-Ying; Garnero, E.; Williams, Q.; and Lognonn´e, P. 2011. “SeismicDetection of the Lunar Core.” Science , Vol. , Issue 6015, pp. 309 - 312[72] Williams, J. G., Boggs, D. H., Yoder, C. F., Ratcliff, J. T., and Dickey, J. O. 2001. “Lunarrotational dissipation in solid-body and molten core.”
The Journal of Geophysical Research –Planets , Vol.
Celestial Mechanics and DynamicalAstronomy . arXiv:........[76] Yoder, C. 1982. “Tidal Rigidity of Phobos”.
Icarus , Vol. , pp. 327 - 346[77] Zahn, J.-P. 1966. “Les mar´ees dans une ´etoile double serr´ee.” Annales d’Astrophysique,