Damped Mass-spring model Simulations of Tidal Spin up in a Kelvin-Voigt viscoelastic sphere

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1 Preprint 8 December 2015 Compiled using MNRAS LATEX style file v3.0 Damped Mass-spring model Simulations of Tidal Spin up in a Kelvin-Voigt viscoelastic sphere Julien Frouard 1, Alice C. Quillen 2, Michael Efroimsky 1, David Giannella 2 1 US Naval Observatory, 3450 Massachusetts Ave NW, Washington DC USA 2 Department of Physics and Astronomy, University of Rochester, Rochester, NY USA 8 December 2015 ABSTRACT We use a damped mass spring model within an N-body code to simulate the tidal spin up of a viscoelastic spherical satellite and the semi-major axis decay of a massive perturber. The damped mass spring model represents a Kelvin-Voigt viscoelastic solid. Tidal spin down rates are measured directly from semi-major axis drift within the simulation. We confirm predictions by Efroimsky et al. using a rheological model for the spinning body that the quality function (potential Love number times phase angle or k 2 /Q) is a function of frequency, and peaks at approximately the viscoelastic relaxation time scale (ratio of viscosity to elastic modulus). This study shows that we can directly simulate the tidal evolution of viscoelastic bodies and offers the possibility of easily investigate inhomogeneous and anisotropic bodies with non-spherical shapes. Key words: planets and satellites: dynamical evolution and stability, interiors methods: numerical 1 INTRODUCTION The theory of tidal dissipation in solid planetary bodies has a long and rich history, beginning with a clear mathematical formulation offered by Darwin (1879). The long timescales associated with tidal evolution make direct observations of the phenomenon difficult, requiring either high observational precision (for the Earth-Moon system: see for example Williams & Boggs (2015)) or an extended observational timespan (Lainey et al. 2012). It is a challenge to incorporate tidal evolution into numerical simulations, not by directly adding tidal forces or torques, but by modeling the continuum mechanics problem in such a way that tidal dissipation causes spin down and orbital drift, and is a natural outcome of the simulation. Efficient simulations would allow us to test and confirm analytical predictions for the tidal torque on a spinning body. If we can efficiently and accurately simulate the tidal evolution of a uniform body, then we can go further and use simulations to study tidal evolution of inhomogeneous non-spherical bodies, bodies with inhomogeneous material properties and non-linear rheologies, allowing study of tidal evolution in settings that are more difficult to predict analytically. In a mass-spring model simulation, massive particles are connected with a network of massless springs. Because of their simplicity and speed, mass-spring models are a popular Contact jfrouard@federatedit.com 2 QUALITY FUNCTION FOR A KELVIN-VOIGT VISCOELASTIC SOLID method for simulating soft elastic bodies (e.g., Nealen et al. 2006). Ostoja-Starzewski (2002); Kot et al. (2014) have shown that mass spring models can accurately model elastic materials. Because mass-spring models are a particle based simulation technique, they can be incorporated into N-body simulations and viscoelastic material properties simulated simply and rapidly along with gravitational forces. After a presentation of the theory of tides for viscoelastic rheologies in section 2, we will present in section 3 the mass-spring simulation model. We compare semi-major axis drift rates measured from the simulations to the predictions of tidal theory. We will conclude this study in the last section. We consider a near-spherical extended body of mass M and radius R, tidally deformed by a perturber of mass M residing at a position r, where r R. The binary orbit has semi-major axis a with mean motion n = G(M + M )/a 3 where G is the gravitational constant. For a distant perturber, a R, the quadrupole term in the expansion for the perturbing potential W is dominant and this depends on the semi-diurnal frequency ω = 2(n θ) (1) c 0000 The Authors

2 2 Frouard et al. where θ is the body s spin in the orbital plane. We refer the reader to the appendix for a derivation that includes the other Fourier components (Equation 1 is equation A14). Considering the quadrupole component alone, the torque on the spinning body T = 3 2 GM 2 R 5 k2(ω) sin ɛ2(ω) (2) a6 (following equation A15) where k 2 is a unit-less Love number and ɛ 2 is a phase lag. The product k 2 sin ɛ 2 is often called the quality function (Makarov 2013; Efroimsky 2015) or kvalitet (Makarov 2015; Makarov et al. 2016), see section A4. Conservation of momentum allows an estimate of the secular drift rate in semi-major axis at low inclination and eccentricity from the spin down rate (see section A6) ȧ n a = 2 T a GM M ( ) ( ) M 5 R = 3 k 2(ω) sin ɛ 2(ω), (3) M a (This is equation A17 in the appendix). The quality function depends on the composition and rheology of the body. Below we measure ȧ in a simulation and compare it to that predicted using a homogenous Kelvin-Voigt viscoelastic rheology. 2.1 The shape of the quality function Taking into account the indices of the Fourier components, the quality function is more generally written k l (ω lmpq ) sin ɛ l (ω lmpq ) (as described in the appendix). The form of a quality function depends on the degree l, the composition of the body, the rheology of its layers, and the overall size and mass of the body. The size and mass are important, because the tidal response is defined not only by the internal structure and rheology, but also by self-gravitation. The process of deriving the quality function for any linear viscoelastic rheology is described by Efroimsky (2015) and we introduce that derivation here. Starting from the expression of the static Love number for an homogeneous, incompressible, self-gravitating elastic sphere k (static) l = where 3 1 2(l 1) 1 + B l /J B l = 3(2l2 + 4l + 3) = 1 4π(2l 2 + 4l + 3), (5) 4lπGρ 2 R 2 e g 3l where e g GM 2 R 4 (6) is to order of magnitude the gravitational energy density of the satellite, J = 1/µ r is the compliance and µ r is the body s rigidity. We pass into the Fourier domain and use the equivalence principle for viscoelastic materials to obtain the complex Love number 3 1 k l (χ) = 2(l 1) 1 + B l / J = k l (χ) e iɛ l(χ) where χ ω is the tidal mode and J(χ) is the complex (4) (7) compliance of the material. Once J(χ) is prescribed with the choice of a rheological model, we can compute k l (χ) sin ɛ l (χ) = Im[ k l (χ)] (8) which gives [k l sin ɛ l ](ω) = 3 B l Im( ( ) J(χ)) 2(l 1) 2 Re( J(χ)) + Bl + ( Im( J(χ)) ) 2 sign ω (9) The shape of the quality function is very similar for different but linear viscoelastic materials and exhibits a sharp kink with two peaks having opposite signs (see Noyelles et al. 2014,Efroimsky 2015). From a frame rotating with the body spin, the tidal force of the perturber appears to rotate at the frequency ω. When ω is small compared to the viscoelastic relaxation time, the body deformation is almost exactly aligned with the tidal perturbation, there is no torque on the deformed body and so the quality function is very small. At the opposite high frequency limit, if the body is extremely viscous then it barely deforms during the period P o = 2π/ω. 2.2 Quality function for a Kelvin-Voigt solid The Kelvin-Voigt model for a viscoelastic solid can be represented by a purely viscous damper and purely elastic spring connected to two mass elements in parallel. If we connect these two elements in series rather than parallel, we have a Maxwell material model (see Figure 1). The Kelvin-Voigt rheology is easier to model within a mass-spring model as the spring and damping forces are both directly applied to each particle. This can also be seen from the general expression of the stress tensor σ pq as a function of the strain rate tensor ε pq in the time domain (see Efroimsky 2012b) σ pq(t) = 2 t µ(t t ) ε pq(t )dt (10) where µ(t t ) is the stress-relaxation function (also called the relaxation modulus) whose values for different linear rheologies is well-known and can be found in continuum mechanics textbooks. In the context of a mass-spring model where the force between particles is likened to an uniaxial stress, the norm of the normal force applied on the particle i due to the particle j is then: F i(t) = 2 t µ(t t ) ε(t )dt (11) The Kelvin-Voigt model has (see for example Mase et al. 2010) µ(t t ) = µ + ηδ(t t ) (12) where µ and η are the unrelaxed shear rigidity and viscosity of the link between i and j. Inserting that rheology into Eq.11, we obtain F i(t) = 2µε(t) 2µε( ) + 2η ε(t) (13) which, if we neglect the strain at t =, is equivalent to what we use in the code in Eqs , below, and what is used to model anelastic collisions in granular simulations of rubble piles (Richardson et al. 2009; Sánchez & Scheeres 2011). Unfortunately the Maxwell model cannot be integrated as easily within the context of a mass spring model.

3 Evaluating this for l = 2 (using equation 5) j ( χ, µ) = eg µ 3 1 i χ 1 + χ 2 (19) The above is a function of shear modulus µ in units of e g and frequency ω in units of the inverse of the relaxation time τ. Using our unitless frequency parameter χ equation 9 becomes Figure 1. Green circles are mass elements. In a Kelvin-Voigt model (top illustration) spring and damping forces are applied in parallel to the mass elements. In a Maxwell solid model (bottom illustration) the two forces are applied in series. Efroimsky (2015) predicted the quality function for the Maxwell model. Because it is easier to model a Kelvin-Voigt viscoelastic material with a massspring model, we compute the quality function for the Kelvin- Voigt model, allowing a direct comparison between predicted and measured (from simulations) quality functions. In a mass-spring model simulations, massive particles are connected with a network of massless springs. The rheology of a Kelvin-Voigt solid can be simulated with randomly distributed particles connected by damped springs (Lloyd et al. 2008; Quillen et al. 2015). To each particle that is connected to a spring, a damping force is applied that depends on the spring strain rate (see section 2.2 Quillen et al. 2015). The elastic modulus, µ I, can be computed for the isotropic random mass-spring model from the strength, lengths and distribution of the springs (Kot et al. 2014). The shear viscosity, η I, can be estimated from the strain rate dependent force law. The behavior of the simulated resolved body is that of a Kelvin-Voigt solid with elastic modulus µ = µ I, viscosity η = η I and relaxation time τ = η µ. (14) There may be a difference between the estimated values for any physical quantity for the simulated system than what is actually simulated (Kot et al. 2014). We will discuss this issue later when we compare measurements from the simulations to those predicted analytically. We now derive the quality function (Eq.9) for a Kelvin- Voigt rheology. Equation 9 depends on the complex compliance of a Kelvin-Voigt body J(χ, µ, η) µ iχη µ 2 + χ 2 η = 1 iχτ 2 µ 1 (15) 1 + χ 2 τ 2 where we can either describe J as a function of variables shear modulus and viscosity, µ, η or shear modulus and relaxation time, µ, τ. Using a unitless frequency χ ω τ (16) the complex compliance can be written J( χ, µ) µ 1 1 i χ 1 + χ 2 (17) We define a unitless function j l ( χ, µ) B 1 l J( χ, µ) = (B l µ) 1 1 i χ 1 + χ 2 (18) [k l sin ɛ l ]( χ, µ) = 3 Im( j l ( χ)) 2(l 1) (Re( j l ( χ)) + 1) 2 + (Im( j l ( χ))) 2 This function (equation 20) has extrema at ωpeak l = ± µb l + 1 = ± 1 ( ) B l η τ µb l For l = 2 this peak frequency can be written ( ) χ peak = ωpeak τ = eg µ sign ω (20) (21) For elastic bodies µ > e g otherwise they would collapse due to self-gravity. This implies that we expect ωpeakτ 1. Using shorthand y(µ, χ) = 0.025e gµ 1 (1 + χ 2 ) 1 equation 20 for l = 2 can be written [k 2 sin ɛ 2]( χ, µ) = 3 y χ (22) 2 y 2 (1 + χ 2 ) + 2y + 1 We have dropped the sign here, but it can be taken into account by allowing χ to have the sign of ω. Before we describe our numerical model in the next section, we discuss compressibility. The classic tidal theory is valid for incompressible materials only (Poisson ratio ν = 0.5), which is why the classic formulation of k 2 depends only on the shear modulus of rigidity. However mass-spring models approximate a material with Poisson s ratio ν = 0.25 (Kot et al. 2014). A.E.H. Love derived a general formula for the potential Love number k 2 for a compressible material 1 (Love 1911). We numerically checked that his formula for k 2 in the compressible case is only very weakly dependent on the Poisson s ratio for a large range of body size and density. So we are confident in the use of the classic tidal equations, directly substituting the shear modulus for the rigidity, when considering a compressible material with Poisson s ratio ν = DAMPED MASS-SPRING MODEL SIMULATIONS Because of their simplicity and speed, we use a mass-spring model simulation, rather than a finite-element method, to compare the tidal spin down rate of a simulated viscoelastic body to that predicted analytically. We use the massspring model by Quillen et al. (2015) that lies within the modular N-body code rebound (Rein & Liu 2012). For our simulations, we work in units of the extended body radius and mass R = 1, M = 1. Time is specified in units of t grav = R 3 /GM and we refer to this as a gravitational 1 But keeping in mind that the assumptions of homogeneity and compressibility of Love s theory are contradictory and that it can only be used as an approximation (Melchior 1972).

4 4 Frouard et al. with damping coefficient γ I that is equivalent to the inverse of a damping time scale. The damping parameter γ I is independent of k I. Because all our particles have the same mass, in equation 25 we do not use the reduced mass as did Quillen et al. (2015). How does the local spring constant k I and damping parameter γ I relate to the global rigidity and viscosity of the body? The Young s modulus (or elastic modulus) is computed following Kot et al. (2014) with a sum over the springs Figure 2. A snap shot of one of our simulations as viewed in the open-gl viewer of rebound. We only show the resolved body as the perturbing one is distant from it. time scale. Pressure, energy density and elastic moduli are given in units of GM 2 /R 4 or e g. In these units, the velocity of a massless particle in a circular orbit just grazing the surface of the body is 1, and the period of this orbit is 2π. For the resolved body, we randomly generate an initial spherical distribution of equal mass particles (as described in section 2.2 by Quillen et al. 2015), see Figure 2 for an illustration. Each pair of particles must have initial distance between them greater than d I and each particle must lie within a radius of R = 1 from the body center. Springs with spring constant k I are placed between each pair of particles closer than d S and with rest length equal to its initial length. They act in compression or dilatation around their rest length. The number of springs and the spring network does not vary during our simulations. The springs have different rest length so we record their initial mean rest length L I. The total number of particles is N I and total number of springs is NS I. The mass of each particle m I = 1.0/N I and the initial mass density is approximately uniform. The elastic force from a spring between two particles i, j with coordinate positions x i, x j, on particle i is computed as follows. The vector between the two particles x i x j gives a spring length L ij = x i x j that we compare with the spring rest length L ij,0. The elastic force from a spring between two particles i, j on particle i is computed as F elastic i = k I(L ij L ij,0)ˆn ij (23) where k I is the spring constant and the unit vector ˆn ij = (x i x j)/l ij. The force has the opposite sign on particle j. The strain rate of a spring with length L ij is ɛ ij = L ij 1 = (x i x j ) (v i v j) (24) L ij,0 L ijl ij,0 where v i and v j are the particle velocities and L ij is the rate of change of the spring length. To the elastic force on particle i we add a damping force proportional to the strain rate F damping i = γ I ɛ ijl ij,0m I ˆn ij (25) E I = 1 6V k IL 2 i (26) i where L i is the spring rest length for spring i, and V is the total volume. For the random isotropic mass spring model the Poisson ratio is ν = 0.25 (Kot et al. 2014). The Young s modulus computed with equation 26 has been used to predict flexure of a simulated beam under the force of gravity, illustrating that the mass-spring model can accurately represent physical materials (Kot et al. 2014). Our mass spring model allows us to estimate the Young s modulus, not the elastic rigidity which is commonly used for tidal evolution calculations. The relation between the shear modulus, µ I, and the Young s modulus, E I, is µ I = E I 2(1 + ν) = EI 2.5 (27) where we have used Poisson ratio ν = With non-zero damping coefficients, the stress is sum of an elastic term proportional to the strain and a viscous term proportional to the strain rate so the model should locally approximate a linear Kelvin-Voigt rheology. The mass spring model is compressible so when damped springs are used it exhibits a bulk viscosity (in analogy to the bulk modulus) and a shear viscosity (in analogy to the shear modulus). For a mass-spring model comprised of equal masses m I, damping coefficients γ I, and spring constants k I, we can estimate the shear viscosity η I by taking the ratio of the damping and elastic forces (equation 25 and equation 23) and assuming that η I scales similar to µ I as computed using equations 27 and 26. The simulated viscosity of the material we estimate as ( ) γim I η I µ I (28) k I The time scale for relaxation of the Kelvin-Voigt solid we estimate as τ relax ηi µ I γimi k I (29) While the fidelity of the computed elastic modulus of a mass-spring model has been tested with numerical measurements (Kot et al. 2014), the viscosity of a damped massspring model has not been tested. Thus we consider equations 28 and 29 as approximate. Likely they must both be corrected by factors of order unity. We will discuss this possibility later when we compare measurements of our model to predictions based on tidal evolution. The resolved body was set to an initial spin σ 0 by setting the initial velocities for each particle in the resolved body to v i = x i σ 0ẑ (30)

5 5 where v i and x i are the velocity and position vectors of particle i with respect to the resolved body s center of mass and the initial spin vector σ 0ẑ is perpendicular to the orbital plane. The rotation was mostly retrograde (σ 0 < 0) in our simulations to allow for a larger range of the tidal frequency ω to be simulated. The semi-major axis was always decreasing (see Table 3) and the body s spin rate was accelerated. We checked that the initial particle distribution creates only negligeable non-diagonal terms in the inertia tensor. From the rebound code version 2 from Nov 2015, we use the open-gl display, open boundary conditions, the direct all pairs gravitational force computation and the leap-frog integrator to advance particle positions. To the gravitational particle accelerations we added the additional spring and spring damping forces, as described above. To maintain numerical stability the time step is chosen to be smaller than the elastic oscillation frequency of a single particle in the spring network. The gravitational softening length is chosen to be 1/100-th of the minimum inter-particle spacing. 3.1 Measuring the semi-major axis drift rate Common parameters for our simulations are listed in Table 1 along with their chosen or computed values. A list of varied and measured parameters are listed in Table 2 and the values for them during different simulations listed in Table 3. We chose values for mass ratio M /M and initial semimajor axis a 0 common in two sets of simulations, however in each set the spring damping rate γ I and initial body spin σ 0 were chosen to sample a range of values for frequency ω and viscoelastic relaxation time τ relax. Using equation 26 we compute the Young s modulus for all springs with midpoint radius less than 0.9 and using the radius out to 0.9 to compute the volume. A fairly soft body under a strong tidal force (giving a large mass ratio and weak springs) was used to reduce the integration time required to see a significant tidally induced change in semi-major axis. We made sure that the Young s modulus was sufficently large so that the body was strong enough to maintain a nearly constant density radial profile. As there is no pressure force; the body is held up against self-gravity by spring forces alone and so the body is compressed and the springs in the interior are under compression. In all the simulations the particles are displaced in the body frame by at most a few percents of the unit length R. We chose the initial semi-major axis a 0 large enough to ensure that the quadrupole tidal potential term would dominate and set the initial relative velocity of M and M so that the two bodies are in a circular orbit. In the body frame of the resolved body (taking into account its spin), the perturbing body orbits on a period of P o = 2π/ω. We integrated each system a length of time t = 11P o, recording the semi-major axis only 10 times and at an interval of P o. Because the springs initially are at their rest lengths, gravity causes the system to bounce at the beginning of the simulation. During the first time interval, we integrate the body with a high damping parameter to dissipate the vibrational oscillations and we do not use the semi-major axis value computed during this interval of time. Subsequently, we recorded the semi-major axis at an interval of P o so that the irregularities of the particle distribution do not affect the measurement of the slowly drifting semi-major axis. The semi-major axis was computed using the distance between center of mass of the satellite and primary body and their velocity difference. To measure the semi-major axis drift rate ȧ, we fit a line to the 10 measurements of semi-major axis at the 10 time intervals we recorded the simulation output. Each fit was individually inspected to ensure that the 10 points lay on a line. The standard error of the fitted slope value giving ȧ was 1%. We measured the total momentum vector during the integrations, checking that all its components are accurately conserved. We also checked that measured spin and semi-major axis were tightly anti-correlated implying that angular momentum is conserved. We computed the difference in total angular momentum (measured from the center of mass of the binary) at the end of simulations compared to that at the beginning and find that its absolute value is below Total energy is not conserved because of the spring damping forces. 3.2 Comparison of numerically measured and predicted quality functions Inverting equation 3 [k 2 sin ɛ 2]( χ) = ȧ 3na ( M M ) ( a ) 5 (31) R where we have placed the quantities chosen and measured in the simulation on the right hand side and the quality function on the left hand side. From the measured semi-major axis drift rates measured in our simulations we use equation 31 and the quantities listed in table 3 to measure the quality function for a range of values of frequency. These values are plotted as a function of the unit-less frequency χ in Figure 3 and 4. The unit-less frequency was computed using the relaxation time computed for each simulation individually. The numerically generated points in Figure 3 show that the numerically measured quality function is linearly proportional to the frequency at small frequencies but decays at large frequencies. This behavior was previously predicted for different rheologies (Noyelles et al. 2014; Efroimsky 2015), but has not previously been seen in simulations. The points used to make this plot were measured from simulations with different spins and relaxation times. Nevertheless they all lie near the same curve, suggesting that the function is primarily dependent on the relaxation time. We have numerically confirmed the expected behavior and sensitivity of the quality function on frequency and viscoelastic time scale. On top of the points measured from the simulation we plot the predicted value for the quality function using Eq.22. To compute the quality function in Eq.22 we set the elastic modulus µ = µ I using equation 27 for the mean shear modulus of all the simulations (and this is proportional to the Young s modulus listed in Table 1). The ratio of semimajor axis and radius of the body, and mean motion are taken from Table 1. The result is the grey line shown at the bottom of Figure 3. While the shape of the curve implied by the points looks correct, the numerically measured points are too high. The blue curve on Figure 3 shows the predicted model multiplied by s = 1.3. Our measured semi-major axis drift is larger than that we have predicted. Our simulated body has particles out to a radius of 1 from its center of mass and the semi-major axis drift rate is strongly sensitive to the ratio of semi-major

6 6 Frouard et al. axis to body radius a/r. We consider whether we have underestimated the drift rate because we don t have the correct ratio of semi-major axis to body radius, a/r, used in our measured value for the quality function. The factor we multiply function 22 to match our measured points is 1.3 and the fifth root of this is The difference of 5% is smaller than our initial minimum distance between particles d I, so our simulated body actually has effective radius somewhat smaller than 1. However, as our body is somewhat smaller than we had intended, the tidal force on it should be smaller, not larger than estimated. We should have measured a smaller semi-major axis drift rate than we predicted, but we measured the opposite trend. Even though the fifth root of 1.3 is not large, if we lowered the body radius in equation 31 we would have further increased our measured quality function making the discrepancy between our measured and predicted values worse. Figure 3 shows that the peak frequency for our semimajor axis drift rate is approximately at χ = 1.0 and consistent with that predicted. If the relaxation time scale τ relax had been miscomputed for our simulations (and equation 29 is incorrect by a factor) then the curve would only have matched the data points if we had corrected the frequency scale. Because the numerical measured peak frequency matches that predicted, it is likely that we have estimated the shear viscosity (and associated relaxation time) for the mass spring model correctly. Both peak and height of quality function may shift if higher order terms (higher order values of l) are included in the tidal equations and we have not taken these into account in our comparison. As this might explain the difference in height of our numerically measured quality function compared to that predicted, we test this possibility by running simulations at a larger semi-major axis. Simulations with larger initial semi-major axis are also listed in Table 3 and the quality function for these runs plotted in Figure 4. We find that the amplitude correction factor required to match the numerical results is the same as for the previous set of simulations and again the frequency scale does not need to be rescaled. Higher order terms in the tidal potential don t explain the amplitude discrepancy between our numerical simulations model and our predictions. Because we are using a random mass-spring model, the particle distribution and spring network differs between each simulation. We computed the Young s modulus and relaxation time for each simulation, and these quantities are listed in Table 3. We find that there are variations in the elastic modulus between simulations. Shown in Figures 3 and 4 are predicted quality functions (using equation 22) multiplied by 1.3 and for a shear modulus that is higher and lower than 10% than that for the mean value used to plot the blue and grey curves. Points with higher values of E I, corresponding to harder bodies, systematically lie lower than nearby points with lower values of E I and so correspond to softer bodies that experience stronger tidal deformation. Much of the scatter in our points above and below the blue line we can attribute to variations in the particle distribution and associated spring network. We don t fully understand the source of the discrepancy in amplitude but we can discuss a variety of possible explanations. Near the body surface, the number of springs per particle is lower and the spring network is anisotropic. The Table 1. Common Simulation Parameters N I 800 Number of particles in resolved body NS I 9254 Number of interconnecting springs L I Mean rest spring length k I 0.08 Spring constant E I 2.3 Mean Young s modulus d I 0.15 Minimum initial interparticle distance d S Spring formation distance dt timestep N I, NS I, E I and L I vary slightly between simulations as particle distributions are randomly generated. E I is computed using equation 26 and is the average value for all the simulations. simulated body is weaker (and floppier) than we have estimated from the value for the Young s modulus integrating over the volume. Because the surface is softer than the interior, its tidal response would be larger than we predicted. The ratio of the number of springs per node is about 11.5 and is somewhat lower than the level recommended by Kot et al. (2014) (see their figure 5). In this regime, Kot et al. (2014) found that the spring network behaved 10% stronger than estimated using equation 26. This exacerbates our discrepancy in amplitude as our simulated body should have a slightly weaker tidal response than we have estimated. We generated the sphere of particles with springs at their rest wavelengths, however after the simulation begins the body is compressed by self-gravity. The resulting density profile is not perfectly flat as the center is more compressed than particles near the surface. The initial spin of the body causes the body to deviates from a sphere. To sample a large range of frequency (in units of relaxation time) and have a large tidal response (and so reduce the time required for the simulations) we use a soft body which is particularly prone to deformation when spun and compressed. While we did not see a difference in amplitude when we ran simulations at larger semi-major axis, we have neglected higher order terms in the quality function and these would increase its predicted amplitude, were we to take them into account. The computed value for the shear and Young s modulus is sensitive to the exact volume used to calculate it and this too could affect the amplitude of our predicted quality function. Because our simulated body is comprised of randomly distributed point particles it is not exactly round or uniform. The body principal axes (from the moment of inertia tensor) are initially randomly oriented and the three moments of inertia not exactly equal, so the body angular momentum initially is not oriented exactly vertically. We measured the size of the x and y components of spin angular momentum, finding them to be a few hundredths of the initial z component. This is small enough that the low level of spin about another axis is probably not the cause of our amplitude discrepancy.

7 7 Figure 3. A comparison of numerically measured quality functions, shown as points, to that predicted for a uniform Kelvin-Voigt rheology (the grey curve, equation 22) and a rescaled version of this curve (with rescaling factors listed on top right) shown as the blue curve for simulations with perturber mass M = 100, and initial semi-major axis a 0 = 10. The green curves show the effect of raising and lowering the shear modulus by 10%. We find that the numerically measured quality function is a function of frequency with form consistent with that predicted by Noyelles et al. (2014); Efroimsky (2015) using a rheological model for the deforming body. The predicted function for a Kelvin-Voigt model is consistent with the numerical measurements if the predicted function is multiplied by a moderate factor 1.3. We attribute the scatter of the points off the line to variations in the value of the shear modulus of the random mass-spring model due to non-uniformity in the particle distribution and spring network. Figure 4. Similar to Figure 3 except for perturber mass M = 200 and initial semi-major axis a 0 = DISCUSSION In this study we have used a damped mass spring model, within an N-body simulation, to directly model the tidal acceleration or deceleration of the spin of a viscoelastic body. We have directly measured the quality function (Love number divided by Q) in the simulations from the semi-major axis drift rate and have confirmed that it is a strong function of frequency and viscoelastic relaxation time for a simulated Kelvin-Voigt rheological model. Our work confirms previous analytical predictions for the form of the quality function that are based on linear viscoelastic rheology (e.g., Efroimsky 2015). Mass-spring models have not yet compared an estimate for the material viscosity derived from the spring damping coefficients, spring lengths, spring constants and network to one measured numerically to improve the accuracy of the value computed from the simulations. Consequently we were uncertain of our estimated shear viscosity and associated computed relaxation time. However, a comparison between our computed quality factor and that predicted for a Kelvin- Voigt solid suggests that we have estimated the simulated shear viscosity and viscoelastic relaxation time scale correctly. The amplitude of our numerically predicted quality function is about 30% larger than that predicted. By comparing simulations at two different semi-major axes, we concluded that the cause is unlikely to be the neglect of higher order terms in the quadrupole expansion for the potential. We are uncertain of the root of the discrepancy but suspect the non-uniformity of the spring network at the surface of the body that would give it a larger tidal response.

8 8 Frouard et al. Table 2. Description of Varied and Measured Simulation Parameters a 0 M /M χ ȧ γ I σ 0 τ relax ω P o E I initial semi-major axis mass ratio Unitless frequency Rate of orbital decay in semi-major axis Spring relaxation time Initial spin Estimated relaxation time of viscoelastic solid Initial tidal frequency (semi-diurnal) Time between integration outputs Computed Young s modulus The viscoelastic relaxation time τ relax is computed using equations 26 and 29. The frequency ω is defined in equation A14 and χ in equation 16. The period P o = 2π/ω is also that of simulation outputs. Table 3. Quantities either set or measured from different mass spring N-body simulations. χ ȧ γ I σ 0 τ relax ω P o E I with a 0 = 10, M = 100, n = e e e e e e e e with a 0 = 20, M = 200, n = e e e e e e e e e The rightmost column is the value of the Young s modulus computed using equation 26 for each simulation. This study demonstrates that we can directly simulate tidal evolution of viscoelastic bodies. It would be hard to simulate tidal damping over a billion year time scale as the time step must be chosen from the number of particles within the body and the spring constants for the springs connecting them, restricting it to be much smaller than the orbital time scale. However mass-spring models could be used to explore spin down for inhomogeneous and anisotropic bodies and study how their quality functions depend on rheological properties and their distribution within the body. We could also simulate a variety of phenomena that are more difficult to calculate analytically such as capture into spin orbit resonance, tidal circularization, orbital or normal mode resonance crossing, the distribution of tidally generated heat and the effect of non-linear rheological behavior. Acknowledgements We thank Harry Braviner and Moumita Das for helpful discussions. This work was in part supported by NASA grant NNX13AI27G. REFERENCES Clavet, S., Beaudoin, P. and Poulin, P. 2005, Particle-based Viscoelastic Fluid Simulation, Eurographics/ACM SIGGRAPH Symposium on Computer Animation (2005) K. Anjyo, P. Faloutsos (Editors) Darwin, G. H On the precession of a viscous spheroid and on the remote history of the Earth. Philosophical Transactions of the Royal Society of London, Vol. 170, pp Efroimsky, M. 2015, AJ, 150, 98, Tidal Evolution of Asteroidal Binaries. Ruled by Viscosity. Ignorant of Rigidity. Efroimsky, M., Williams, J. G. 2009, CeMDA, 104, 257, Tidal torques: a critical review of some techniques Efroimsky, M. 2012a, ApJ, 746, 150 Efroimsky, M. 2012b, CMDA, 112, 283 Efroimsky, M. and Makarov, V. V. 2013, ApJ, 764, 26, Tidal Friction and Tidal Lagging. Applicability Limitations of a Popular Formula for the Tidal Torque Ferraz-Mello, S. 2013, CeMDA, 116, 109, Tidal synchronization of close-in satellites and exoplanets. A rheophysical approach Goldreich P. 1963, MNRAS, 126, , On the eccentricity of satellite orbits in the solar system Kaula, M. 1964, RvGeo, 2, 661, Tidal Dissipation by Solid Friction and the Resulting Orbital Evolution Kot, M., Nagahashi, H., Szymczak, P. 2014, The Visual Computer, DOI /s , Elastic moduli of simple mass spring models Lainey, V., Karatekin, Ö., Desmars, J., Charnoz, S., Arlot, J.- E., Emelyanov, N., Le Poncin-Lafitte, C., Mathis, S., Remus, F., Tobie, G., Zahn, J.-P ApJ, 752, 14, Strong Tidal Dissipation in Saturn and Constraints on Enceladus Thermal State from Astrometry Lloyd, B. A., Kirac, S., Szekely, G., Harders, M. 2008, EURO- GRAPHICS Short papers, editors K. Mania and E. Reinhard, Publisher: The Eurographics Association, Identification of Dynamic Mass Spring Parameters for Deformable Body Simulation, DOI = /egs A.E.H. Love, Some problems of geodynamics, first published in 1911 by the Cambridge University Press and published again in 1967 by Dover, New York, USA. Makarov, V.V., and Efroimsky, M ApJ, 764, 27, No pseudosynchronous rotation for terrestrial planets and moons. Makarov, V. V.; Berghea, C.; and Efroimsky, M ApJ, 761, 83, Dynamical evolution and spin-orbit resonances of potentially habitable exoplanets. The case of GJ 581d. Makarov, V. V., Frouard, J., and Dorland, B. 2016, MNRAS, in press, Forced libration of tidally synchronized planets and moons. Makarov, V. V. 2013, MNRAS, 434, L21 - L25, Why is the Moon synchronously rotating?

9 9 Makarov, V. V. 2015, ApJ, 810, 12, Equilibrium rotation of semiliquid exoplanets and satellites. Makarov, V. V. 2016, in preparation, Perpetual long libration of terrestrial planets in tidal resonances Mase, G.T., Smelser, R.E., Mase, G.E. 2010, CRC Press, Taylor & Francis Group. Continuum Mechanics for Engineers, 3rd edition Melchior, P. 1972, Vander éditeur, Bruxelles. Physique et Dynamique planétaires Nealen, A., Muller, M., Keiser, R., Boxerman, E., Carlson, M., Ageia, N. 2006, Comput. Graph. Forum 25(4), , Physically based deformable models in computer graphics. Noyelles, B., Frouard, J., Makarov, V. V., Efroimsky, M. 2014, Icarus, 241, 26, Spin-orbit evolution of Mercury revisited Ogilvie, G. 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Tides on the Moon: Theory and determination of dissipation APPENDIX A: THE QUALITY FUNCTION A1 Static tides In a surface point R of the body, the potential due to the perturber is 2 W (R, r) = W l (R, r). (A1) where the inputs W l (R, r) are proportional to the appropriate Legendre polynomials P l (cos γ), with γ being the angle between the vectors r and R pointing from the body centre. The integers l are termed the degrees. The l-degree term W l (R, r) of the perturber s potential causes a tidal deformation of the perturbed body, assumed to be linear. Then the resulting l th addition U l to the perturbed body s potential is also linear in W l : ( ) l+1 U(r ) = U l (r R ) = k l W r l (R, r), r being an exterior point, and k l being the static Love numbers. Distorting the extended body, the perturber experiences 2 The reason why summation in the equation (A1) goes over l 2 is explained, e.g., in Efroimsky & Williams (2009, Eqns. 5-11). its response in the form of the incremental potential U taken at the point r = r : ( ) l+1 R U(r) = U l (r) = k l W l (R, r). (A2) r As the perturber is exterior ( r > R ), the quadrupole part of the expansion for the perturbing potential W is dominant. The same pertains to U. A2 Evolving tides The case of evolving tides is more complicated. Owing to the internal friction, the tidal deformation (and the resulting additional potential U ) always lags in time 3 behind the perturbation W. To take into account different lagging at different frequencies, it is necessary to expand both the perturbing potential W and the response U in Fourier series. The linearity of response implies that the same frequencies should emerge in both spectra, when W and U are observed at the same point of space. From the cornerstone work by Kaula (1964), it is easy to derive that the Fourier tidal modes read as ω lmpq = (l 2p) ω + (l 2p + q) n + m ( Ω θ) (l 2p + q) n m θ (A3) where θ and θ are the rotation angle and rotation rate of the extended body, introduced in the equatorial plane. In neglect of the equinoctial precession, θ can be identified with the sidereal angle. As ever, the notations ω and Ω stand for the perturber s argument of the pericentre and the longitude of the node, as seen from the extended body. The formula also includes the mean anomaly M and the anomalistic 4 mean motion n Ṁ (with M = 0 at the pericentre). Derivation of the expression (A3) is explained in Section 4.3 of Efroimsky & Makarov (2013). The modes ω lmpq can be of either sign, while their absolute values χ lmpq = ω lmpq (l 2p + q) n m θ, (A4) have the meaning of positive definite forcing frequencies of stresses and strains in the distorted body. The Fourier modes are parameterised with the four integers l, m, p, q. The integers l and m are the degree and order of the spherical harmonics employed in the expansion. 5 The dynamical analogue to the formula (A1) is: W (R, r, t) = W l (R, r, t) = W lmpq (R, r, t), (A5) lmpq 3 The caveat in time is important. Lagging in time does not necessarily imply geometric lagging of the bulge. The lunar orbit being above synchronous, the main (semidiurnal) tide created by the Moon on the Earth always leads, not lags. This, however, gets along well with causality. 4 With a being the semimajor axis and G the gravity constant, the mean anomaly M(t) = M 0 (t) + t dt G(M + M )/a 3 renders the anomalistic mean motion as n Ṁ = Ṁ 0 + G(M + M )/a 3. In neglect of external perturbations, Ṁ 0 0 and the anomalistic mean motion can be approximated with the Keplerian mean motion: n G(M + M )/a 3. 5 Sometimes m is also referred to as the azimuthal wavenumber (Ogilvie 2014).

10 10 Frouard et al. where a term W lmpq is proportional to cos (ω lmpq t +... ), with ellipsis denoting some phase: W lmpq (R, r, t) = A lmpq (R, r, t) cos (ω lmpq t +... ). (A6) Both the static formula (A1) and its dynamical analogue (A5) render the value of the perturbing potential at a surface point R. Writing down a dynamical analogue to the static expression (A2) turns out to be a highly nontrivial problem. Above we stated that, owing to the linearity of the problem, the spectrum of U should contain the same frequencies as that of W, provided both U and W are observed at the same point of space. Therefore, a Fourier series for U would contain terms proportional to cos (ω lmpq t +... ), had it been written for the (evolving in time) value of U at the same surface point R. We however are interested in the values of U in a different point, the point r where the moving perturber is located. There, the spectrum of U(r, t) will be richer than that of W (R, r, t), and will be parameterised with six indices lmpqhj : U(r, t) = U l (r) = U lmpqhj (r, t), (A7) lmpqhj see Efroimsky (2012b, Sections 7 & 8). As was pointed out yet by Kaula (1964), U(r, t) contains a secular part and that part is parameterised with the four indices lmpq : U(r, t) = U l (r, t) = U lmpq (r), (A8) lmpq where the angular brackets... denote time-averaging, and the terms on the right-hand side are given by U lmpq (r) = k l (ω lmpq ) cos ɛ l (ω lmpq ) ( ) l+1 R A lmpq(r, r), (A9) r where A lmpq are the magnitudes from the formula (A6), while k l (ω lmpq ) and ɛ l (ω lmpq ) are the degree-l dynamical Love numbers and phase lags written as functions of the Fourier modes. A3 The secular part of the tidal torque acting on the spin of the extended body The negative gradient of the secular potential (A8) renders the secular part of the orbital torque wherewith the extended body is acting on the perturber. An equal but opposite torque is acting on the extended body and is influencing its spin. The polar component of the secular torque reads as where T (z) = T (z) lmpq = k l (ω lmpq ) sin ɛ l (ω lmpq ) T (z) l = lmpq T (z) lmpq, (A10) ( ) l+1 R m A lmpq(r, r). (A11) r We see that an lmpq component of the torque may be either decelerating or accelerating the spin, dependent upon the sign of the phase lag ɛ l (ω lmpq ) which always coincides with the sign of the Fourier mode ω lmpq. A4 The quality function ( kvalitet ) The product k l (ω lmpq ) sin ɛ l (ω lmpq ) is sometimes termed as the quality function (Makarov 2013; Efroimsky 2015) or kvalitet (Makarov 2015; Makarov et al. 2016). In the literature, it is conventional to write it as k l (ω lmpq ) sin ɛ l (ω lmpq ) = k l(ω lmpq ) Q l (ω lmpq ) Sgn ω lmpq where the quality factors are introduced via, (A12) 1 Q l (ω lmpq ) = sin ɛ l(ω lmpq ), (A13) and where it is taken into account that the sign of a phase lag ɛ l (ω lmpq ) always coincides with the sign of the Fourier mode ω lmpq (e.g., Efroimsky & Makarov 2013). A5 Which terms are leading, and when As the perturber is exterior ( r > R ), the quadrupole part of the expansion for the perturbing potential W is dominant. The quadrupole part comprises all the terms with l = 2. For low inclination and eccentricity, the largest terms in the expansions (A5), (A7), and (A10) are those with {lmpq} = {2200}. They correspond to the so-called semidiurnal Fourier mode ω ω 2200 = 2 (n θ). (A14) When the semidiurnal, or any other lmpq term is leading in the expansion for W, the corresponding lmpq term is leading also in the expansion for the additional tidal potential U. Up to some reservation, this is true also for the expansions of the tidal torque. A reservation comes from the fact that an lmpq term in the expansion for the torque contains as a multiplier the sine of the phase lag ɛ l (ω lmpq ). For example, in the case of small inclination i and eccentricity e, the semidiurnal part of the polar torque operating on the spin of the perturbed body reads as (Efroimsky 2012b): T (z) 2200 = (A15) 3 2 GM 2 R 5 a k2(ω2200) sin ɛ2(ω2200) + 6 O(e2 ɛ) + O(i 2 ɛ). The quality function k l (ω lmpq ) sin ɛ l (ω lmpq ) continuously goes through zero (and changes its sign) when the lmpq spin-orbit resonance is transcended, i.e., when ω lmpq goes through zero. So, when a rotator is trapped into an lmpq spin-orbit resonance, the quality function stays zero; so the Fourier mode ω lmpq contributes nothing to the torque. Specifically, in the case of synchronous rotation (known as the 1:1 spin-orbit resonance), the mode ω 2200 vanishes and so does the semidiurnal term of the torque. In the resonance, therefore, it is the higher-than-semidiurnal terms that are leading. This acceding of leadership in resonances, along with its physical consequences for binaries, is described in detail in Makarov & Efroimsky (2013) and Makarov et al. (2012). Here we shall only mention two simple examples. Since the Moon is synchronised, the semidiurnal input into the torque acting on its spin is zero. It is then the other components (mainly, the term with {lmpq} = {2201} ) that define the tidal response of the Moon and influence its libration in longitude (Makarov et al. 2016). As another example, take