MNRAS 000, 000–000 (0000) Preprint 27 July 2016 Compiled using MNRAS LATEX style file v3.0

Tidal spin down rates of homogeneous triaxial viscoelasticbodies

Alice C. Quillen1,2, Andrea Kueter-Young1,3, Julien Frouard4, &Darin Ragozzine51Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA2alice.quillen@rochester.edu3Physics Department, Siena College, Loudonville, NY 12211, USA, ab08kuet@siena.edu4US Naval Observatory, 3450 Massachusetts Ave NW, Washington DC 20392, USA5Brigham Young University, Department of Physics and Astronomy, N283 ESC, Provo, UT 84602, USA

27 July 2016

ABSTRACTWe use simulations to measure the sensitivity of the tidal spin down rate of a ho-mogeneous triaxial ellipsoid to its axis ratios by comparing the drift rate in orbitalsemi-major axis to that of a spherical body with the same mass, volume and simu-lated rheology. We use a mass-spring model approximating a viscoelastic body spinningaround its shortest body axis, with spin aligned with orbital spin axis, and in circularorbit about a point mass. The torque or drift rate can be estimated from that predictedfor a sphere with equivalent volume if multiplied by 0.5(1 + b4/a4)(b/a)−4/3(c/a)−αc

where b/a and c/a are the body axis ratios and index αc ≈ 1.1 is consistent withthe random lattice mass spring model simulations but αc ∼ 4/3 suggested by scalingestimates.

A homogeneous body with axis ratios 0.5 and and 0.8, like Haumea, has orbitalsemi-major axis drift rate about twice as fast as a spherical body with the same mass,volume and material properties. A simulation approximating a mostly rocky body butwith 20% of its mass as ice concentrated at its ends has a drift rate 10 times fasterthan the equivalent homogeneous rocky sphere. However, this increase in drift rateis not enough to allow Haumea’s satellite, Hi’iaka, to have tidally drifted out to itscurrent orbital semi-major axis.

1 INTRODUCTION

The simplicity of analytic formulas for tidal spin down ofspherical bodies and the associated drift rate in semi-majoraxis (Goldreich 1963; Goldreich & Peale 1968; Kaula 1964;Efroimsky & Williams 2009; Efroimsky & Makarov 2013)has made it possible to estimate tidal spin-down rates indiverse settings, including stars, exoplanets, satellites, andasteroids (e.g., Efroimsky & Lainey 2007; Ferraz-Mello et al.2008; Ogilvie 2014). However, there is uncertainty in howto estimate the spin down rate and associated semi-majoraxis drift of non-spherical bodies (though see Mathis & LePoncin-Lafitte 2009 for estimates of tidal drift rates for twoextended homogeneous bodies) and this hampers attemptsto interpret the dynamical history of systems that includeextended spinning bodies like Pluto’s and Haumea’s satellitesystems (e.g., Ragozzine & Brown 2009; Cuk et al. 2013;Weaver et al. 2016).

An extended shape might experience enhanced tidaldistortion and dissipation compared to a stronger sphericalbody. Ragozzine & Brown (2009) suggested that use of theradius of an object with equivalent volume (the volumetric

radius) in classic tidal formulae could lead to an underesti-mate of the tidal torque on Haumea.

A semi-analytical treatment of the gravitational poten-tial inside a triaxial homogeneous body can give a descrip-tion of instantaneous internal deformations (or displace-ments) and stresses as a function of Cartesian coordinates(Dobrovolskis 1982). If the axis of the tidal force and bodylie along a plane of body symmetry, the displacements de-pend on 12 dimensionless coefficients that can be computednumerically. However, analytical computation in the generalcase is more challenging.“The reader is cautioned that in thegeneral case where the tide-raising object does not lie alonga principal axis, as many as 72 unknown coefficients mayappear.” (Dobrovolskis 1982). To semi-analytically computethe tidal torque for a spinning triaxial body we would needto numerically compute all these coefficients and then aver-age over body rotation to compute secular or average driftrates.

Due to the complexity of accurate semi-analytical com-putation, it would be convenient to have scaling relations,dependent on body axis ratios, that would allow one tocorrect analytical formula for tidal evolution from the well

c© 0000 The Authors

2 Quillen et al.

known formulae for homogeneous spherical bodies. Our goalhere is to numerically measure such correction factors usingnumerical simulations.

Because of their simplicity and speed, compared to morecomputationally intensive grid-based or finite element meth-ods, mass-spring computations are an attractive method forsimulating deformable bodies. By including spring dampingforces they can be used to model viscoelastic deformation.We previously used a mass-spring model to study tidal en-counters (Quillen et al. 2016) and measure tidal spin downfor spherical bodies over a range of viscoelastic relaxationtimescales (Frouard et al. 2016). Mass spring models arenot restricted to spherical particle distributions and so canbe used to study triaxial ellipsoids. Our work (Frouard et al.2016) compared orbital semi-major axis drift rates for spher-ical bodies to those predicted analytically. Here we use thesame type of simulations to measure drift rates but work ina setting where analytical computations are lacking and thenumerical measurements may motivate order of magnitudescaling arguments.

2 NUMERICAL MEASUREMENTS

2.1 Description of mass-spring model simulations

To simulate tidal viscoelastic response we use the mass-spring model by Quillen et al. (2016); Frouard et al. (2016)that is built on the modular N-body code rebound (Rein& Liu 2012). Springs between mass nodes are damped andso approximate the behavior of a Kelvin-Voigt viscoelas-tic solid with Poisson ratio of 1/4 (Kot et al. 2015) As byFrouard et al. (2016) we consider a binary in a circular or-bit, with a spinning body resolved with masses and springs.The other body (the tidal perturber) is modeled as a pointmass. The particles are subjected to three types of forces:the gravitational forces acting on every pair of particles inthe body and with a massive companion, and the elasticand damping spring forces acting only between sufficientlyclose particle pairs. Springs have a spring constant k and adamping rate parameter γ. The number density of springs,spring constants and spring lengths set the shear modulus, µ,whereas the spring damping rate, γ, allows one to adjust theshear viscosity, η, and viscoelastic relaxation time, τ = η/µ.For a Poisson ratio of 1/4, the Young’s or elastic modulusE = 2.5µ. The tidally induced spin down rate is computedfrom the drift rate of the semi-major axis measured in thesimulations.

Frouard et al. (2016) measured a 30% difference be-tween the drift rate computed from the simulations and thatcomputed analytically. We do not try to resolve this dis-crepancy here but instead compare the orbital drift rate ofhomogeneous triaxial ellipsoids to that of a spherical bodywith the same mass and volume, assuming that the cause ofthe discrepancy is not strongly dependent on body shape orsimulated rheology.

We consider an extended spherical body of mass M ,tidally deformed by a point mass perturber of mass M∗. Weconsider the two body system with M and M∗ in a circularorbit with orbital semi-major axis ao and mean motion n =√G(M +M∗)/a3o. The body M spins with spin rate σ = θ

and its spin axis is orientated parallel to the orbital angular

momentum vector. Here G is the gravitational constant. Weuse ao to denote orbital semi-major axis so as to differentiateit from body ellipsoid semi-major axis, a. We assume thatao � a.

For spherical bodies and for the simulations describedby Frouard et al. (2016) we worked with mass in units ofM , distances in units of radius R, time in units of tg =√R3/(GM) and elastic modulus in units of eg = GM2/R4

which scales with the gravitational energy density or centralpressure.

For a non-spherical homogeneous ellipsoid body wecould replace the radius R with the body’s semi-major axislength, a, or we could use the volumetric radius, Rv, theradius of a spherical body with the same volume. For oblatesystems we could use the mean equatorial radius, Re = a.We chose the volumetric radius so that it is straightfor-ward to compare simulations with the same number of mass-particles and simulated material properties (shear modu-lus, shear viscosity, viscoelastic relaxation time and type ofspring network). For our triaxial ellipsoids we work withmass in units of M , distances in units of volumetric radiusRv, time in units of

tg =

√R3v

GM(1)

and elastic modulus E in units of

eg =GM2

R4v

. (2)

We assume that the ellipsoid spins about a principal bodyaxis, about its axis of maximum moment of inertia, and withspin axis orientated in the same direction as the orbital an-gular momentum vector.

Our convention for body semi-axis ratios is a > b > cwith c oriented along body spin and orbital spin axes. Thevolume of a triaxial ellipsoid is V = abc 4π

3so a sphere with

the same volume has radius of

Rv = (abc)13 = a(b/a)

13 (c/a)

13 . (3)

We run two types of mass spring models, a randomspring model, (as we described previously; Frouard et al.2016) and a cubic lattice model; for simulation snap shotssee Figure 1. For the random spring model, particle posi-tions are drawn from an isotropic uniform distribution butonly accepted into the spring network if they are within thesurface, x2/a2 +y2/b2 + z2/c2 = 1, and if they are more dis-tant than dI from every other previously generated particle.The parameter dI is the minimum inter-particle spacing. Forthe cubic lattice model, particle positions are generated in3-dimensions at Cartesian coordinates, within the ellipsoidalbody surafce, that are separated in x, y or z by dI . The cubiclattice cell is aligned with the x-axis, the direction betweenthe M and M∗, the z-axis, aligned with the orbital and bodyspin axes and the y-axis along the tangential orbital motion.Crystalline lattices are not elastically isotropic as their stiff-ness depends on the direction on which stress is applied.A cubic lattice was chosen because its elastic behavior isthe same along x, y or z planes. An advantage of a latticemodel is that the spring network is more homogeneous (hasless granularity or porosity) and so can be run with shortersprings and fewer springs per node and this would reducethe affect of a weaker surface layer (due to fewer springs per

MNRAS 000, 000–000 (0000)

3

node near the surface). The advantage of the random springmodel is that its stiffness should not be sensitive to the di-rection on which stress is applied; it should be elasticallyisotropic.

Once the particle positions have been generated, everypair of particles within ds of each other are connected with asingle spring. The parameter ds is the maximum rest lengthof any spring. For the cubic lattice we chose ds so that cubiccell face diagonals and cubic cell cross diagonals are con-nected (see Figure 1 by Kot et al. 2015 for an illustration).For the random spring model we chose ds so that the num-ber of springs per node is greater than 15, as recommendedby Kot et al. (2015).

At the beginning of the simulation the body is not ex-actly in equilibrium because springs are initially set to theirrest lengths. The body initially vibrates. As a result we runthe simulation for a time tdamp with a higher damping rateγhigh. We only measure the semi-major axis drift rate usingmeasurements taken after the high damping rate is finishedand the body is relaxed and in equilibrium. The body is ini-tially a triaxial ellipsoid and if it were not rotating it wouldremain a triaxial ellipsoid. The ellipticity of the body is notdue to its rotation.

The secular part of the semi-diurnal (l = 2) term in theFourier expansion of the perturbing potential (e.g., see ap-pendix by Frouard et al. 2016) gives a torque on a sphericalbody

T =3

2aoGM2

∗R5

a5ok2(ω) sin ε2(ω) (4)

(also see Kaula 1964; Murray & Dermott 1999). When theinclination and eccentricity are small, conservation of angu-lar momentum gives an estimate of the secular drift rate ofthe semi-major axis from the secularly averaged torque

aonao

≈ − 2TaoGMM∗

= −3

(M∗M

)(R

ao

)5

k2(ω) sin ε2(ω) (5)

where the tidal frequency ω = 2(n − θ) = 2(n − σ). Thequality function is k2(ω) sin ε2(ω) and is often approximatedas k2/Q with Q a tidal dissipation factor (e.g., Kaula 1964)and k2 a Love number. For low values of χ = |ωτ | � 1 with τthe viscoelastic relaxation timescale and a stiff homogeneouselastic spherical body

|k2(ω) sin ε2(ω)| ∝ egµχ (6)

(taking the low χ limit of equation 25 by Frouard et al. 2016for the Kelvin-Voigt viscoelastic model) where µ is the shearmodulus or rigidity. This is consistent with k2 ∝ ρgR/µ ∝GM2/(µR4) in the high rigidity limit for a homogeneous andincompressible elastic sphere (Love 1927; Murray & Dermott1999) and tidal dissipation factor Q ∼ χ−1. Here ρ is densityand g = GM/R2 is surface gravitational acceleration.

The semi-major axis drift rate, aonao

, is proportional tothe tidally induced torque T following from angular momen-tum conservation. We compare the orbital drift rate for bod-ies with the same mass and volume but different ellipsoidaxis ratios. We run series of simulations with parameterslisted in Tables 1 and 2. Each simulation series has param-eters similar or the same as given in Tables 1 and 2 but has

Table 1. Common simulation parameters

perturber mass M∗ 10

orbital semi-major axis ao 10

mean motion n 0.1body spin rate σ 0.6

integration time for high damping tdamp 3

high damping rate γhigh 20time step dt 0.003

total integration time Tint 1260

The mass and volume of the spinning body are the same for all

simulations. χ is derived from the viscoelastic relaxation time,spin frequency σ and semidiurnal frequency ω.

different body axis ratios. For comparison we measure as,the semi-major axis drift rate, for a spherical body in thesame series, (with values listed in Table 2) and use it to nor-malize the semi-major axis drift rates for the non-sphericalbodies. Thus the semi-major axis drift rates for the cubiclattice simulations are divided by that of the spherical cubiclattice simulation and the semi-major axis drift rate for therandom spring model simulations are divided by that of thesimilar spherical random simulation.

A fairly low value of the Young’s modulus (in units ofeg) was used so that the body was soft, reducing the integra-tion time required to measure a drift in orbital semi-majoraxis. The frequency χ was chosen to be 0.1 (significantly lessthan 1) so that we remain in the linear viscoelastic regime(e.g., see Ferraz-Mello 2013; Noyelles et al. 2014; Efroimsky2015; Frouard et al. 2016) where the quality function andtidal torque are linearly proportional to χ (equivalently pro-portional to phase lag). The simulations were run 200 timesthe period associated with the semi-diurnal frequency or fora total time Tint = 200Posc with Posc = 2π/ω. This length oftime is chosen so that we can average over variations causedby body rotation and compute a secular drift rate in orbitalsemi-major axis. Each simulation in the R and C series re-quired a few hours of computation time on a 2.4 GHz IntelCore 2 Duo from 2010.

For both cubic lattice and R series random lattice mod-els we ran simulations of oblate and prolate systems withaxis ratios c/a =0.4 – 1 in steps of 0.1. The normalized or-bital semi-major axis drift rates ao/as are shown in Figures2 and 3 for oblate and prolate bodies. For the R series ran-dom spring model we also ran a series of triaxial systems andtheir drift rates are shown in Figures 4 and 5. In all cases thebodies rotate about their shortest body axis (correspondingto their principal moment of inertia axis).

Additional numerical tests and comparisons betweensimilar simulations, beyond those previously discussed byFrouard et al. (2016), are described in appendix A.

2.2 Oblate bodies

Bodies with a = b and a > c are oblate and here they arespinning about their short axis. Figure 2 plots as points driftrates measured for the oblate bodies from our simulations,normalized to the sphere with the same volume for the ran-dom R series and cubic C series simulations with parame-ters listed in Tables 1 and 2. With the points we have drawnpower law curves ao/as = (c/a)−4/3. and ao/as = (c/a)−1.1.

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4 Quillen et al.

Table 2. Parameters for series of simulations

Simulation Series R C LR

Particle lattice random cubic random

Young’s modulus E/eg 3.1 3.1 3.0

frequency χ 0.10 0.10 0.10number of mass nodes N 1150 1240 2900

Springs per node NS/N 16 11 15spring constant k 0.06 0.1 0.0475

spring damping rate γ 7.2 13 15

minimum particle spacing dI 0.135 0.15 0.1maximum spring length ds 2.48dI 1.8dI 2.38dI

drift rate for the sphere as 1.16 ± 0.06 × 10−6 1.521 ± 0.003 × 10−6 1.45 ± 0.01 × 10−6

Numbers of mass nodes, springs, frequency χ and Young’s modulus are average values for each series. Parameters ds and dI are fixed in

each series and used to generate the spring network. The spring constant k and γ are fixed in each series and set to achieve elastic

modulus E/eg ∼ 3 and χ ∼ 0.1. The drift rate as for the sphere in the series is measured from the simulation output by fitting a line tothe orbital semi-major axis as a function of time. The error is the rms value of the deviation of the simulation measurements from the

fitted function.

The power law with index −4/3 is a pretty good approxi-mation to ao/as for the cubic lattice simulations whereas anindex of -1.1 is a better fit to the random lattice simulations.

For an oblate body with spin parallel to orbital axisand aligned with the body’s axis of symmetry, the shapeof the tidally deformed body is independent of time. Thissuggests that we should consider how the drift rate dependson the equatorial radius, Re = a. For the oblate ellipsoid,the volumetric radius

Rv = a(c/a)13 = Re(c/a)

13 (7)

We consider the hypothesis that equation 5 is appro-priate for our oblate body but substituting the equatorialradius for the volumetric radius. Equation 5 gives ao/(nao)as a unitless parameter so that it is independent of time, andwe need not correct for the dependence of our unit of timeon body radius (equation 1). The shear modulus, viscosityand viscoelastic relaxation timescales are the same for eachsimulation as are the body spin rate σ, semi-diurnal fre-quency ω and semi-diurnal frequency normalized with theviscoelastic relaxation time χ. Equation 5 shows that theorbital drift depends on radius to the 5-th power times thequality function. However equations 2 and 6 and imply thatwhen χ < 1 the quality function is inversely proportionalto radius to the 4-th power, due to the normalization ofthe elastic modulus (with eg ∝ R−4

v ). Taking into accountthe dependence of eg on radius R we expect ao/(nao) ∝ R.This implies that the ratio of the drift rate predicted usinga classical tidal formula and replacing radius with equato-rial radius when normalized with that using the volumetricradius is ao/as = (c/a)−1/3 as Rv/a = (c/a)1/3. Our nu-merical measurements are not consistent with this scalingas we measured a power-law index 3 to 4 times larger inmagnitude than 1/3.

In appendix B1 we crudely estimate body strain withHooke’s law by applying the tidal force to the body surfaceand estimate that the drift rate of an oblate body normalizedto that of the sphere with the same volume(aoas

)oblate

≈( ca

)− 43

(8)

(equation B13). The dependence on c arises because thestresses and strains depend on the body cross sectional area.

The index -4/3 is closer to the -1.1 that we measured forthe random lattice model than the -1/3 estimated using theequatorial radius alone in the classical tidal formula.

The body surface area is larger for more extreme axisratio ellipsoids than the equivalent sphere. The soft surfacelayer present in the simulations should increase the driftrates compared to what is expected in the continuum limit.We can consider our numerically measured points an up-per limit for the value in the limit of an infinite number ofsimulated particles as we expect that numerically generatedsurface softness would increase the drift rates. However wemust keep in mind that our hard-edged bodies (see discus-sion in appendix A) also have slightly higher density nearthe surface than a homogeneous body and we are not surehow this would have affected the numerical measurements,though we suspect that the weak surface has a stronger in-fluence on the drift rates. As our numerical measurementsare likely to be upper limits, we suspect that a more rigor-ous scaling argument (better than in appendix B1) shouldpredict a reduced exponent.

2.3 Prolate Bodies

For the prolate systems, the long axis of the body rotatesin the orbital plane and b = c. Figure 3 is similar to Figure2 but here we only plot the prolate bodies. Power law func-tions of any index give poor fits to either cubic or randomlattice prolate simulations. Order of magnitude scaling es-timates (see appendix B2), based on averaging tidal stressand strain, suggest that the drift rates should scale with thefunction(aoas

)triaxial

≈ 1

2

(1 +

b4

a4

)(b

a

)− 43 ( c

a

)− 43

(9)

(repeated here from equation B23). The dependence on 1 +b4

a4arises from averaging over body rotation whereas the

dependence on (b/a)−43 (c/a)−

43 is from the dependence of

tidal stress on cross sectional area. For a prolate system withb = c this implies that(aoas

)prolate

∼ 1

2

(1 +

c4

a4

)( ca

)− 83, (10)

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Figure 1. Two simulation snapshots. The top one shows a triaxial

body from the random mass-spring model R-series with b/a = 0.6and c/a = 0.5 as seen looking down on the orbital plane. The bot-

tom snapshot shows a cubic lattice simulation for an oblate body

with b/a = 1.0 and c/a = 0.7. This simulation is seen from an in-clined angle. The long green lines point to the tidally perturbing

mass. A single mass node is colored red so that body librationcan be viewed during tidal lock. The mass nodes are renderedas spheres with radius equal to 1/4 the minimum interparticledistance dI and spring connections are shown as thin green lines.

The mass nodes are kept in their positions by spring elastic forcesbetween nodes, not because the spherical surfaces repel.

however if we take into account that we measured a powerlaw dependence of -1.1 for the oblate systems then we mightexpect(aoas

)prolate

∼ 1

2

(1 +

c4

a4

)( ca

)−2.4

(11)

as 4/3 + 1.1 ∼ 2.4. We plot both of these curves on Figure3, finding that both functions are adequate matches to the

Figure 2. We show the drift rates in orbital semi-major axis

due to tidal torque as a function of c/a body axis ratio for sim-ulated homogeneous viscoelastic oblate bodies (a = b) spinning

about a minor axis with spin axis aligned with the orbital rota-

tion axis. Measurements from two sets of simulations are shownas points, one based on a cubic lattice, the other using the ran-

dom mass/spring model (C and R series with parameters listed

in Table 2). The simulations in each series have approximatelythe same numbers of particles, springs per node, body volume

and mass, shear modulus, viscoelastic relaxation timescale, initialspin, semi-major axis and perturber mass ratio. The differences

are in the axis ratio of the simulated oblate ellipsoid. The vertical

axis is a/as where as is that measured for the spherical body inthe simulation series. The horizontal axis shows body axis ratio

c/a. The curves are power laws with index listed in the key. The

cubic lattice simulation rates are similar to (c/a)−4/3 whereas therandom models scale somewhat shallower, as (c/a)−1.1.

prolates from the random spring model simulations but notthe cubic lattice ones.

At extreme axis ratios, the drift rates for the cubic lat-tice are much higher than that for the random lattice (whencompared to their matching spherical body). The c/a = 0.4prolate body cubic lattice model drift rate has a ratio ofao/as = 11.6 and lies above the limits of the plot in Figure3. This value is more than twice the value for the random-spring model prolate with c/a = 0.4.

In appendix B2 we estimated the drift rate by averag-ing the torque with body orientation along the tidal axisand that oriented perpendicular to the tidal axis. While thestrain for the cubic lattice in x and y directions (with re-spect to the orientation of a cubic cell in the lattice) is thesame, the body is weaker when stresses are applied along adirection 45◦ from an edge of the cubic cell and in a planecontaining a cubic cell face. Consequently our averaging pro-cedure underestimates the average tidal deformation. Thisis likely a stronger affect when the axis ratios are high evenwhen normalizing to the matching sphere, that is also af-fected by the elastic anisotropy of the cubic lattice. The cu-bic lattice has a shallower but softer surface (due to a lowernumber of springs per node) than the random spring modeland this too might contribute to the stronger sensitivity ofthe drift rate to axis ratio compared to the random springmodel. We can attribute the high drift rates at low axis ra-

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6 Quillen et al.

Figure 3. We show drift rates in orbital semi-major axis as a

function of body axis ratio c/a for simulated viscoelastic prolatebodies (with b = c) spinning about minor axis and with spin

axis aligned with the orbital axis. Measurements from two sets

of simulations are shown as points, one based on a cubic lattice(the C series), the other using the random mass/spring model

(the R series). The simulations in each series have approximately

the same numbers of particles, springs per node, body volumeand mass, shear modulus, viscoelastic relaxation timescale, initial

spin, semi-major axis and perturber mass ratio. The differencesare in the axis ratio of the simulated prolate ellipsoid. The vertical

axis is a/as where as is that measured for the spherical body in

the simulation series. The horizontal axis shows body axis ratioc/a = b/a. The drift rates are poorly fit by a power law. Shown

as two curves are equations 10 amd 11 based on the estimate

via order of magnitude stress/strain estimates in appendix B2.These relations are good matches to the random spring model

simulations.

tios for both oblate and prolate bodies (and a stronger affectfor the prolates) to the anisotropy of the lattice.

The strong dependence of drift rates on axis ratios em-phasizes that the drift rates are strongly influenced by theweakest part of the simulated body. When prolate, the bodyis easiest to deform along its long axis, when a cubic latticeis present, the drift rate is strongly influenced by the elas-tic anisotropy and softness in the simulated body surfaceis likely our largest source of error in estimating tidally in-duced drift rates. A real asteroid or Kuiper belt object ifit contains soft materials, discontinuities or fractures maybe poorly approximated by a homohomogeneous strengthviscoelastic model.

2.4 Triaxial bodies

Figure 4 shows (as points) the numerically measured nor-malized drift rates for the R-series of random spring modelsand for triaxial bodies with c/a = 1.0, 0.9, 0.8, 0.7, 0.5, 0.4and b/a covering the same range but with b/a > c/a so theyare stable as the bodies are rotating about the short axis.On this figure the point type labels c/a and the drift ratesare plotted versus b/a. The oblate bodies lie on the righthand side of the plot and the prolate bodies lie along thelower left diagonal. Taking the power law approximations

Figure 4. Orbital semi-major axis drift rates for triaxial bodies

measured from the random spring model simulations from the Rseries. The vertical axis is (a/as) where as is that measured for

the spherical body. The horizontal axis shows b/a body axis ratio.

Bodies with different values of c/a have different point types. Thekey on the upper right shows the body axis ratio c/a for each

point type. Oblate bodies lie on the far right. The curves show

equation 12, each line with a different value of c/a and with linecolor matching the numerically measured points with the same

value.

Figure 5. We show the orbital semi-major axis drift rates mea-sured from the R-series of random spring model simulations but

corrected by (c/a)1.1 with power index based on the power lawthat fit the oblate simulations. The vertical axis is a/as×(c/a)1.1

where as is that measured for the spherical body. Once corrected

for the axis ratio c/a, the drift rates for the triaxial bodies resem-ble those of the prolate bodies.

for the oblate bodies we correct the drift rates by (c/a)1.1

and replot the points in Figure 5. In Figure 5 we see thatthe points lie on a curve that is similar to that of the prolatebodies (see Figure 3). This implies that the drift rate can beconsidered a product of two functions; one that depends on

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Table 3. Information about Haumea and Hi’iaka

Haumea:

Semi-major axis of ellipsoid aH 960 kmAxis ratio bH/aH 0.80

Axis ratio cH/aH 0.52

Volumetric radius RvH 716.6 kmMass of Haumea mH 4 × 1021 kg

Energy density scale eg,H 4.05 GPa

Gravitational timescale tg 1171 sSpin rate σH tg 0.52

Tidal frequency ω ∼ 2σH 0.9 × 10−3 Hz

Mass ratio q = MHi/MH 0.0045

orbital semi-major axis Hi’iaka aHi 49880 km

Body semi-major axis and axis ratios are by Lockwood et al.

(2014). Mass of Haumea, mass ratio, q, of Hi’iaka and Haumeaand semi-major axis are by Ragozzine & Brown (2009). The

volumetric radius RvH is the radius of a sphere with the samevolume as the triaxial ellipsoid. Spin rate, gravitational timescale,

tg and energy density scale, eg , are computed using the volumet-

ric radius and equations 1 and 2. The spin rate was computedusing the spin period PH = 2π/σH = 3.91531 ± 0.00005 hours

measured by Lockwood et al. (2014).

c/a and is approximately a power law, and the other thatdepends on b/a.

Our scaling argument presented in appendix B2 sug-gested that the drift rate should be a product with form inequation 9. That correcting for c/a puts the triaxial driftrates on the same line as the oblates supports the expecta-tion (based on the order of magnitude estimates) that thedrift rates can be approximated by a product of functions,one depending on c/a and the other on b/a.

The oblate random lattice models were better fit witha function proportional to (c/a)−1.1 rather than (c/a)−4/3,so in Figure 4 we have plotted on top of the numericallymeasured points curves generated with(aoas

)triaxial

≈ 1

2

(1 +

b4

a4

)(b

a

)− 43 ( c

a

)−1.1

. (12)

This function is a pretty good fit to all the simulations shownin Figure 4. The slight increase at b/a near 1 is reproducedby both simulations and the estimating scaling behavior.Had we plotted curves using equation 9, the curves wouldhave matched the prolate points on the left but would havediverged from the oblate plots on the right hand side ofthe plot. We conclude that the random spring models areprobably accurate and that the function estimated in theappendix 9 is a good approximation though the power-lawindex for c/a may be somewhat shallower than -4/3, as givenin equation 12.

2.5 Notes on Haumea

The dwarf planet Haumea (Brown et al. 2005) is an ex-tremely fast rotator with density higher than other objectsin the Kuiper belt; it is consistent with a body dominated byrock (Rabinowitz et al. 2006; Lacerda et al. 2008; Lellouch etal. 2010; Kondratyev 2016). Visible and infrared light curvefits (Lockwood et al. 2014) find the body consistent with

a rapidly rotating oblong Jacobi ellipsoid shape in hydro-static equilibrium with axis ratios listed in Table 3 (also seeLellouch et al. 2010) and a density of ρ = 2.6 g cm−3. Fordiscussion on formation scenarios for the satellite system seeLeinhardt et al. (2010); Schlichting & Sari (2009); Cuk et al.(2013). Parameters based on Haumea and Hi’iaka are listedin Table 3.

The pressures in the body at depth for a body as mas-sive as Haumea would lead to ductile flow giving long-term deformation allowing the body to approach a figureof equilibrium (a Jacobi ellipsoid). Even if Haumea’s shapeis consistent with a hydrostatic equilibrium figure, on shorttimescales the body should behave elastically. For tidal evo-lution, the relevant tidal frequency is ω ∼ 2σH ∼ 10−3Hz,comparable to vibrational normal mode frequencies in theEarth. Our simulations do not allow ductile flow on longtimescales, but can approximate the faster tidal deforma-tions if we model the body as a stiff elastic body with itscurrent shape.

We ran a simulation in the LR random lattice serieswith axis ratios b/a = 0.8 and c/a = 0.5, consistent withmeasurements for Haumea. From the simulation we mea-sure ao/as = 2.04 or drift rate approximately twice that ofthe equivalent volume sphere. Equation 12 predicts a value2.034, consistent with the numerical measurement, whereasequation 9 gives 2.39. In their section 4.3.1 Ragozzine &Brown (2009) speculated that using the volumetric radiusleads to an underestimate of the tidal evolution. Howeverfor the axis ratio of Haumea b/a ≈ 0.8 and c/a ≈ 0.52, wefind here that the drift rate would only be about twice asfast as estimated using the volumetric radius.

Kondratyev (2016) proposed that stresses between icyshell and core and associated relaxation would cause ice toaccumulate at the ends of Haumea. He proposed that the icyends could separate forming the two icy satellites Namakaand Hi’iaka. Estimates for the fraction of ice in a differen-tiated Haumea range from 7% (Kondratyev 2016) to 30%(Probst et al. 2015).

Young’s modulus of ice is estimated to be a few GPa(Nimmo & Schenk 2006; see Collins et al. 2010 for a review)and this is about 10 times lower than the Young’s modulusfor rocky materials. To explore the affect of softer icy endson the semi-major axis drift rate, we ran a simulation of abody that is not homogeneous. Using the same axis ratiosof b/a = 0.8 and c/a = 0.5 and parameters of the LR series,we reduced the spring constants to 1/10th the value in thebody core at radii greater than 1 (in units of volumetric ra-dius) from the body center. About 20% of the springs havereduced spring constants. This has the effect of lowering thesimulated Young’s modulus at the ends of the ellipsoid by afactor of 10. We did not vary the density as the difference indensity between ice and rock is much lower than their dif-ference in elastic modulus. The spring damping parameterγ does not vary, so τ , the viscoelastic relaxation time-scale,and tidal frequency, χ, is the same in both regions. Themeasurements of semi-major axis for this simulation and forthe homogeneous one with the same axis ratios are shownin Figure 6 along with linear fits that measure the seculardrift rate. We measured the drift rate in semi-major axis inthis simulation, finding that it is about 5.2 times faster thanthe homogeneous ellipsoid with the same axis ratios and 10times faster than the equivalent homogeneous sphere. Even

MNRAS 000, 000–000 (0000)

8 Quillen et al.

Figure 6. We compare the drift rate in orbital semi-major axis

for two simulations in the LR series and with axis ratios b/a =

0.8 and c/a = 0.6, similar to Haumea. The blue points show asimulation of a homogeneous body, whereas the black points show

a simulation with weaker ends (where springs have a lower spring

constant), mimicking a rocky body with icy ends. The lines showlinear fits measuring the secular drift rate. The x axis shows time

in units of tg (equation 1) and the y axis shows semi-major axis

in units of volumetric radius, Rv , measured from the initial value.The drift rate of the body with soft ends is about 5.2 times faster

than the homogeneous ellipsoid with the same axis ratios andabout 10 times faster than the equivalent volume homogeneous

sphere.

a small fraction of softer material can significantly affectthe simulated drift rates. Perhaps this should have been ex-pected based on the strong sensitivity to elastic anisotropywe saw with the cubic lattice model simulations.

The classical tidal formula for the tidal drift rate insemi-major axis

aonao

=3k2HQH

M∗MH

(RvHao

)5

(13)

(e.g., Murray & Dermott 1999) for perturbing object M∗(here Hi’iaka) due to tidal dissipation in the spinning bodyMH , where kH and QH are the Love number and dissipationfactor for Haumea. We can describe corrections to the tidaldrift rate by multiplying the right hand side by a parame-ter fcorr > 1. The above equation implies that ao ∝ a−5.5

o

(taking into account dependence on nao ∝ a−1/2o ). We inte-

grate the above equation for Haumea to estimate the timeit takes Hi’iaka to tidally drift outwards to its current semi-major axis. Putting unknowns on the left hand side

k2HQH

fcorr ∼1

nτa

(aHiRv

)52

39q−1 ∼ 0.1 (14)

with mass axis ratio q = MHi/MH and where we have usedvalues from Table 3 and an age τa = 4 Gyr for the timescaleover which tidal migration is taking place. Here n and aHiare the mean motion and orbital semi-major axis of Hi’iakaat its current location. If k2H for Haumea is as large as 0.01(at the border of what would be consistent with rigidityfor rocky material) and we use corrections for shape andcomposition increasing the drift rate by fcorr = 10 then

we have equality only if the dissipation parameter is large;Q ∼ 1. We conclude that it is unlikely that Hi’iaka alonetidally drifted to its current location even if tidal drift rateis larger by a factor of 10 than estimated using the equivalentvolume rocky sphere.

One explanation for the origin of Haumea’s satellitesand compositional family is a collisional disruption of a pastlarge moon of Haumea (Schlichting & Sari 2009). The ‘ur-satellite’ would have formed closer to Haumea and becauseof its large mass, could have migrated more quickly thanHi’iaka outward during the lifetime of the Solar system. Thefailure of our enhanced tidal drift rate estimate to accountfor Hi’iaka’s current position would suport the ’ur-satellite’proposal (also see discussion by Cuk et al. 2013).

3 SUMMARY AND DISCUSSION

Motivated by the discovery of spinning elongated bodiessuch as Haumea, we have carried out a series of mass-springmodel simulations to measure the tidally induced drift rate(in orbital semi-major axis) of homogeneous spinning vis-coelastic triaxial ellipsoids in a circular orbit about a pointmass. We have restricted this initial study to bodies spin-ning about the shortest principal body axis aligned with theorbital axis and with tidal frequency times the viscoelas-tic timescale χ � 1 (equivalent to tidal dissipation factorQ� 1).

Our simulations and order of magnitude estimates showthat the tidal torque or associated orbital semi-major axisdrift rates, when normalized by that of a spherical body ofequivalent volume, are described by

aoas≈ 1

2

(1 +

b4

a4

)(b

a

)− 43 ( c

a

)−αc

(15)

with αc ∼ 1.1 consistent with our random lattice simulationsbut αc = 4/3 predicted via order of magnitude estimates.This function is a good match to the prolate simulationsusing either value of αc but better matches the oblate andall the triaxial ones with αc = 1.1.

For a homogeneous body with axis ratios equal to thoseof Haumea (b/a = 0.8, c/a = 0.5) we estimate that thedrift rate in orbital semi-major axis is about twice as fastas that estimated for a spherical body with the same massand volume. Motivated by the proposal that ice could haveaccumulated at Haumea’s ends (Kondratyev 2016) we alsoran a simulation of a non-homogeneous body with 20% ofthe springs (those at the ends of the body) set at 1/10th thestrength of those in the core, approximating a body com-prised of two materials, ice and rock. This simulation has adrift rate 10 times higher than the equivalent homogeneoussphere. Reexamining the tidal evolution of Hi’iaka, we findthat even this increase by 10 is insufficient to have allowedHi’iaka to have drifted tidally to its current location viatidal interaction with Haumea alone. We have only consid-ered the behavior of a solid body with fixed axis ratios anda static viscoelastic rheology and we have neglected the roleof Namaka, so more complex models could reexamine thisconclusion.

We experimented with using a cubic lattice distribu-tion for simulated mass nodes, but suspect that the ran-dom spring model is more accurate because it is elastically

MNRAS 000, 000–000 (0000)

9

isotropic, even though the cubic lattice is more homogeneousand can be set up with shorter springs at the same numberof mass nodes. The random spring model is hampered by asoft and weak surface region with depth set by the maximumlength of the springs. Until we speed up the gravity compu-tation (perhaps using a multipole method) we cannot on asingle processor increase the number of particles past a fewthousand so as to reduce the effect of the soft surface layer.Spin orbit resonances have been neglected from this studyand the order of magnitude scaling estimates rely on crudeapproximation for the stresses associated with tidal acceler-ation. Future work, both analytical and numerical, will berequired to improve upon the accuracy of tidal computationsfor bodies with extreme axis ratios.

Acknowledgements.We thank Valery Lainey and Dan Scheeres for helpful

discussions and correspondence. This work was in part sup-ported by the NASA grant NNX13AI27G and NSF awardPHY-1460352.

APPENDIX A: NUMERICAL COMPARISONSAND TESTS

The orbital parameters and mass ratio for our simulationsare similar to those used by Frouard et al. (2016). In that pa-per we measured the sensitivity of the predicted to measuredorbital semi-major axis drift rate to the size of the initial or-bital semi-major axis. The ratio was independent of orbitalsemi-major axis, implying that for our adopted semi-majoraxis of 10 the torque is independent of the higher order termsin the expansion of the tidal gravitational potential.

For the spherical random R-series simulations, we rana comparison simulation using the adaptive time step 15th-order IAS15 integrator (Rein & Spiegel 2015). The differencebetween measured drift rates was less than 0.02% implyingthat rebound’s faster and less accurate second order leap-frog integrator for our chosen step-size is sufficient for ourstudy.

Gravitational interactions are still computed using thecomputationally intensive but accurate all particle pairs di-rect gravity routine in rebound as initial tests with the fasterbut less accurate tree-code were not promising. (A soft bodythat was barely strong enough to withstand self-gravity us-ing the direct gravity computation imploded with the tree-code). Even at the semi-major axis of 10 and mass ratio of10 (see Tables 1 and 2) the deformations are small and theorbital semi-major axis drift rates, listed for the spheres inTable 2, are of order as ∼ 10−6. The gravity computationmust be done accurately to measure tidal deformation andevolution. We leave development of tree or multipole gravityacceleration methods for future work.

Each time we run a simulation with a random springnetwork, a new set of particles is generated and this meansthere are variations in the spring network between simula-tions. To estimate the variation in measured semi-major axisdrift rates due to differences in the spring network we ran 3simulations with identical parameters each in the R series ofsimulations for the spherical body, and the oblate and pro-late bodies with c/a = 0.5. The standard deviation of dao/dtcomputed from each set of three simulations was less than3% and smaller than the differences between the drift ratesfor simulations with body shapes that differ in axis ratios by0.1. The R-series of random lattice simulations has sufficientnumbers of particles that differences in the generated springnetworks only cause small variations in orbital semi-majoraxis drift rate.

In the random spring model, as particles are never gen-erated outside the ellipsoid surface, there is a higher prob-ability of generating particles just within the surface thannear planar surfaces embedded within the body. This meansthat the surface is stronger and denser than the interior.A way around this problem is to generate a random distri-bution of particles in the same manor but in a larger regionthat contains our desired ellipsoid and then remove particlesthat lie outside the ellipsoid. We refer to models generatedthis way as soft-edged random spring models and those gen-erated with our original procedure as hard-edged randomspring models. The soft-edged distribution is more uniformbut the surface is more porous and weaker. Because there arefewer particles near the surface boundary the density gra-dient is shallower near the surface than for the hard-edgeddistribution (that is shown in Figure 1). The softer surfaceincreases the drift rate, whereas the softer density profilewould decrease it. In comparisons between these differentparticle distributions we find that the soft-edged bodies havehigher orbital semi-major axis drift rates so the softer edgeis a more important factor affecting tidal drift. For spher-ical bodies, the difference in drift rate (using the R-seriesof parameters and between hard- and soft-edged bodies) is20%, however the difference (between hard and soft) in driftrate for the prolate with c/a = 0.5 is 43%. The differencedepends on the body surface area and so the prolate modelsare more affected by the softer edge. Because the hard edgesomewhat reduces the effect of the soft surface layer and howit affects the drift rates, we opted to run the random springmodel simulations using our original method (hard-edged)for generating the random lattice particle distribution.

Even though the maximum spring length is smaller forthe cubic lattice simulations (and so the thickness of the softregion reduced), because the number of springs per node islower than for the random spring lattice, the surface layerfor the cubic lattice can be quite weak. We could strengthenthe surface layer by increasing the maximum spring length(so that there are more springs per node) but this has theeffect of increasing the depth that is weaker than the interiorand any advantage of using the cubic lattice model.

We also ran a series of random lattice model simulationswith larger numbers of particles than the R series that wecall the LR series. The LR series has about 2.5 times thenumber of mass nodes as the R series and takes about 6times longer to run as gravity computations are done usingall pairs of mass nodes (direct rather than using a tree codeor a multipole algorithm). In the LR series we ran a spher-

MNRAS 000, 000–000 (0000)

10 Quillen et al.

ical model, and oblate and prolate models with axis ratioc/a = 0.5. We also ran a triaxial model with b/a = 0.8 andc/a = 0.5. When normalized to the drift rate of the spher-ical simulation in the same series we measured a differencebetween the ratio ao/as computed with 1200 particles (Rseries) and those computed with 3000 particles (LR series)that is less than 4% (when divided by the ratio computedwith 3000 particles). There was no trend; the LR series ratiosdid not increase or decrease with axis ratio. This test sug-gests that we are running sufficient numbers of particles toensure that the measured drift rates are not strongly sensi-tive to the structure of the surface spring network. Howeverwe must keep in mind that the R and LR series only differby 2.5 in the number of particles and the maximum springlength (serving as a skin depth) is a significant faction of thevolumetric radius in both cases (ds = 0.33 for the R seriesand 0.24 for the LR series).

APPENDIX B: SCALING ESTIMATES FORTHE TIDAL DRIFT RATES OFHOMOGENEOUS TRIAXIAL ELLIPSOIDS

The quadrupole moment of a mass distribution in Cartesiancoordinates

Qij =

∫ρ(3xixj − δijr2)d3x (B1)

where ρ is the mass density. For a uniform triaxial ellipsoidwith semi-axes a, b, c and in coordinates aligned with bodyaxes

Q =M

5

2a2 − b2 − c2 00 2b2 − a2 − c2 00 0 2c2 − b2 − a2

(B2)

with mass M = 4π3ρabc. Q is a tensor so we can rotate

it; Q′ = R(θ)QR(θ)−1 with R(θ) a rotation matrix. Afterrotation of the body by angle θ in the xy plane we transformQ to Q′ finding off diagonal term

Q′xy =3

10M sin(2θ)(a2 − b2). (B3)

Distant from the object the quadrupolar contributionto the gravitational potential

V2(x) = GQij2

xixjr5

. (B4)

If a mass M∗ is located at x = ao, y = z = 0 then the zcomponent of the torque on it due to the quadrupolar forceis

T = M∗x∂V2

∂y

∣∣∣∣x=ao,y=0,z=0

=GM∗Q

′xy

a3o

=GMM∗a3o

3

10sin(2θ)(a2 − b2). (B5)

This is equivalent to an application of MacCullagh’s formulafor the instantaneous torque exerted by a planet on the per-manent figure of an extended satellite. If M∗ causes tidaldeformation of M and θ is a lag angle (due to viscoelas-tic response) then this formula can be used to estimate thetorque and associated spin down rate and semi-major axisdrift rate (e.g., see Murray & Dermott 1999).

A three dimensional version of Hooke’s law relatingstress applied on three Cartesian coordinates, σx, σy, σz tostrain in the three directions

εx =1

E(σx − ν(σy + σz)]

εy =1

E(σy − ν(σz + σx)]

εz =1

E(σz − ν(σx + σy)] (B6)

where ν is the Poisson ratio and E the Young’s modulus.More generally a linear relation between stress and strainor Hooke’s law in tensor form is σij = λ(trε)δij + µεij withLame constants λ, µ. In a coordinate system that diagonal-izes the stress and strain tensors, Hooke’s law reduces toequations B6 as off-diagonal terms are zero.

The tidal acceleration on M from distant mass M∗ withM∗ located at x = ao, y = z = 0 is

aT ≈GM∗a3o

(2x,−y,−z) (B7)

taking only the quadrupolar term.Instead of applying tidal force throughout the body

(Dobrovolskis 1982), we roughly approximate it as an in-stantaneously applied stress on the body surface.

We approximate the three stresses as scaling with thetidal acceleration, aT , on the surface, times mass,M , dividedby cross sectional area (from the midplane perpendicular tothe direction applied);

σx ∼ GM∗M

a3o

2a

bc

σy ∼ −GM∗Ma3o

b

ac

σz ∼ −GM∗Ma3o

c

ab, (B8)

where we orient the body with semi-major axis a along the xaxis, b along the y axis and c along the z axis. Using Hooke’slaw (in the form in equations B6) this gives surface strainsof order

εx ∼ GM∗M

a3o

1

Eabc(2a2 + ν(b2 + c2))

εy ∼ GM∗M

a3o

1

Eabc(−b2 + ν(c2 − 2a2))

εz ∼ GM∗M

a3o

1

Eabc(−c2 + ν(b2 − 2a2)). (B9)

B1 Oblate bodies

An oblate body with unperturbed semi-major axes has a =b and volumetric radius Rv = a(c/a)1/3. Due to the tidalforce the body becomes elongated in the equatorial planewith new semi-major and minor axes a′ ≈ a(1 + εx) andb′ ≈ a(1 + εy). We note that the moment Q′xy depends ona′2−b′2 ∝∼ εx−εy. Inserting a′, b′ into the formula (replacinga′, b′ for a, b) for the torque (using equation B5) we find that

To ∝∼GMM∗a3o

GM∗M

a3o

1

Ea2ca4(1 + ν). (B10)

MNRAS 000, 000–000 (0000)

11

We neglect the dependence on lag angle θ. Using the torqueto estimate the semi-major axis drift rate (see equation 5)

aonao

∝∼

(M∗M

)(GM2

Ea4

)(a

ao

)5a

c(B11)

=

(M∗M

)(GM2

ER4v

)(Rvao

)5 ( ca

)− 43

(B12)

where on the first line we are using the equatorial radiusa and on the second line the volumetric radius Rv is usedfor scaling. The first line suggests that using the equato-rial radius alone in the classic tidal formulae for sphericalbodies would lead to an underestimate of the drift rate bythe factor a/c. The dependence on c arises because the esti-mated stresses (equations B8) and strains εx, εy (equationsB9) depend on the inverse of the cross-sectional area.

This series of approximations suggests that the driftrate for oblate bodies should scale with the axis ratio c/a tothe -4/3 power when normalized to a sphere with equivalentvolume(aoas

)oblate

≈( ca

)− 43

(B13)

This approximation is valid when the viscoelastic relaxationtime times the tidal frequency χ < 1. If the tidal frequencyis large compared to the inverse of the viscoelastic relax-ation timescale then the scaling would be different as thenthe quality function is not proportional to shear modulusnormalized to eg.

B2 Triaxial bodies

Orienting the long axis of a triaxial body along the x axis(setting the tide) and using equation B9 gives us strain val-ues for a tidally deformed triaxial body. The aligned buttidally deformed body has semi-major axis a′ = a(1 + εx)and b′ = b(1 + εy) giving torque (inserting a′ and b′ for a, binto equation B5)

Ta ∝∼ a2 − b2 + 2a2εx − 2b2εy (B14)

and we have neglected terms dependent on the Poisson ratio.The term independent of the strain should average to zero(and we will see that makes sense below when we estimatethe torque for the same body but rotated by 90◦ with respectto M∗). We compute

a2εx − b2εy ∼GMM∗a3o

1

Eabc(2a4 + b4) (B15)

We now consider the same body but rotated by 90◦ (asit can be if rotating about the z axis). If the middle axis ofthe body b is oriented along the x axis then

σx ∼ GM∗M

a3o

2b

ac

σy ∼ −GM∗Ma3o

a

bc

σz ∼ −GM∗Ma3o

c

ab(B16)

(recall x is along the tidal axis) giving strains

εx ∼ GM∗M

a3o

1

Eabc(2b2 + ν(a2 + c2))

εy ∼ GM∗M

a3o

1

Eabc(−a2 + ν(c2 − 2b2))

εz ∼ GM∗M

a3o

1

Eabc(−c2 + ν(a2 − 2b2)). (B17)

For the prolate body oriented with b axis along x we havedeformed axes a′ = b(1+εx), b′ = a(1+εy) and the resultingtorque (replacing a′, b′ for a, b in equation B5) is

Tb ∝∼ b2 − a2 + 2b2εx − 2a2εy (B18)

We compute

b2εx − a2εy ∼GMM∗a3o

1

Eabc(2b4 + a4) (B19)

Taking an average of the two torques Ta and Tb we see thatthe terms independent of strain cancel. The torque averagedover rotation (and neglecting angular dependence of phaseangle θ), is approximated from the average of the two orien-tations,

T ∝∼GMM∗a3o

GM∗M

a3o

1

Eabc(a4 + b4). (B20)

The associated drift rate for a triaxial body we estimate as

aonao

∝∼

(M∗M

)(GM2

Ea4

)(a

ao

)5

×(

1 +b4

a4

)(b

a

)−1 ( ca

)−1

(B21)

=

(M∗M

)(GM2

ER4v

)(Rvao

)5

×(

1 +b4

a4

)(b

a

)− 43 ( c

a

)− 43. (B22)

When normalized to the drift rate for an equivalent vol-ume sphere, the triaxial bodies should have drift rates(aoas

)triaxial

≈ 1

2

(1 +

b4

a4

)(b

a

)− 43 ( c

a

)− 43

(B23)

where the factor of 1/2 is so that when a = b = c the ratiois 1.

These order of magnitude scaling estimates do not com-pute the stress and strain fields accurately nor do they ap-propriately average over body rotation.

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