the dependence of surface tidal stress on the internal structure of europa: the possibility of...

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doi:10.1016/j.ps

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(Y. Harada).

Planetary and Space Science 54 (2006) 170–180

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The dependence of surface tidal stress on the internal structureof Europa: The possibility of cracking of the icy shell

Yuji Haradaa,b,�, Kei Kuritaa

aEarthquake Research Institute, The University of Tokyo, 1-1-1 Yayoi, Bunkyo, Tokyo 113-0032, JapanbMizusawa Astrogeodynamics Observatory, National Astronomical Observatory of Japan, 2-12 Hoshigaoka, Mizusawa, Iwate 023-0861, Japan

Received 13 January 2005; received in revised form 30 November 2005; accepted 2 December 2005

Available online 19 January 2006

Abstract

This study shows dependence of the surface diurnal tidal stress on the internal structure of Europa. Its purpose is to investigate

possibility of cracking of the icy shell. The stress is evaluated under the plausible model of the internal structure constrained by the

gravity field data. The possible effective stress at the sub-Jovian point decreases with thickening of the shell, while it does not depend on

the core radius and the thickness of the H2O layer. The range of this value is from 0.095 to 0.161MPa, which does not exceed the tensile

strength of ice. The stress required for the surface cracking would be mainly due to longer period deformations, especially non-

synchronous rotation. And/or the actual strength of the ice at the surface would be smaller because of the preexisting cracks than that at

a laboratory of the same temperature.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Europa; Internal structure; Tidal stress; Icy shell; Cracking

1. Introduction

Europan surface topography reflects action of tectonicprocesses. Voyager (1979) and Galileo (1995–2003) mis-sions, especially high resolution imaging data of Galileo’ssolid state imager have shown existence of a global networkof numerous lineaments on the icy surface. Most of thelineaments are described as double ridges (Greenberg et al.,1998).

There are two main reasons for believing that Europanlineaments represent cracks. The first reason is that salt isdistributed along the lineaments. Darker, reddish materialis distributed in the peripheral region of the double ridges,giving rise to an appearance of triple bands at Voyager’slow resolution images. From the spectroscopic data ofGalileo’s Near Infrared Mapping Spectrometer, this color-ing is due to contamination by heavily hydrated com-

e front matter r 2005 Elsevier Ltd. All rights reserved.

s.2005.12.001

ing author. Mizusawa Astrogeodynamics Observatory,

nomical Observatory of Japan, 2-12 Hoshigaoka, Mizusa-

861, Japan. Fax: +81 197 25 6619.

resses: [email protected], [email protected]

pounds (McCord et al., 1998, 2002). Plausible candidatesfor these are frozen hydrated sulfuric acid and hydratedsalt minerals. Probably these non-ice materials result fromupward transport of brine through the cracks.The second reason is that the morphology of the

lineaments requires preexisting cracks. Two differentformation scenarios for this interesting structure have beenproposed. One is a squeezing model with a thin icy shell onthe order of 1 km (Greenberg et al., 1998), and another is adiapirism model with a thick icy shell on the order of 10 km(Head et al., 1999; Nimmo and Gaidos, 2002; Nimmoet al., 2003). Greenberg et al. (1998) suggested a formationmodel based upon diurnal tidal working on preexistingcracks. During repeated opening and closing of cracks, seawater would be carried from the subsurface ocean, andsqueezed as ice debris or slurry to the surface. Theseeruptions would construct the double ridges along cracks.Head et al. (1999) considered the icy shell to be divided intotwo layers; a cold brittle layer and a warm ductile layer. Ifdiapirism occurs in the ductile layer, the warm ice wouldrise at the bottom of preexisting cracks. The brittle surfacewould be likely to flex upward, and parallel ridges would beproduced. Nimmo and Gaidos (2002) proposed a similar

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ARTICLE IN PRESSY. Harada, K. Kurita / Planetary and Space Science 54 (2006) 170–180 171

two-layered model, and estimated frictional heating atstrike-slip faults caused by the diurnal tide. They explainedthe double ridge structure by diapiric upwarp of the upperlayer and percolation of melt water at the shear zone. Theexistence of cracks prior to formation of the double ridgesis assumed in either of these models.

As for the origin of the cracking, tidal stress isconsidered as a major driving force. In several previousstudies (e.g. Helfenstein and Parmentier, 1980, 1983, 1985;Leith and McKinnon, 1996; Greenberg et al., 1998),correlation between orientations of the lineaments andthose of the principal axes of the tidal stress have beeninvestigated. Summarizing these studies (especially Green-berg et al., 1998), orientations of new lineaments, whichcrosscut older lineaments, tend to be perpendicular topresent directions of the maximum tensile stresses of tide.Although orientations of old lineaments are inconsistentwith these directions, most of these orientations can beexplained in terms of non-synchronous rotation (Green-berg et al., 1998). From these previous studies, it isconcluded that both the new and old lineaments resultfrom the cracking by the tidal stresses due to diurnalvariation and non-synchronous rotation.

The existence of cracks constrains not only directions ofthe surface stress field but also magnitude of the stress. Ifthe tidal force really fractures the icy surface, magnitude ofits maximum tensile stress ought to be greater than thefailure strength of ice. Magnitude of the surface tidal stressdepends on the internal structure as well as the orbitalelements. Therefore, whether cracking of the shell occurs ornot would be also dependent on the nature of the Europaninterior.

In several previous studies (Helfenstein and Parmentier,1980, 1983, 1985; Leith and McKinnon, 1996; Greenberg etal., 1998), tidal deformation has been modeled for Europaassuming a two-layered sphere; a uniform elastic icy shellfloating on a water ocean. Helfenstein and Parmentier(1980) have made an estimate of the directions of theprincipal axes on the Europan surface based on elasticdeformation of the icy shell. Subsequent investigations(Helfenstein and Parmentier, 1980, 1985; Leith andMcKinnon, 1996; Greenberg et al., 1998) have estimatedalso the value of the principal stress. For instance,Greenberg et al. (1998) evaluated the maximum principalstress as approximately 0.15MPa (including the effect ofboth diurnal tide and non-synchronous rotation). In theirmodels, the internal structure of Europa is simplified asconsisting of the shell and the ocean. In fact, Europa has asolid core, which is possibly differentiated into metal androck. Giving a more realistic internal structure, themaximum principal stress could be different from theabove value. Moreover, dependence of tidal stress onthe internal structure (i.e. the shell thickness) has not beeninvestigated in these studies, although the shell thicknessvaries the surface stress.

In other previous studies (Moore and Schubert, 2000;Wu et al., 2001), tidal deformation has been modeled

assuming a four-layered sphere; from inside to outside,metal, silicate, water, and ice. Their main purpose was todetermine tidal amplitude in order to discriminate theexistence of a subsurface ocean. They calculated the tidalamplitude and the tidal Love numbers, including theirdependence on the shell thickness. They did not mention,however, the tidal stress and its dependence on the internalstructure.In this study, the tidal stress is reevaluated for the

plausible model of the internal structure of Europa. Thepurpose is to investigate the possibility of cracking ofthe icy shell. The admissible range of the internal structureis constrained by the gravity field determined fromGalileo’s Doppler radio tracking data (Anderson et al.,1997, 1998). The focus is particularly on how themagnitude of the tidal stress depends on the thicknessesof the four layers, especially the shell thickness. The tidalstress treated in this study is due to diurnal variation,although Greenberg et al. (1998) commented both diurnalvariation and non-synchronous rotation.

2. Model

2.1. Layered structure

In the present study, a density profile for Europa isobtained by a method similar to that of Sohl et al. (2002).The Europan interior is modeled simply as a sphericallysymmetric multi-layer. Each of the layers is chemically andmineralogically homogeneous, so each layer has a uniformdensity. The layers are shown as follows; a metallic core ofFe (iron) and FeS (iron sulfide) alloy, a rocky mantle ofFe2SiO4 (fayalite) and Mg2SiO4 (forsterite) solid solution,a water ocean, and an icy shell of ice Ih. High pressurephases of H2O (e.g. ice III, V, VI) are not likely tocrystallize at the bottom of the H2O layer since the pressureat the mantle–ocean boundary is too low to induce thesephase transitions.There are two differences between the model of Sohl

et al. (2002) and that of the present study. First, in themodel of Sohl et al. (2002), H2O is a single layer, whereasin the model of the present study, H2O is divided into twolayers; the solid shell and the liquid ocean. Because thedensity of ice Ih and that of liquid water are very close, thediscrimination is not possible by the present gravityanalysis. Tidal response is, however, quite differentbetween the shell and the ocean. In this study, the thicknessof the shell is treated as a free parameter.Second, the model of Sohl et al. (2002) includes effects of

pressure and temperature on the densities of the layers. InSohl et al. (2002), these effects are evaluated using theisothermal Murnaghan equation of state corrected forthermal pressure (Fei et al., 1990) under volume-averagedpressure and temperature of each layer. These effects areignored here as they have little effect on tidal deformation.

ARTICLE IN PRESSY. Harada, K. Kurita / Planetary and Space Science 54 (2006) 170–180172

2.2. Liquid phases

From the observation of the magnetic field, it isconsidered that the interior of Europa includes a probablewater ocean. Galileo’s magnetometer data (Kivelson et al.,1997, 2000; Khurana et al., 1998) indicate that Europa hasan induced magnetic field, which is generated by currentinduced in an electrically conducting layer in Europa as itorbits in the magnetosphere of Jupiter. The existence of theinduced magnetic field is the most convincing argument forthe subsurface ocean.

For simplicity, the core is assumed to be liquid metal, inagreement with Moore and Schubert (2000). AlthoughGalileo’s magnetometer data indicate that the Europanmagnetic field is not intrinsic but induced, the data cannotnecessarily resolve whether the core is liquid or solid. AsMoore and Schubert (2000) point out, the effect of a solidcore on reducing tidal deformation is relatively small.

2.3. Free parameters

To construct the density profile, the densities and thethicknesses of the layers are given as parameters in order tosatisfy the following constraints for the mass M, radius R,and moment of inertia I of the satellite (Sohl et al., 2002):

M ¼4

3pX4i¼1

riðr3i � r3i�1Þ, (1)

R ¼X4i¼1

ðri � ri�1Þ, (2)

I ¼8

15pX4i¼1

riðr5i � r5i�1Þ, (3)

where ri and ri are the density and outer radius of the ithlayer, respectively, beginning with the core, and r0 ¼ 0 forconvenience. The outer radius of the shell is constant,r4 ¼ R, from Eq. (2). The densities of the ocean and theshell are also constant, r3 ¼ rwater: the density of liquidwater and r4 ¼ rice: the density of ice Ih, respectively, inthe present model. Unknown are the following fiveparameters; the core radius: r1; the mantle outer radius:r2; the ocean outer radius: r3; the core density: r1; themantle density: r2.

These five parameters are not mutually independent.Above equations can be rewritten as a matrix equation interms of r1 and r2:

43pr31

43pðr32 � r31Þ

815pr51

815pðr

52 � r51Þ

24

35 r1

r2

" #

¼

M � 43p rwaterðr

33 � r32Þ þ riceðR

3 � r33Þ� �

I � 815p rwaterðr

53 � r52Þ þ riceðR

5 � r53Þ� �

24

35. ð4Þ

When r1, r2, and r3 are given, r1 and r2 are uniquely fixedby Eq. (4) whenever r1or2. In this study, the three radii areregarded as free parameters to construct the density profile.In other words, tidal stress is calculated for models ofvariable values of these three parameters.In the present model, the five unknown parameters

obviously must satisfy the inequalities below:

0or1or2or3oR, (5)

rFeXr1XrFeS, (6)

rfaXr2Xrfo, (7)

where rFe, rFeS, rfa, and rfo are the densities of iron, ironsulfide, fayalite, and forsterite, respectively. Under radiisatisfying (5), when densities determined from Eq. (4) areout of the range of (6) or (7), these density profiles areexcluded from the present model.Inequalities (5)–(7) allow the range of the thickness of

the icy shell as

1pR� r3p159 (8)

and the range of the thickness of the total H2O layer (i.e.the icy shell plus the water ocean) as

100pR� r2p160 (9)

and the range of the radius of the metallic core as

150pr1p700, (10)

where, in the present model, R� r3, R� r2, and r1 arechanged at regular intervals of 1, 10, and 50km, respectively.

3. Method

3.1. Tidal potential

In the present study, Europan tidal deformation iscalculated for a self-gravitating elastic multi-layered spherebased on the Takeuchi’s formulation (Takeuchi, 1950;Takeuchi and Saito, 1972; Saito, 1974). The externalforcing assumed here is a diurnal tidal potential inhydrostatic equilibrium. The diurnal tide is consideredfor the double ridge formation (e.g. Greenberg et al., 1998;Nimmo and Gaidos, 2002). In the calculation here, theorbit is assumed to be synchronous with zero inclination.In an Europan reference frame, the time-varying tidal

potential is given as follows (e.g. Kaula, 1964; Poirier et al.,1983; Segatz et al., 1988; Moore and Schubert, 2000):

F ¼ r2o2e �32

P02ðcos yÞ cosot

�þ1

4P22ðcos yÞ½3 cosot cos 2fþ 4 sinot sin 2f�

�, ð11Þ

where r: radius from the center of mass, o: the orbitalangular frequency, e: the orbital eccentricity, y: colatitudewith zero at the north pole, f: longitude with zero at thesub-Jovian point, t: time with zero at the peri-Jove point.In Eq. (11), the potential is approximated simply to firstorder in e and lowest order in r=A, where A: the orbitalsemimajor axis (see also Murray and Dermott, 1999).


3.2. Governing equations

The governing equations for isotropic elasticity areexpanded into surface spherical harmonics. Although thesolids in the satellite are assumed to be Maxwell visco-elastic bodies in Moore and Schubert (2000), these solidsare assumed to be Hooke elastic bodies here. The validityof this elastic approximation will be mentioned in thediscussion. Under quasi-static approximation, the sphericalharmonic expansion yields six first-order linear differentialequations (Takeuchi and Saito, 1972):

dy1

dr¼

1

lþ 2my2 �

lr½2y1 � nðnþ 1Þy3�

� �, (12)

dy2

dr¼

2

rldy1

dr� y2

� �þ

1

r

2ðlþ mÞr� rg

� �

�½2y1 � nðnþ 1Þy3� þnðnþ 1Þ

ry4

� r y6 �nþ 1

ry5 þ

2g

ry1

� �, ð13Þ

dy3

dr¼

1

my4 þ

1

rðy3 � y1Þ, (14)

dy4

dr¼ �

lr

dy1

dr�

lþ 2mr2½2y1 � nðnþ 1Þy3�

þ2mr2ðy1 � y3Þ �

3

ry4 �

rrðy5 � gy1Þ, ð15Þ

dy5

dr¼ y6 þ 4pGry1 �

nþ 1

ry5, (16)

dy6

dr¼

n� 1

rðy6 þ 4pGry1Þ þ

4pGrr½2y1 � nðnþ 1Þy3�,

(17)

where y1, y2, y3, y4, and y5 are the radial factors of theradial displacement, the radial stress, the tangentialdisplacement, the tangential stress, and the potentialperturbation, respectively, and y6 is a quantity related tothe radial gradient of the potential perturbation. Note thatthe definition of y6 is different from that by Alterman et al.(1959). And r, l, m, g, n, and G are density, Lame’s elasticmodulus, elastic rigidity, gravitational acceleration, degree,and the universal gravitation constant, respectively. In thesolid layers, the above equations are integrated simulta-neously.

In liquid case, however, the governing equationsdegenerate (Dahlen, 1974). Under quasi-static approxima-tion, the spherical harmonic expansion yields two first-order linear differential equations (Saito, 1974):

dy5

dr¼

4pGrg�

nþ 1

r

� �y5 þ y7, (18)

dy7

dr¼

2ðn� 1Þ

r

4pGrg

y5 þn� 1

r�

4pGrg

� �y7, (19)

where y7 is a quantity related to the radial gradient of thepotential perturbation instead of y6. Note that thisdefinition of y7 is different from that by Longman (1963).In the liquid layers, the above equations are integratedsimultaneously.Under the profiles on rðrÞ, lðrÞ, mðrÞ, and gðrÞ, the above

ordinary equations are integrated numerically using theRunge–Kutta–Fehlberg scheme. The initial values and theboundary conditions for integrating the function y aregiven in detail in Saito (1974). Brief summaries of them areshown in Appendix B.

3.3. Failure criterion

The possibility of cracking in the icy shell is consideredusing von Mises’s failure criterion. It is based on thetheory of shear strain energy. For the application of thiscriterion, firstly, principal stresses are evaluated from thestress tensor calculated through the method of Takeuchi.Secondly, an effective stress is evaluated from these princi-pal stresses.As a value compared with the strength of ice, an effective

stress (also called an equivalent stress) is introduced. Thatis defined as (e.g. Schulson, 2002):

se ¼1ffiffiffi2p ðs1 � s2Þ

2þ ðs2 � s3Þ

2þ ðs3 � s1Þ

2� �1=2

, (20)

where se: effective stress, s1: maximum principal stress, s2:intermediate principal stress, s3: minimum principal stress.The way of computing the principal stress is shown inAppendix B. From von Mises’s failure criterion, failureoccurs when this effective stress reaches the uniaxial tensilestrength of ice.

4. Result

4.1. Standard point

First of all, a specific point on the Europan surface wasselected for considering the dependence of the tidal stresson the internal structure. The variation of the internalstructure does not shift the surface stress field pattern, butdoes increase or decrease principal stresses and an effectivestress at each point on the surface. The influence of thevariation in the internal structure on the surface effectivestress can be evaluated from calculating an effective stressat a certain colatitude and longitude with varying theinternal structure. As this standard point, the coordinateswhere the highest effective stress is achieved should bechosen in order to address the possibility of cracking at thesurface. To decide the coordinates, the stress field on theglobal surface is calculated.The surface tidal stress field at the peri-Jove is shown in

Figs. 1 and 2, which express the principal axes and theeffective stress, respectively. The pattern of the principalstress field is almost identical with those of the previous

ARTICLE IN PRESS

Fig. 1. The orientation of the principal axes of the surface tidal stress field

under the peri-Jove. The internal structure is that of the shell thickness of

1 km, the total H2O layer depth of 130 km, and the core radius of 600 km.

The black and gray bars indicate the tensile and compressive stress,

respectively. The scale bars at the lower right side indicate 0.1MPa.

Fig. 2. The distribution of the effective stress of the surface tidal stress

field under the peri-Jove. The internal structure is the same as that of Fig.

1. The unit of the legend on the right side is MPa.

0.180.170.160.150.140.130.120.110.100.09

Effe

ctiv

e S

tres

s [M

Pa]

0 20 40 60 80 100 120 140 160Shell Thickness [km]

Core Radius = 600 km, H2O Thickness = 130 km

Fig. 3. The variation of the effective stress with respect to the shell

thickness at the sub-Jovian point under the peri-Jove.

0.180.170.160.150.140.130.120.110.100.09

95 100 105 110 115 120 125 130 135H2O Thickness [km]

Effe

ctiv

e S

tres

s [M

Pa]

120 km90 km

60 km

30 km

Core Radius = 600 km

Fig. 4. The variation of the effective stress with respect to the internal

structure, especially the H2O depth at the sub-Jovian point under the peri-

Jove. The values attached to the lines indicate the thickness of the shell.

Y. Harada, K. Kurita / Planetary and Space Science 54 (2006) 170–180174

studies (e.g. Greenberg et al., 1998, 2002, 2003). In thesestudies, diurnal variation of the surface tidal stress fieldover one orbital period is illustrated in detail. Onexamination of these patterns at the peri-Jove, the effectivestress is highest at the points on the meridian planeperpendicular to the semimajor axis of the tidal deforma-tion (i.e. longitudes of 90� and 270�). However, the vonMises’s failure criterion tends to be inapplicable to a stressfield including compression. In this meaning, these pointsare inappropriate for the standard.

As the standard coordinate, the sub-Jovian point at theperi-Jove is settled:

r ¼ R, (21)

y ¼p2, (22)

f ¼ 0, (23)

t ¼ 0. (24)

This standard point is always on the semimajor axis of thetidal bulge whenever at the peri-Jove. The radial displace-

ment at this point is necessarily a maximum value, whilethe tangential displacement is zero. There the stress field isa horizontally isotropic tensile field. Setting the anti-Jovianpoint instead of the sub-Jovian point as the standard haspractically the same result because of the axial symmetry oftidal deformation.

4.2. Structural dependences

The variation of the effective stress with respect to theinternal structure at the standard point defined above isshown in Figs. 3–5. The calculation here indicates threedistinct trends. Firstly, thickening of the shell decreases thestress under constant H2O thickness and core radius(Fig. 3). Secondly, variation in the H2O thickness hardlychanges the stress under constant shell thickness and coreradius (Fig. 4). Thirdly, variation in the core radius alsoscarcely changes the stress under constant shell thicknessand H2O thickness (Fig. 5). Thus, the stress dependsmainly on the thickness of the icy shell. The series ofcalculations in this study indicates that the principal stressat the standard point ranges between 0.095 and 0.161MPaunder the present model constrained by the total mass, thesurface radius, and the moment of inertia of the satellite.

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0.180.170.160.150.140.130.120.110.100.09

400 450 500 550 600 650Core Radius [km]

Effe

ctiv

e S

tres

s [M

Pa]

H2O Thickness = 130 km

30 km

60 km

90 km120 km

Fig. 5. The variation of the effective stress with respect to the internal

structure, especially the core radius at the sub-Jovian point under the peri-

Jove. The values attached to the lines indicate the thickness of the shell.

Y. Harada, K. Kurita / Planetary and Space Science 54 (2006) 170–180 175

5. Discussion

5.1. Interpretation for dependences

The effective stress at the standard point is dependent onthe shell thickness, and independent on the H2O thicknessand the core radius. These dependences can be mostlyinterpreted in terms of two effects as follows. One is thegravitational potential perturbation at the bottom of theshell, and another is the elastic support of the shell. This isbecause the radial stress loading at the bottom of the shelland the resistance of the shell to this radial stress controlboth displacement and stress in the shell. The radial stressis provided by the deformation of the ocean, which followsthe gravitational potential perturbation (i.e. the tidalpotential plus the perturbation due to deformation). Theresistance means the elastic support of the shell.

On the basis of these effects, the dependences areexplained as below. First, suppose that the H2O thicknessand the core radius are fixed and only the shell thickness isvaried. When the shell is thicker, its elastic support isgreater, and the tidal potential at its bottom is smaller.These effects reduce the displacement and the stress in theshell. As a result, thickening of the shell decreases the stressat the surface.

Second, suppose that the only the shell thickness is fixedand the H2O thickness and/or the core radius are varied.Since the shell thickness is fixed, the effect of its elasticsupport is also unchangeable. As to the potential perturba-tion at its bottom, although the tidal potential is notvaried, the perturbation due to the deformation can bevaried. Variation of the core radius and/or the mantlethickness changes the displacement of these layers. Theperturbation due to the deformation of these layers is,however, too small to effect the deformation of the ocean.This is because the elastic support of the mantle is muchgreater than that of the shell. The range of the mantlethickness allowed by the constraints is 750–1250 km, whichgives a fairly small displacement in the core and the mantle.As a result, the stress does not depend on the core radiusand the H2O thickness.

Here, notice the following two points on the firstexplanation. One point is that the body force directlyworking on the shell itself hardly contributes to thedeformation of the shell. Certainly, the body force on theshell as a whole varies with thickening of the shell. Thisinfluence is, however, much smaller than the surface forceworking at the bottom of the shell by the deformation ofthe ocean. It is clear since the displacement of the surface inthe case of the absence of the ocean is also much smallerthan that in the case of the presence of the ocean (Mooreand Schubert, 2000; Wu et al., 2001). This variation in thebody force is negligible in the case of the interior with theocean.Another point is that the shell is approximated as a

perfectly elastic material here. The relaxation time of iceclose to its melting point is, however, equivalent to theperiod of the Europan diurnal tide (e.g. Ojakangas andStevenson, 1989a; Moore and Schubert, 2000; Tobie et al.,2003). At least near the ocean interface, therefore, ice ispossible to behave as a visco-elastic material, especially inthe convective shell thicker than about 10–20 km. In thiscase of the relaxation of ice, the bottom of the shell as aviscous material practically acts like the ocean under thediurnal forcing. In other words, the top of the shell as anelastic material is thinner than the actual shell in total. As aresult, the negative slope of the curve in Fig. 3 may becomesmaller in fact. The maximum value in Fig. 3 in the case ofvisco-elasticity is, however, same as that of perfectelasticity. From this viewpoint, the elastic approximationdoes not affect the validity of the discussion in the nextsubsection.Notice also the following point on the second explana-

tion. The point is that the value of the effective stress inFig. 3 is greater than that in Greenberg et al. (1998) asmentioned in the next subsection. It seems strange resultbecause their model consists of only the shell and the ocean(see the introduction) whereas the present model includesalso the mantle (the elastic layer) and the core. Theperturbation due to the deformation of the model withoutthe elastic layer should be larger than that with the elasticlayer (Moore and Schubert, 2000). The reason for thisdiscrepancy is uncertain since the detail of the model andthe method is not mentioned in Greenberg et al. (1998). Itmay result from the difference between the physicalproperties employed in their model and those in thepresent model and/or possible exclusion of the perturba-tion due to the deformation from their model.

5.2. Possibility of cracking

The maximum value in the effective stress at thestandard point is 0.161MPa. The value is greater thanthat in Greenberg et al. (1998). This stress is about0.1MPa, although it increases to about 0.15MPa coupledby the effect of non-synchronous rotation. They comparetheir values with the uniaxial tensile strength of ice asobtained from laboratory measurement. This strength is


about 0.3MPa at the strain rate of 10�8 s�1 (Mellor, 1986).Under this strain rate, however, the behavior of ice tends tobe ductile rather than brittle (Hawkes and Mellor, 1972).Furthermore, this tensile strength was determined at a hightemperature of �7 �C. The actual surface temperature ofEuropa is about 100K (Spencer et al., 1999). At such a lowtemperature, much higher tensile strength is expected fromvery low-temperature experiment. This value is about10MPa, which is expected from the compressive strengthat the same temperature (Arakawa and Maeno, 1997).

With any internal structure allowed by the aboveconstraints, the effective stress cannot exceed the strengthof ice. This implication apparently contradicts the evi-dence that, as mentioned in the introduction, the surfacelineaments of Europa are due to tidally driven cracks.

If brittle failure related to possible cracking of the icysurface still occurs under such conditions, three possibi-lities can be considered. The first possibility is variation inthe amplitude of the diurnal tidal potential over the historyof the Jovian system. In the present study, the orbitalelements of the satellite are assumed to be constants; hencethe potential amplitude is also constant. Actually, theorbital angular frequency and orbital eccentricity havevaried in the Jovian history.

The second possibility is a long-term deformation. In thepresent study, only the short-term diurnal tide is regardedas a deformation process. Actually, much longer-perioddeformations may have operated. Possible candidates forthese are a deformation due to potential variation otherthan the diurnal tide (e.g. non-synchronous rotation, truepolar wander, spinning down) and a deformation otherthan that due to potential variation (e.g. volumetricchange). When its orbit is non-synchronous, or withvarying inclination, the equilibrium tidal bulge itself shiftsaccording to the equilibrium configuration of the satellite.In this case, the non-synchronous rotation (cf. Helfensteinand Parmentier, 1985; Greenberg et al., 1998) and/or thetrue polar wander (cf. Ojakangas and Stevenson, 1989b;Leith and McKinnon, 1996) must be also considered as thetidal deformation. When the rotational rate is changed, theequilibrium bulge also deforms because of the variation ofits centrifugal potential. When its surface radius changes, adeformation can occur without these potential variation(Nimmo, 2004). This change of the radius can be generatedby phase transitions of H2O (e.g. liquid to ice Ih)accompanying the thermal evolution. This factor mayhave significant effects on the surface stress field.

The third possibility is the local stress field associatedwith regional cracks, which is impossible to estimate from aglobal tidal deformation model. In the present study, theicy shell is considered as a perfectly intact layer from thetop to the bottom. Actually, the shell is likely to includepreexisting cracks. Especially in the surface brittle layer,the strength of ice cannot recover in a short time because ofits low temperature. Cracks in the brittle layer caused by,for example, large impacts remain for a long time. Since thestress concentrates at these weak points, an effective stress

far exceeding the tensile strength of ice may arise locally.Besides, because diurnal tidal stress is periodically loadedon these weaknesses, the parts where the stress concen-trates are gradually fatigued according to the cyclic stress.The ice in the brittle layer is probably weaker than that in alaboratory.The first possibility is weak in terms of celestial mechanics

and crater chronology. The Laplace resonance with Io andGanymede was probably achieved at an early stage in theorbital evolution (e.g. Showman and Malhotra, 1997; Pealeand Lee, 2002). The crater age of the Europan surface isyoung, on the order of about 100My, depending onestimates of the impact flux (e.g. Zahnle et al., 1998) andpossibly on effect of crater retention. The orbit would settleat the resonance at the earlier than the age of the presentsurface features. In this case, the amplitude variation of thepotential does not have to be taken into account.Indeed it is not entirely excluded that the orbital elements

of Europa evolve due to the thermal–orbital coupling withIo and Ganymede even though the satellites orbitresonantly (Hussmann and Spohn, 2004). For example, itis possible that the eccentricity enlarges from 10�2 to 10�1

suddenly in a short term. In this phase, the surface stressmay become larger than that in the present on themagnitude of also about one order. Even in this case,however, this stress is smaller than the strength of ice asmentioned above on the magnitude of about one order.The second and third possibilities are relatively reason-

able. With regard to the second, the diurnal tidal stress mayplay a role of a trigger imposed on a background stresscaused by other deformations as above. There is oneproblem in this explanation. Long-term deformations arenecessarily accompanied by viscous relaxation of the stress.If the diurnal tide is the trigger, the time scale of build-up inthe background stress must be much shorter than the timescale of relaxation of ice.Non-synchronous rotation is the most plausible candi-

date among the background stresses (Greenberg andWeidenschilling, 1984). One reason is that the relationbetween the orientation of the lineaments and the principalstresses becomes more consistent by considering not onlythe diurnal tide but also the non-synchronous rotation(Greenberg et al., 1998). Another reason is that the periodof non-synchronous rotation is relatively short, on theorder of from 0.01 to 100My. The lower limit is estimatedfrom comparing the position relative to the terminator orthe limb of features of Europa as imaged by Voyager andby Galileo (Hoppa et al., 1997). The upper limit is thecrater age because crater statistics on Europa do not showany detectable leading and trailing hemispheric asymmetry.

6. Conclusion

The dependence of the surface diurnal tidal stress on theinternal structure of Europa is investigated in order toexplore the possibility of cracking of the icy shell. Underthe internal density profiles satisfying the constraints of


total mass and moment of inertia of the satellite, thepossible effective stress at the standard point decreases withthickening of the shell, and the stress does not depend onthe core radius and the H2O thickness. The reasons forthese dependences are interpreted in terms of the gravita-tional potential perturbation at the bottom of the shell andthe elastic support of the shell.

The range of this value is from 0.095 to 0.161MPa underthe internal structure allowed by the constraints in thepresent model. The range does not exceed the tensilestrength of ice. One possibility is that the most of thesurface stress required for cracking the ice would derivefrom the longer period deformations, especially non-synchronous rotation. Another possibility is that the actualstrength of the ice at the surface would be smaller becauseof the preexisting cracks than that at a laboratory of thesame temperature.

Acknowledgements

The two anonymous reviewers are greatly appreciatedfor their careful reviews and constructive comments on thisstudy. Yuji Harada obtained valuable advice and usefulsuggestions about tidal deformation from Prof. ShuheiOkubo (Earthquake Research Institute, the University ofTokyo), and about fracture mechanics of ice from Prof.Masahiko Arakawa and Dr. Yasuyuki Yamashita (Grad-uate School of Environmental Studies, Nagoya Univer-sity). For this study, Yuji Harada has used the computersystems of Earthquake Information Center of EarthquakeResearch Institute, The University of Tokyo.

Appendix A

The constants used in the present study are listed inTables 1 and 2.

Appendix B

An arbitrary solution for the yi ð1pip7Þ is expressed bya linear combination of the three integrations in the solidlayers ð1pip6Þ and the one integration in the liquid layersði ¼ 5; 7Þ. A second subscript j is introduced on theyij ð1pjp3Þ in the solid layers in order to distinguish thethree independent integrations. Complete solutions in

Table 1

The physical properties of Europa

Parameter Unit Value Name

Total mass kg 4:800� 1022 M

Surface radius km 1560 R

Moment of inertia kg km2 4:042� 1028 I

Orbital angular frequency rad/s 2:05� 10�5 oOrbital eccentricity — 9:3� 10�3 e

The former three and the latter two are referred from Sohl et al. (2002) and

Moore and Schubert (2000), respectively.

the core, the mantle, the ocean, and the shell are given,respectively, by

yi ¼ Qcore1 yl

i1, (25)

yi ¼X3j¼1

Qmantlej ys

ij , (26)

yi ¼ Qocean1 yl

i1, (27)

yi ¼X3j¼1

Qshellj ys

ij , (28)

where QðlayerÞj is a constant of integration determined by the

boundary conditions and

Qcore1 ¼ Qmantle

1 ¼ Qocean1 ¼ Qshell

1 . (29)

The superscripts s on ysij and l on yl

i1 refer to values in thesolid layers and the liquid layers, respectively.At the center, the integration of yl

i1 can be started frominitial values

yl51 ¼ rn, (30)

yl71 ¼ 2ðn� 1Þrn�1. (31)

At the core–mantle boundary, the boundary conditionsdecompose into three independent sets. The first set is

ys11 ¼ 0, (32)

ys21 ¼ �r

lyl51, (33)

ys31 ¼ 0, (34)

ys41 ¼ 0, (35)

ys51 ¼ yl

51, (36)

ys61 ¼ yl

71 þ4pGrl

gyl51, (37)

the second set is

ys12 ¼ 1, (38)

ys22 ¼ rlg, (39)

ys32 ¼ 0, (40)

ys42 ¼ 0, (41)

ys52 ¼ 0, (42)

ys62 ¼ �4pGrl , (43)

and the third set is

ys13 ¼ 0, (44)

ys23 ¼ 0, (45)

ys33 ¼ 1, (46)

ARTICLE IN PRESS

Table 2

The densities and the elastic constants

Parameter Unit Layer Phase Value Name

Density kg=m3 Core Iron 7800 rFeIron sulfide 5330 rFeS

Mantle Fayalite 4457 rfaForsterite 3222 rfo

Ocean Liquid water 1000 rwaterShell Ice Ih 937 rice

Lame’s elastic modulus GPa Mantle Olivine 29 lolivineShell Ice Ih 19 lice

Elastic rigidity GPa Mantle Olivine 65 molivineShell Ice Ih 9.2 mice

The former and the latter are referred from Sohl et al. (2002) and Turcotte and Schubert (2002), respectively.

Y. Harada, K. Kurita / Planetary and Space Science 54 (2006) 170–180178

ys43 ¼ 0, (47)

ys53 ¼ 0, (48)

ys63 ¼ 0. (49)

The superscripts l on rl refer to values in the liquid layers.At the mantle–ocean boundary, the boundary conditionsare

yl51 ¼

X3j¼1

Qmantlej

Qocean1

ys5j, (50)

yl71 ¼

X3j¼1

Qmantlej

Qocean1

ys6j þ

4pG

gys5j

� �, (51)

where the ratios of Qmantle2 =Qocean

1 and Qmantle3 =Qocean

1

are

X3j¼1

Qmantlej

Qocean1

ys2j ¼ rl

X3j¼1

Qmantlej

Qocean1

ðgys1j � ys

5jÞ, (52)

X3j¼1

Qmantlej

Qocean1

ys4j ¼ 0. (53)

The above equations determine the ratios of Qmantle2 =Qocean

1

and Qmantle3 =Qocean

1 and hence all the yi’s at the boundaryare determined except for a free multiplier Qocean

1

ð¼ QðlayerÞ1 Þ. It is to be determined by the surface boundary

condition. At the ocean–shell boundary, the boundaryconditions are same as those at the core–mantle boundary.The surface boundary conditions are as follows:

X3j¼1

Qshellj ys

2j ¼ 0, (54)

X3j¼1

Qshellj ys

4j ¼ 0, (55)

X3j¼1

Qshellj ys

6j ¼2nþ 1

R. (56)

After solving the surface boundary conditions for the threeconstants of integration, stresses at arbitrary place can becalculated.At the surface, the stress tensor is

srr ¼ 0, (57)

sry ¼ 0, (58)

srf ¼ 0, (59)

syy ¼ ly1 þlþ 2m

r½2y1 � nðnþ 1Þy3� �

2mr

y1

� �Y m

n

�2mr

y3 cot yqqyþ

1

sin2 y

q2

qf2

� �Y m

n , ð60Þ

syf ¼2mr

y3

1

sin yq2

qyqf�

cos y

sin2 y

qqf

� �Y m

n , (61)

sff ¼ ly1 þlþ 2m

r½2y1 � nðnþ 1Þy3� �

2mr

y1

� �Y m

n

�2mr

y3

q2Y mn

qy2, ð62Þ

where Y mn is surface spherical harmonics. The diagonaliza-

tion of the matrix ðsyysyfsyfsffÞ yields two eigenvalues and their

corresponding eigenvectors (i.e. principal stresses and itsprincipal axes, respectively). When syfa0, the principal


stresses and the principal axes are

sp ¼

syy þ sff þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsyy � sffÞ

2þ 4s2yf

q2

then�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðsp � sffÞ2þ s2yf

q ðsp � sff;syfÞ;

syy þ sff �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðsyy � sffÞ

2þ 4s2yf

q2

then�1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðsp � syyÞ2þ s2yf

q ðsyf;sp � syyÞ:

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

(63)

where sp: principal stress. The norms of the principal axesabove are normalized to unity. When syf ¼ 0, these are

sp ¼syy then� ð1; 0Þ;

sff then� ð0; 1Þ:

((64)

The norms above are again normalized.Each of the two principal stresses above corresponds to

one of s1, s2, and s3 in Eq. (20). The rest one of s1, s2, ands3 is zero at the surface.

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