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Icarus 192 (2007) 41–55www.elsevier.com/locate/icarus

Time-varying interaction of Europa with the jovian magnetosphere:Constraints on the conductivity of Europa’s subsurface ocean

Nico Schilling ∗, Fritz M. Neubauer, Joachim Saur

Institut für Geophysik und Meteorologie, Universität zu Köln, Albertus-Magnus-Platz, 50923 Köln, Germany

Received 22 December 2006; revised 21 June 2007

Available online 15 August 2007

Abstract

We study the conductivity distribution inside Europa by using a time-dependent 3D-model of the temporal periodic interaction between Europaand the jovian magnetosphere. The temporal variations are caused by periodic variations of the magnetospheric plasma and magnetic fields atEuropa. We develop a model which describes simultaneously the 3D plasma interaction of Europa’s atmosphere with Jupiter’s magnetosphereand the electromagnetic induction in a subsurface conducting layer due to time-varying magnetic fields including their mutual feedbacks. Wefind that inclusion of the magnetic field perturbations caused by the interaction with Jupiter’s magnetospheric plasma is important for interpretingGalileo’s magnetic field measurements near Europa. This leads to improved constraints on the conductivity and thickness of Europa’s subsurfaceocean. We find for the conductivity of Europa’s ocean values of 500 mS/m or larger combined with ocean thicknesses of 100 km and smaller tobe most suitable to explain the magnetic flyby data. In summary, this results in the following relation: electrical conductivity × ocean thickness� 50 S/m km. It is shown that the influence of the fields induced by the time variable plasma interaction is small compared to the induction causedby the time-varying background field, although some aspects of the plasma interaction are changed appreciably.© 2007 Elsevier Inc. All rights reserved.

Keywords: Europa; Interiors; Magnetic fields; Jupiter, magnetosphere

1. Introduction

Galileo measurements of Europa’s gravitational field andmodeling show Europa to be a differentiated satellite consist-ing of a metallic core, a silicate mantle and a water ice–liquidouter shell. The minimum water ice–liquid outer shell thick-ness is about 80 km for plausible mantle densities (Andersonet al., 1998). High resolution data obtained with the SolidState Imaging (SSI) system show evidence of a young andthin, cracked and ruptured ice shell (e.g., Belton et al., 1996;Carr et al., 1998). The geological observations imply that warm,convecting material existed at shallow depths within the subsur-face at the time of its recent geological deformation. Global-scale tectonic patterns can be explained by nonsynchronousrotation and tidal flexing of a thin ice shell above a liquid wa-ter ocean (Geissler et al., 1998; Greenberg et al., 2000). The

* Corresponding author. Fax: +49 221 4705198.E-mail address: [email protected] (N. Schilling).

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water layer is likely comprised of three sub-layers: an outer,brittle/elastic ice layer, an underlying ductile layer of poten-tially convecting ice, and a lower layer of liquid. Estimatesof the thickness of the ice layer (including the lower ductilelayer) range from a few km to 60 km (e.g., Schenk, 2002;Hussmann et al., 2002). However, while the evidence for liquidwater in the past is favorable, there is no unambiguous indica-tion from spacecraft imaging that such conditions exist today(Pappalardo et al., 1999).

Thermal models indicate that a significant portion of theouter water shell could be liquid today (e.g., Squyres et al.,1983; Schubert et al., 1986; Spohn and Schubert, 2003). Oneenergy source for maintaining a liquid water ocean is tidalheating caused by the three-body Laplace resonance with Ioand Ganymede. This process could offset the freezing of thewater ocean by subsolidus ice convection (e.g., Cassen et al.,1979). The major uncertainty in modeling is the rheology of ice(Durham and Stern, 2001). Also, the rate of freezing of the in-ternal ocean depends on its composition, since the occurrenceof minor constituents in the ice and ocean such as hydrated salt

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42 N. Schilling et al. / Icarus 192 (2007) 41–55

minerals (McCord et al., 1998) and ammonia (Kargel et al.,1991; Deschamps and Sotin, 2001) effect the rheology of theice and the freezing temperature of the ocean.

The strongest indication for a liquid water ocean at presenttime comes from the magnetic field observations (Kivelsonet al., 1999, 2000; Khurana et al., 1998). Europa has no de-tectable permanent internal magnetic field (Schilling et al.,2004). The magnetic field data show evidence for electromag-netic induction taking place in the interior of Europa due to thetime-varying external magnetic field (Neubauer, 1998a, 1998b;Khurana et al., 1998). Therefore, the observations support theidea, that a global subsurface conducting layer may be present.In addition, there are magnetic field perturbations owing tothe interaction of Europa’s atmosphere with the jovian mag-netospheric plasma. These time-varying magnetic field pertur-bations are permanently there on top of the induction signaturecompromising their interpretation. While induction signaturesare clearly visible in the data when Europa is well outsideJupiter’s current sheet, the strong plasma interaction dominatesand hides the induction effect when Europa is close to the cen-ter of the current sheet (Kivelson et al., 1999).

An atmosphere at Europa was detected by Hall et al. (1995)with the Hubble Space Telescope Goddard High-ResolutionSpectrograph. The observations imply molecular oxygen col-umn densities in the range of ∼(2−14)× 1018 m−2 (Hall et al.,1998). The atmosphere of Europa is produced by the interac-tion of energetic charged particles with Europa’s surface (e.g.,Johnson et al., 2004). Shematovich et al. (2005) find by using acollisional 1D Monte Carlo model of Europa’s atmosphere thatthe atmosphere is more strongly structured in the radial direc-tion than previously assumed and that the near-surface region ofthe atmosphere is determined by both water and oxygen photo-chemistry. Recent images of Europa’s atomic oxygen emissionobtained with the HST Space Telescope Imaging Spectrograph(STIS) indicate that the surface is not icy everywhere, but thatthe composition varies with longitude (McGrath et al., 2004).Atomic Na and K are observed in the extended atmosphere(Brown and Hill, 1996; Brown, 2001). They occur in a ratiodifferent from that at Io (Na/K is 25 for Europa and 10 for Io),and from meteoritic or solar abundance ratios (Brown, 2001;Johnson et al., 2002). Therefore a subsurface source of alkalisis suggested (Johnson et al., 2002; Leblanc et al., 2002). A re-cent review on Europa’s atmosphere can be found in McGrathet al. (2004).

Kliore et al. (1997) detected an ionosphere on Europa byusing Galileo radio occultation measurements. They deriveda maximum electron density of about 10,000 cm−3 with ascale height of 240 km. By assuming a spherically symmetricionosphere for each derived electron density profile, Kliore etal. (1997) found a strongly asymmetric ionosphere with max-imum densities on the flanks and minimum densities down-stream. The main source of Europa’s ionosphere is electronimpact ionization (Saur et al., 1998).

Zimmer et al. (2000) investigated the implications of the ob-served induced magnetic fields for the electrical structure ofEuropa’s interior. By using a simple shell model they set boundson the characteristics of the current carrying layer. Zimmer et al.

(2000) find that the magnetic signature at Europa is consistentwith more than 70% of the induced dipole moment expected fora perfectly conducting sphere. Therefore, currents are requiredwhich flow in a shell with conductivity of at least 60 mS/m andclose to the surface (within a 200–300 km depth). Zimmer etal. (2000) argue that solid ice, an ionosphere or a conductingcore cannot reproduce the amplitude of the observed magneticperturbation. In addition, Zimmer et al. (2000) argue that itseems to be very unlikely that the magnetic signature can be ex-plained by induction taking place in a conducting mantle only.They therefore support the idea of a subsurface ocean. However,they do not take into account the plasma interaction of Europawith the jovian magnetosphere, which we show to be importantfor the interpretation of Galileo’s magnetic field measurementsnear Europa. Instead they treat magnetic field perturbations dueto local plasma currents as a systematic error.

Previous models of Europa’s interaction with the jovianmagnetosphere considered either the plasma interaction only(Saur et al., 1998), included an intrinsic dipole moment as afree parameter but did not include the induced magnetic fieldself-consistently (Kabin et al., 1999; Liu et al., 2000), or theyfocused on the electromagnetic induction process inside Eu-ropa (Zimmer et al., 2000). For example, Kabin et al. (1999)included an intrinsic dipole moment as a free parameter, wherethey fitted their model results to the data of Galileo’s first Eu-ropa flyby. Depending on their model parameters they find dif-ferent values for the internal dipole moment. When rotating theplasma flow upstream of Europa by 20◦, their intrinsic dipolemoment is close to the direction of an induced dipole moment,but with a small magnitude of about 70% of that obtained byKivelson et al. (1999).

In this paper we combine the electromagnetic inductionprocess inside Europa and the interaction with the ambientmagnetospheric plasma and consider their interdependency.This allows us to improve the results of Zimmer et al. (2000)and thus to further constrain the nature of the internal con-ducting layer. The time-dependent interaction between Europaand the jovian magnetosphere includes the local plasma inter-action of Europa’s atmosphere and ionosphere as well as theinteraction of a potential internal ocean with the magnetosphereof Jupiter. Due to Jupiter’s rotation with respect to Europaand the inclination of Jupiter’s magnetic dipole moment, themagnetospheric plasma density and the background magneticfield vary at the position of Europa. The time-varying magneticfields induce currents in an electrically conducting ocean belowthe Europan ice crust. These currents generate a time-varyinginduced magnetic field which influences the plasma interac-tion. In addition, the periodic variations of the magnetosphericplasma and the interaction with the atmosphere lead to a sec-ondary induction effect. To study this time-dependent interac-tion, we develop, for the first time, a three-dimensional single-fluid MHD model which includes periodic magnetic fields fromthe interior of the Moon.

In Section 2, we give a basic overview of the “classical”electromagnetic induction problem applied to Europa. The in-duced magnetic fields derived in this section are part of ourinteraction model. In Section 3, we introduce our model of the

Conductivity of Europa’s subsurface ocean 43

plasma interaction and describe how the induction is imple-mented self-consistently. We also describe the procedure usedto determine the plasma induced magnetic fields. In Section 4,we then present and discuss the results of our full numericalmodel. We investigate the influence of the plasma interactionon the induction process. By comparing our model results withthe Galileo spacecraft in situ measurements we determine the sofar strongest constraints on the conductivity of Europa’s ocean.In Section 5, we summarize our results.

Before proceeding we define the coordinate system used inthis work. We use the EPhiO coordinate system centered at Eu-ropa with its z-axis along Jupiter’s spin axis, the y-axis alongthe radius vector towards Jupiter (positive inward) and the x-axis azimuthal with respect to Jupiter, i.e., in the ideal corota-tion direction.

2. The electromagnetic induction problem

We begin with a description of the electromagnetic induc-tion process at Europa. In this section we describe the modelof Europa’s internal electrical structure used in this paper. Wederive the analytical solution for the internal induced magneticfield which is part of our interaction model presented in the fol-lowing section.

For studying the interior of the Galilean satellites electro-magnetic induction sounding is well suited because Jupiter pro-vides a strong primary signal. The orbit of Europa is located atthe outer edge of the Io plasma torus close to the transition tothe middle magnetosphere of Jupiter. At the position of Europa,Jupiter’s magnetic field consists of a multipole field of inter-nal origin and the magnetic field of an equatorial plasma sheet(Connerney et al., 1981). The primary time-varying magneticfield experienced by Europa is due to the rotation of Jupiter’stilted dipole. Together with the satellites orbital period this re-sults in a uniform alternating magnetic field at the positionof Europa with a period of 11.1 h. The background magneticfield varies mainly in the By - and to a minor degree in theBx -component at the position of Europa while Bz is almost con-stant (Khurana et al., 1998). If one assumes a conducting sub-surface layer at Europa, the time-varying inducing field drivescurrents in the conducting layer which generate a time-varyinginduced magnetic dipole field. Simultaneously, magnetosphericplasma and magnetic fields interact with Europa’s tenuous at-mosphere and ionosphere and also with the time-varying mag-netic field from the interior of the Moon. The induced magneticfields therefore influence the plasma interaction as it is dis-cussed in Neubauer (1999). The plasma interaction generatescurrents, e.g., Alfvénic and ionospheric currents, in the vicinityof Europa. Due to the time-varying magnetospheric conditionsat the position of Europa during one synodical rotation, thecurrents generated by the plasma interaction in the vicinity ofEuropa are also time-dependent. This leads to a secondary in-duction effect, where the magnetic fields induced by the plasmacurrents contain also higher order moments (Neubauer, 1999).

The depth of penetration of a time-varying magnetic field offrequency ω in a conductor of electrical conductivity σ is given

by the skin depth δ = (σμ0ω/2)− 12 . Here μ0 is the permeabil-

ity of vacuum. While the skin depths for inducing fields withJupiter’s rotational period in a core similar to the Earth corewould be only ∼180 m, it would be ∼1000 km in the Earthlithosphere (Stacey, 1992). In Sea water the skin depths wouldbe ∼45 km. A conducting layer with a thickness larger thanthe skin depth in this layer can effectively shield an underly-ing layer even if this one has a large conductivity. For instancea conducting subsurface layer at Europa which is thick enoughcan effectively shield a conducting mantle or core from induc-tion effects.

We model Europa as a sphere with four different conduct-ing layers (Fig. 1) which are radially symmetric: core, mantle,“ocean” and crust. Because of the low electrical conductivityof solid ice at low temperatures, the solid ice crust is assumedto be nonconducting. Similar problems were solved by Lahiriand Price (1939) for an increase of conductivity with depth andby Srivastava (1966) for a sphere made up of concentric shells,each with uniform conductivity.

The fundamental equation of electromagnetic induction instatic conductors is the diffusion equation

(1)∂B∂t

= 1

σμ0∇2B

which can be obtained from Maxwell’s equations. Here we haveignored the displacement currents which for our applicationsare much smaller than the conduction currents. For the sake ofsimplicity the permeability μ inside Europa will be taken equalto the vacuum permeability μ0.

Each component of B(t) can be expressed by a series ofsuperimposed sine waves of various frequencies. Because (1)is linear in B we can examine one sine wave of a single fre-quency. The total field can then be determined by superposition.The potential of the external inducing field can be expandedin spherical harmonics. For a radially symmetric conductivitydistribution each surface spherical harmonic Sm

n of the induc-ing field gives rise to only the same Sm

n of the induced field(Parkinson, 1983). Therefore we can deal with each harmonicseparately and superimpose the solutions.

Without loss of generality we can assume an inducing mag-netic field with a potential:

(2)Ue = bBe

(r

b

)n

Smn (θ,φ)e−iωt ,

where b is the radius of the outer shell (see Fig. 1), Be is a con-stant field, n and m are degree and order of the inducing field,

Fig. 1. Assumed model of Europa’s interior: sphere with three conducting layers(core, mantle, “ocean”) surrounded by an insulating crust.


respectively, and r, θ,φ are spherical coordinates. The induc-tion equation (1) then must be solved within each shell and inthe central region and the boundary conditions at each shell (Br

and Bθ must be continuous) have to be taken into account. Theresulting induced magnetic field outside Europa is then givenby

(3)B indr = (n + 1)

(b

r

)n+2

BiSmn e−iωt ,

(4)B indθ = −

(b

r

)n+2

Bi

∂Smn

∂θe−iωt

with Bi given through the boundary conditions at each shell by

(5)

Bi

Be

= n

n + 1

[F ′1(bk1)

F1(bk1)− (n + 1)

] + F2(bk1)F1(bk1)

DC

[F ′2(bk1)

F2(bk1)− (n + 1)

][F ′

1(bk1)

F1(bk1)+ n

] + F2(bk1)F1(bk1)

DC

[F ′2(bk1)

F2(bk1)+ n

]with

(6)

D

C= F1(ak1)

F2(ak1)

F ′1(ak2)

F1(ak2)− F ′

1(ak1)

F1(ak1)+ F2(ak2)

F1(ak2)GE

[F ′2(ak2)

F2(ak2)− F ′

1(ak1)

F1(ak1)

]F ′

2(ak1)

F2(ak1)− F ′

1(ak2)

F1(ak2)+ F2(ak2)

F1(ak2)GE

[F ′2(ak1)

F2(ak1)− F ′

2(ak2)

F2(ak2)

]

and

(7)G

E= F1(ck2)

F2(ck2)

F ′1(ck2)

F1(ck2)− F ′

1(ck3)

F1(ck3)

F ′1(ck3)

F1(ck3)− F ′

2(ck2)

F2(ck2)

.

Here k1, k2 and k3 are the wave numbers in the ocean, mantleand core, respectively. Thereby the wave number k is defined ask = √

iσωμ0. Further,

(8)F1(r) =√

π

2(rk)−

12 I

n+ 12(rk),

(9)F2(r) =√

π

2(rk)−

12 K

n+ 12(rk)

are the solutions of Bessel’s equation, where In+ 1

2(rk) and

Kn+ 1

2(rk) are the modified spherical Bessel functions of first

and third kind. The induced magnetic field outside a spherewith multiple conducting shells can be related to that of a per-fect conducting sphere by defining an amplitude A and a phaselag Φ:

(10)Bind(t) = AB∞ind

(t − Φ

ω

).

The solution of the induction problem derived in this chapterserves as a basis for our interaction model described in the nextsection.

With the analytical solution derived above, we investigatedthe influence of the core size, the mantle conductivity and thethickness of a conducting outer layer on the induction signatureoutside Europa depending on the conductivity of the subsur-face layer (for more details, see Schilling, 2006). We find thata conducting core of reasonable size is almost not detectableoutside Europa for conductivities of the subsurface layer largerthan 60 mS/m as suggested by Zimmer et al. (2000) when using

Jupiter’s synodical period as excitation period for the induction.In addition, we support the finding of Zimmer et al. (2000) thata conducting mantle alone cannot explain the induction signa-ture found in the Galileo magnetic field data. An influence onthe induction signature, when using Jupiter’s synodical periodas the excitation period for the induction, is only expected foran ocean conductivity σoc � 100 mS/m. By varying the thick-ness and the conductivity of Europa’s subsurface layer, we findthat for ocean conductivities larger than 100 mS/m a resolutionof the oceans lower boundary is almost not possible if the oceanthickness is larger than 100 km. In addition, the lower bound-ary of the ocean cannot be resolved if the ocean is thicker than10 km and the ocean conductivity is larger than 1 S/m.

3. Satellite plasma interaction and the concept of a virtualplasma

3.1. Description of the model

3.1.1. ConceptThe basic idea of this paper is to study the time-dependent

plasma interaction of Europa’s atmosphere and its proposed in-ternal ocean with the jovian magnetosphere. We describe theplasma environment of Europa with a three-dimensional single-fluid MHD model and solve the MHD flow problem and theinternal induction problem simultaneously. Since both effectsinteract, the MHD equations are rewritten such that the inducedmagnetic field is explicitly included.

3.1.2. Mass balanceMagnetospheric plasma is convected into the atmosphere of

Europa where the density is modified by ionization and recom-bination processes. An ionospheric singly charged ion popula-tion with mO2 = 32 amu is produced in our model. FollowingKivelson et al. (2004), we use for the determination of the mag-netospheric electron density an effective ion charge of the up-streaming plasma of z = 1.5. In our model the plasma consistsof one ion species. We describe the evolution of the bulk densityρ by

(11)∂ρ

∂t+ ∇ · ρu = (P − L)mi,

where P is the production rate, L the loss rate and u the bulkvelocity. Saur et al. (1998) find that electron impact ionizationis the dominant process to generate Europa’s ionosphere, beingover an order of magnitude larger than photoionization. There-fore, we only include electron impact ionization as a sourceprocess for the ionospheric plasma. The ionization cross sec-tions σj are taken from the NIST database (Kim et al., 1997).

Ionospheric electrons which are produced by electron im-pact are much cooler than the magnetospheric electrons andtherefore are not involved into the ionization process. Whilethe number of ionospheric electrons is determined by (11), weassume a separate continuity equation for the magnetosphericelectrons. Electron impact ionization does not change the num-ber of the magnetospheric electrons and the loss of the magne-tospheric electrons due to dissociative recombination is negli-gible due to their low density and high temperature. We assume


the magnetospheric electrons to move with the bulk velocity.Thus, we solve the following continuity equation for the mag-netospheric electrons

(12)∂ne,ms

∂t+ ∇ · ne,msu = 0.

The loss rate L in (11) is given by

(13)L = αDn2

with the number density n and a dissociative recombination co-efficient (Torr, 1985)

(14)αD = 2 × 10−13(300

Te

)0.7m3/s.

3.1.3. Induction equation and momentum equationThe magnetic field at Europa can be subdivided into sev-

eral portions. First, the time-varying background magnetic fieldof Jupiter and the magnetospheric current sheet B0. Second,the magnetic field caused by the interaction of the magne-tospheric plasma with Europa’s atmosphere BP . Finally, wehave the induced magnetic field from the interior due to thetime-varying background field and the time-varying plasma cur-rents Bind = Bind(B0) + Bind( jP ). We assume the backgroundmagnetic field B0 to be a time-varying homogeneous magneticfield at Europa. The periodicity of the background field is givenby the synodic rotation period of Jupiter. Doing so, we do notaccount for other excitation periods, e.g., Europa’s orbital pe-riod.

The total magnetic field Btot at Europa can thus be written

(15)Btot = B0 + BP + Bind(B0) + Bind( jP ).

We assume the outer ice crust of Europa to be electricallynonconducting, which causes the current systems outside andinside Europa to be isolated against each other. Thus, B0 andBind(B0) are potential fields outside the Moon.

Generally, the total magnetic field has to fulfill the inductionequation, which can be written as

(16)∂Btot

∂t= ∇ × (u × Btot) − ∇ × (η∇ × Btot),

where η is the magnetic diffusivity. Inside Europa the dynamoterm (first term on the right-hand side) vanishes and (16) thendescribes the diffusion of the magnetic field into the Moon.

For the variation of the Bx - and By -components of Jupiter’smagnetospheric field at Europa’s location we fit an ellipse tothe data given in Kivelson et al. (2000) (Fig. 2). Thus, the time-varying inducing background magnetic field Bi

0 is calculatedanalytically by

(17)Bi0 = −84nT sin (Ωt)x − 210nT cos (Ωt)y.

Ωt is the angle between the rotating dipole moment of Jupiterand the line of sight Jupiter–Europa (see Fig. 1 in Neubauer,1999). The induced magnetic field Bind(B0) is then given by(3) and (4). The inducing field is strongest when the rotatingdipole moment of Jupiter points towards (Ωt = 0◦) or awayfrom Europa (Ωt = 180◦). We use a constant B0,z = −410 nT.

Fig. 2. Background magnetic field at Europa. Values are obtained by fitting anellipse to the magnetic field data given by Kivelson et al. (2000).

Using (15) and (16) results in the following induction equa-tion for the magnetic field caused by the interaction of the mag-netospheric plasma with Europa’s atmosphere and the therebyinduced magnetic field which we solve with our model:

∂(BP + Bind( jP ))

∂t= ∇ × [

u × (BP + Bind( jP )

)]− ∇ ×

[me

ne2

(νen + me

mi

νin + L

n

)∇

× (BP + Bind( jP )

)]+ ∇ × [

u × (B0 + Bind(B0)

)](18)− ∂(B0 + Bind(B0))

∂t.

Thereby we account for resistivity due to loss processes, ion-neutral and electron-neutral collisions in the generalized Ohm’slaw. We use a constant initial plasma velocity with v0,x =104 km/s. The collision frequency of the ions with the neutralsis

(19)νin = 2.6 × 10−9nn

√α0

μa

s−1,

where α0 is the polarizability of the neutral gas in units of10−24 cm−3, nn is the neutral gas density in cm−3, and μa isthe reduced mass in amu. We use α0 = 1.59 for O2 (Banksand Kockarts, 1973). For the electron-neutral collision fre-quency we use νen = 10−9nn s−1 which we calculated by usingthe momentum-transfer cross sections given by Itikawa et al.(1989). Induction effects also have to be accounted for in themomentum equation, which then reads

ρ

(∂u∂t

+ u · ∇u)

= (∇ × (BP + Bind( jP )

))× (

BP + Bind( jP )) − ∇p

−(

meνen + νin + P

)ρu

mi n


+ [∇ × (BP + Bind( jP )

)]× (

B0 + Bind(B0))

+ [∇ × (B0 + Bind(B0)

)](20)

× (B0 + BP + Bind(B0) + Bind( jP )

).

3.1.4. Plasma temperatureThe interaction of the plasma with the neutral gas (inelastic

as well as elastic processes) changes the internal energy e of theplasma. Here, we deliberately use a simplified energy equation

(21)∂e

∂t= −∇ · eu.

Including additional terms in (21), such as frictional heat-ing from the relative motion between plasma and neutral at-mosphere or Joule heating, does not affect the results in thispaper as tested in multiple runs. The only process which isstrongly temperature dependent is the production of ionosphericelectrons by electron-impact ionization. For a proper descrip-tion of this process, the temperature of the magnetosphericelectrons Te,ms is required. While the evolution of the densityof this population is given by (12), we use the following de-scription for their temperature dependence: The plasma torusaround Europa represents an extensive energy reservoir, whichcan provide energy via electron heat conduction along the mag-netic field lines. This process is extremely effective at Europa(Saur et al., 1998). Although this energy reservoir is strictlyspeaking limited, it is legitimated to assume an unlimited en-ergy reservoir for our purposes. Moreover, we assume infiniteheat conductivity along the field lines. Saur et al. (1998) showthat the electron temperature is reduced strongest close to thesurface and on the flanks where the electron density reachesits maximum because of the longer transport time for a plasmafluid element through the dense part of the ionosphere wheremost of the inelastic collisions occur. By maintaining a con-stant temperature T 0

e,ms for the magnetospheric electrons oneoverestimates the production rate in these regions. Hence, theplasma density is overestimated. In order to compensate for thisdeficiency, we use a simple parameterization and implementa calibration factor for the temperature Te,ms of the magne-tospheric electrons. The temperature of the magnetosphericelectrons then has the following spatial dependence:

(22)Te,ms = T 0e,ms

(1 −

(1 − t0 cos

φ

2

)e− h

HT

).

The azimuthal angle φ varies in the xy-plane from 0◦ (up-stream) to 180◦ (downstream), h is the height above the sur-face. The factor t0 and the scale height HT are the calibrationparameters to be determined. We determine these parametersby comparing our model with measurements of flybys duringwhich Europa was located in the middle of the current sheet(e.g., E12), i.e., where induction effects are negligible. Doingso, we find t0 = 0.1 and HT = 300 km. Once determined, thecalibration parameters are assumed to be the same for all modelscenarios. Thereby we assume that the atmosphere of Europadoes not have a strong asymmetry. As we will see in Section 4,

the magnitude of the magnetic field is represented very well bythe model for all of the simulated flybys while using constant t0and H , suggesting that this is a proper assumption.

3.1.5. AtmosphereWe use a hydrostatic molecular oxygen atmosphere with a

scale height of 145 km and a surface density n0,0 = 1.7 ×107 cm−3 estimated in Saur et al. (1998) based on HST ob-servations by Hall et al. (1995). This is consistent with an O2column density of Ncol = 5 × 1018 m−2. Pospieszalska andJohnson (1989) demonstrate that sputtering is not uniform overthe surface of Europa, but is decreasing from the trailing to theleading hemisphere. We follow Saur et al. (1998) in assumingthat the surface density n0(φ) varies in direct proportion to thenormalized flux variation calculated by Pospieszalska and John-son (1989). The surface density then has the following spatialdependence:

n0(φ) = n0,0

[1.08H

(π

2− φ

)cosφ

(23)+ 0.885

(cos

φ

2+ 1.675

)],

where H(π2 −φ) is the Heaviside step function and the angle φ

varies from 0◦ (upstream) to 180◦ (downstream).

3.1.6. Magnetospheric properties at EuropaThe magnetospheric electrons at the location of Europa can

be divided basically in two populations, a thermal Maxwellianand a suprathermal non-Maxwellian population. For the supra-thermal population we use a density of nsth = 2 cm−3 atTe,sth = 250 eV (Sittler and Strobel, 1987). For the thermal pop-ulation we use Te,th = 20 eV (Sittler and Strobel, 1987), whiletheir density varies with the position of Europa in the plasmasheet. For the upstream magnetospheric plasma we use an ionmass mi = 18.5 amu (Kivelson et al., 2004). The variation ofthe background plasma density stems from the tilt of the plas-masheet against Europa’s orbital plane by about 7◦ (Dessler,1983). In addition, the rotational velocity of the plasma is largerthan the orbit velocity of Europa. Thus, Europa passes throughdifferent plasma regimes during one synodic rotation of Jupiter.We assume the plasma to be symmetric around the centrifu-gal equator and to vary periodically, with a minimum electronnumber density of ne = 18 cm−3 when Europa is outside theplasma sheet and a maximum value of ne = 250 cm−3 whenEuropa is in the center of the plasma sheet. In between we as-sume the density to fall of with exp(−(z/H)2), where H is thescale height of the plasma and z is the distance of Europa fromthe center of the plasma sheet (Thomas et al., 2004). With thevalues above, the scale height of the plasma in our model isH = 0.7 RJ .

3.1.7. NumericsThe kernel of our model consists of Zeus3D (Stone and

Norman, 1992a, 1992b), a three-dimensional time-dependentcode which solves the ideal MHD equations. We extended theZeus3D code to nonideal MHD. We use a Cartesian grid di-vided into four regions with different spatial resolution: a very


high resolution region from −1.5 RE to 1.5 RE in all three di-rections in space with a grid size of 80 km, a high resolutionregion up to a distance of 3 RE in all three directions in spacewith a grid size of 157 km, followed by a medium resolutionregion with a grid size of 795 km and finally a low resolutionregion with a grid size of 1569 km. The total grid volume is[[−10,10], [−10,10], [−60,60]] RE .

3.2. The concept of a virtual plasma

In contrast to other bodies, e.g., Titan, the thin atmosphereof Europa cannot completely shield the surface of Europa fromthe streaming plasma flow. Plasma which hits the surface of Eu-ropa is absorbed by the surface and interacts with it. Thus, theinterior of Europa is free of plasma, while the magnetic fieldcan diffuse into the Moon. This fact can be implemented intwo ways: First, inner boundary conditions could be set at thesurface to connect the external regimes to the internal one de-scribed by (16). This solution is impractical, because it wouldrequire a correct description of the diffusion into the Moonthrough the boundaries. Secondly, the interior of Europa canbe part of the simulation. As the description of a solid body isnot included in the single-fluid equations a priori, we describethe interior of Europa as a “virtual” plasma to mimic the realrelevant properties of a solid body. This leads to a consistentdescription of the interior and the exterior of Europa. For nu-merical reasons the volume of the solid body cannot be free ofplasma to make the Zeus3D-code applicable.

The “virtual” plasma is defined as to fulfill Eqs. (11), (18),(20), etc., with parameters chosen such that the magneticReynolds number RM in the interior is as small as possible.Therefore we artificially fill the interior with neutral gas nvirtual

n .The magnetic diffusivity in the interior ηi is proportional tonvirtual

n /ρ. Hence, we control ηi by adjusting the neutral gasdensity in every time step at every grid point in the interiordepending on the local plasma density. Simultaneously, lossprocesses are implemented in order to reduce the plasma den-sity in the interior of the Moon. For the plasma momentumequation, (20) is applied in the interior. It must be controlled insuch a way that the velocity is small enough to guarantee RM

to be small.

3.3. Procedure

In principle, the interaction problem could be solved by solv-ing the set of single-fluid equations in the exterior and thediffusion equation in the interior of the Moon simultaneously.However, because of the different time scales of the processesinvolved and the Courant–Friedrich–Lewy (CFL) criteria, thesimulation time needed for a stable periodic solution wouldbe unrealistically long. Therefore, we develop a different ap-proach. For this approach two time scales are important: thetime it takes the plasma interaction to adjust to new external andinternal conditions, which is in the order of a few minutes andthe timescale of the electromagnetic induction, i.e., the diffu-sion time scale, which is on the order of the synodical period ofJupiter (Neubauer, 1998b). Thus, we can use a quasi-stationary

approach (Neubauer, 1998b). The plasma interaction pictureadjusts to the slowly varying internal and external conditions ina quasi-stationary way. Therefore, the plasma interaction pic-ture as a function of time is given by consecutive stationarystates, where each steady state is given by the current upstreamconditions and the present internal magnetic field.

We then solve the external MHD problem and the internalelectrodynamic problem by using our 3D-MHD model in an it-erative process. As a starting point we include the induction by ahomogeneous background field only. Thereby we use the time-varying components of the background magnetic field givenby (17). We then have the analytical solution of (1) as an ini-tial condition for the internal magnetic field. Then we solve astationary problem for 8 different times ti which are equally dis-tributed over the synodic period of Jupiter (Ωt = 0◦, 45◦, 90◦,etc.). This is done by using our Fluid-Code described above.Note that the background plasma conditions vary considerablyat the different times ti .

In the second step the induced magnetic field Bind( jP ) due tothe time-varying currents in the exterior is determined. There-fore, we first acquire the time variable part of the external fieldB′

p on the surface of the conducting sphere at a given time:

(24)B′p = (

Bp + Bind( jP )) − ⟨

Bp + Bind( jP )⟩

where the constant part averaged over one synodic period iscalculated by

(25)⟨Bp + Bind( jP )

⟩ = 1

Tsyn

Tsyn∫0

Bp(t) + Bind( jP )dt.

Subsequently, we determine the harmonic coefficients of the in-duced magnetic fields for each time ti . A Fourier expansionof these harmonic coefficients determines the plasma inducedmagnetic fields Bind( jP ) for any time t . The analytical descrip-tion of the plasma induced fields together with the induced fieldBind(B0) from the first iteration yields the new initial internalmagnetic field. We then repeat the above procedure until wereach convergence for the determined harmonic coefficients.

4. Results and discussion

4.1. Harmonic coefficients of the plasma induced magneticfields

After deriving the spherical harmonic coefficients (alsocalled Gauss coefficients) of the plasma induced magnetic fieldsfor each time ti , we do a Fourier expansion of these coeffi-cients. This enables us to calculate the harmonic coefficients atany time t . Hence, we are able to calculate the plasma inducedmagnetic field at any point and at any time t . Fig. 3 shows theharmonic dipole and quadrupole coefficients of the plasma in-duced magnetic fields during Jupiter’s synodical period for anassumed ocean thickness of 100 km, an ocean conductivity of5 S/m and a thickness of the ice crust of 25 km. The deriva-tion of the spherical harmonic coefficients was done by usingthe normalization according to the convention of Schmidt forthe associated Legendre polynomials (e.g., Blakely, 1995).


Fig. 3. Harmonic coefficients of the plasma induced magnetic fields duringJupiter’s synodical period. The upper panel shows the dipole, the lower panelthe quadrupole coefficients.

The Gauss coefficients gmn and hm

n for a given expansion canbe grouped under three headings; zonal, sectoral, and tesseralharmonics. Zonal harmonics are those of order zero. Note thatnone of the h coefficients can be of order zero. The zonal har-monics represent fields whose moments are aligned with thez-axis. Therefore, the g0

1 term describes the moment of an ax-ial Europa-centric dipole aligned with the z-axis, and the g0

2term, an axial, Europa-centric, quadrupole. Sectoral harmonicsare those for which degree and order are equal. They repre-sent fields which have their moments in the equatorial plane. Inthe EPhiO coordinate system the g1

1 term describes a Europa-centric dipole aligned with the y-axis while the dipole associ-ated with the h1

1 term is aligned with the −x-axis.The maximum dipole coefficient is found at Ωt = 270◦ with

a value of g11 ≈ 12 nT (see Fig. 3). This coefficient remains

nearly constant with a small value of ≈−3 nT between 0◦ and180◦. The g0

1 coefficient varies with a period of π but theamplitude is not exactly symmetric. The h1

1 term also showsan asymmetric behavior. The quadrupole coefficient with thelargest values is the g1

2 term. It is π -periodic with a maximumvalue of ≈14 nT. The g0

2 can also reach values up to 10 nT andis 2π -periodic.

None of the harmonic coefficients shows a purely π -periodicbehavior. This feature can be explained by the time-varying

Table 1Constant part of the Gauss coefficients of the external field (order 1 and 2)generated by plasma currents. The multipole coefficients are in nT

G01 G1

1 H 11 G0

2 G12 G2

2 H 12 H 2

2

12 −8 2 −2 −29 5 −2 2

magnetospheric background conditions in our model. Althoughwe use a plasma density model which is symmetric aroundthe centrifugal equator and a background magnetic field whichvaries symmetric in time, the Bx component of our backgroundmagnetic field breaks the symmetry.

Please note, that the expected differences in the measuredmagnetic field near Europa when Europa is located in the centerof the plasma sheet compared to outside the plasma sheet is notsimply reflected in the difference of the harmonic coefficients.The time-varying magnetic field caused by the interaction ofEuropa’s atmosphere with the jovian magnetospheric plasmadepends on the external coefficients, which are related to theinternal Gauss coefficients by (5), and the location on the con-ducting surface given by (2). For example, the difference inthe g1

2 term between Europa in the center and outside of theplasma sheet is 25 nT. Upstream of Europa on the surface ofthe conducting layer this corresponds to a difference in Bz of130 nT. Other coefficients give additional contributions. Theoverall magnetic field due to the interaction of Europa’s at-mosphere with the jovian magnetospheric plasma is given bya Fourier synthesis of all coefficients. Since the source of theplasma induced fields is outside Europa, the magnetic fieldstrength in the source region is even higher. The decrease infield strengths between source region and surface of the con-ducting layer becomes even larger for higher order moments(octupole, etc.) which therefore may be more important outsideof Europa.

We also show lower order moments of the constant part ofthe Gauss coefficients of the external field calculated by (25)in Table 1. The G1

2 term has the largest value and accounts forthe upstream enhancement and the downstream decrease of Bz

at Europa. Again the constant magnetic field of the plasma cur-rents at Europa is then given by a Fourier synthesis of all theGauss coefficients and using (2) and (5). We remind the readerthat higher order moments become important outside Europaclose to the source region of those currents, and the extrapola-tion is only valid in the “source free” region below the externalcurrent region. Therefore, a magnetic field calculated by usingonly the extended coefficients given in Table 1 maybe inaccu-rate away from the surface.

The contribution of the single multipoles to the plasma in-duced magnetic field can be displayed by the spatial powerspectrum of the internal magnetic field Rn (Blakely, 1995),where Rn is defined as the scalar product Bn · Bn averaged overthe spherical surface S(r):

(26)Rn(r) = 1

4πr2

2π∫ π∫Bn · Bnr

2 sin θ dθ dφ

0 0


with

Bn = −∇(

Rc

(Rc

r

)n+1 n∑m=0

(gm

n cosmφ

(27)+ hmn sinmφ

)P m

n (θ)

)

and

(28)B =∞∑

n=1

Bn

where r = Rc represents the surface of the ocean. Using theorthogonality property of spherical harmonics, Eq. (26) can bereduced to (Backus et al., 1996):

(29)Rn(r) =(

Rc

r

)2n+4

(n + 1)

n∑m=0

[(gm

n

)2 + (hm

n

)2].

The set of values of Rn for n = 1,2,3, . . . at a fixed radius r

is sometimes called the Mauer–Sberger–Lowes spectrum (e.g.,Backus et al., 1996). Thereby the averaged squared field overany sphere is the sum over the spectrum:

(30)⟨|B|2⟩

S(r)=

∞∑n=1

⟨|Bn|2⟩S(r)

=∞∑

n=1

Rn(r).

Fig. 4 shows the spectral coefficients for dipole, quadrupoleand octupole terms at the surface of the ocean. It is obviousthat the main spectral power is in the quadrupole field. Theplasma induced fields are strongest when Europa is located inthe center of the plasma sheet. Smallest values are found whenEuropa is between the two extreme conditions (inside and out-side the plasma sheet), i.e., at Ωt = 45◦, 135◦, 225◦, and 315◦.The power in the octupole terms is small compared to dipoleand quadrupole contributions. Higher-order multipoles (n > 3)are even less important. The spectral coefficients for the back-ground field induced dipole term varies between ∼14,000 and∼88,000 nT2 (not shown).

We would like to point out that the external interaction fieldsare mainly of order 2 and therefore induce mainly quadrupolefields in a spherically symmetric conductor. This is in agree-ment with our results shown in Fig. 4. In a perfectly sym-metric case, there would be no plasma induced dipole com-ponents. However, the following features break the symmetry:The Alfvén wings are bend back and the background bulk ve-locity is not always perpendicular to the background magneticfield. During a synodical rotation period of Jupiter, the strengthof the currents, which flow in the Alfvén wings, varies period-ically. This also leads to time varying external uniform fieldsthat induce dipole components in the conductor. In addition,differences between upstream and downstream ionospheric cur-rents create an asymmetry. The asymmetric current system canbe illustrated by superposing the symmetric ionospheric cur-rents with a closed current loop. While the symmetric currentsinduce quadrupole fields, the closed loop gives a uniform mag-netic field in the center of Europa and thus induces a dipolemagnetic field.

Fig. 4. Spectral coefficients of the Mauersberger–Lowes spectrum for the dipole(n = 1), the quadrupole (n = 2), and the octupole part (n = 3) of the plasmainduced magnetic field.

Once we have calculated the plasma induced magneticfields, we include them as initial internal magnetic fields intoour model in addition to the background field induced dipole.Coefficients of higher order are only important very close tothe surface since according to (29) Rn(r) ∼ r−(2n+4). There-fore, we only consider dipole and quadrupole coefficients whenincluding the plasma induced fields in our model. In the nextiteration step, we then repeat the procedure described aboveand calculate the plasma induced magnetic fields again. All co-efficients derived after the second iteration differ from thosederived after the first iteration by less than 10%, which is goodenough for our calculations. Therefore, we stop the calculationsafter the second iteration.

Comparing the dipole and quadrupole coefficients of theplasma induced magnetic fields to the dipole coefficients of thebackground magnetic field (Fig. 2), which are on the order of100 nT, it is obvious that the plasma interaction has only a weakimpact on the induction process. However, the coefficients de-rived may have an influence on the lower part of the ionosphereof Europa. Therefore, they may be important when modelingthis part of Europa in detail.

The results presented in this section were derived by assum-ing an almost saturated induction process. Note, that a smallerocean or an ocean which is less conductive leads to even smallerharmonic coefficients.

4.2. New constraints on the conductivity of Europa’ssubsurface ocean

The comparison of our simulation results with the Galileomagnetic field data along the trajectories of different flybys al-lows us to study the conductivity distribution inside Europa.We concentrate on flybys that occurred when the inductive re-sponse is strongest, i.e., when Europa was well outside thecurrent sheet. This was the case for the E4, the E14, and theE26 flyby [see Tables 1 in Schilling et al. (2004) and Kivelsonet al. (2000)]. We remind the reader that the model is parame-


terized such that for a flyby inside the plasma sheet (e.g., E12)where induction effects play no role the agreement with mag-netic field observations is very good.

4.2.1. Europa flyby E4The first close Europa encounter of Galileo was the E4 flyby

on December 19, 1996. This pass occurred in Jupiter’s northernmagnetic hemisphere (Kivelson et al., 1997), i.e., the magneticbackground field at Europa pointed away from Jupiter. Thus,the primary induced dipole moment pointed toward Jupiter. Theflyby was an equatorial pass, oblique through Europa’s wake.Closest approach was at 06:52:58 universal time (UT) at an al-titude of 695 km.

Fig. 5 displays the magnetic field along the trajectory in theEPhiO coordinate system. The red curve shows the magneticfield measured by the Galileo spacecraft (Kivelson et al., 1997).Our model results for a pure plasma interaction without induc-tion in the interior of the Moon are indicated by the dashedblack curve. In addition, the predicted fields including inductioninto our model are shown. We choose different values for theconductivity of Europa’s ocean and analyze what conductivity

Fig. 5. Observed and modeled magnetic field for the E4 flyby in the EPhiO co-ordinate system. From top to bottom: Bx , By , Bz , Bm. The red curve showsthe measured field (Kivelson et al., 1997). The dashed black curve shows thepredicted field when no induction is included in our model. The predicted fieldby including induction is shown for the ocean conductivities σoc: 100 mS/m(blue), 250 mS/m (brown), 500 mS/m (green), and 5 S/m (black). The as-sumed thickness of the crust is 25 km and the assumed thickness of the oceanis 100 km.

fits the measured data best. We start with an ocean thickness of100 km. The smallest ocean conductivity assumed (100 mS/m)is close to the lower limit of 60 mS/m given by Zimmer et al.(2000). The largest conductivity assumed (5 S/m) is the con-ductivity of Sea water found on Earth. For larger values of theσoc, the induction process is saturated, i.e., larger conductivitiesyield effectively the same results.

The dashed black curve in Fig. 5 indicates that a small con-tribution from the plasma interaction can be found in the Bx andBy component of the measured data. Modeling the data with-out induction cannot explain these components. As the E4 flybywas an equatorial pass, the main contribution in the Bx and By

component results from the induced magnetic field in the inte-rior. With the used model of Europa’s interior, we are able to re-produce the Bx and By component for an ocean conductivity of500 mS/m or larger. Note that by using a 100 km ocean thick-ness, the induced field is almost saturated for σoc > 500 mS/m.Thus, we are not able to set an upper limit for σoc.

Our model reproduces the local maximum and minimumin the wake region in the Bx component (between UT 07:00and 07:05) independent from the ocean conductivity used. Thissuggests, that this feature is caused by the plasma interaction.For the agreement between the data and our model, we seeno need for a deviation of the upstreaming plasma flow fromthe nominal corotation direction as it was suggested by Kabinet al. (1999). Both the Bz component and the magnetic fieldmagnitude can almost completely be explained by the plasmainteraction. Panel 3 and 4 of Fig. 5 show that our model re-produces the overall structure as well as the two local maxima.The negative perturbation of Bz and |B| occurs in the down-stream region where the plasma is accelerated and the magneticfield magnitude is decreased. The agreement between the dataand the model for Bz and |B| can also be seen as a test for ourplasma model.

The induced magnetic field depends both on the conductiv-ity and the thickness of the ocean. Thus, the determination ofσoc is ambiguous. To show this ambiguity, we vary the oceanthickness. As an example Fig. 6 shows our simulation resultsfor an assumed ocean thickness of 25 km. As mentioned above,this represents the extreme case of a thin ocean model. Again,the thickness of the crust is 25 km.

In order to explain the measured magnetic field data, largerocean conductivities are necessary when using a thinner ocean.Fig. 6 indicates that in this case ocean conductivities of 1 S/mor less are insufficient to reproduce the Bx and By componentof the magnetic field when an ocean thickness of 25 km is as-sumed. A conductivity of, e.g., 5 S/m is needed. We remind thereader that the induced magnetic field is saturated for conduc-tivities larger than 5 S/m. Hence we can only set a lower limiton the ocean conductivity. In reverse, we can conclude that ifthe ocean conductivity is less than 1 S/m, the ocean has to bethicker than 25 km. In summary, this results in the following re-lation: electrical conductivity × ocean thickness � 50 S/m km.

4.2.2. Europa flyby E14The Europa flyby, E14, occurred in the low plasma density

region above the current sheet (Kivelson et al., 1999). Thus,


Fig. 6. Same as Fig. 5 for an ocean thickness of 25 km. The predicted fieldby including induction is shown for the ocean conductivities σoc: 100 mS/m(blue), 500 mS/m (green), 1 S/m (purple), and 5 S/m (black).

the induced magnetic field is very similar to that during the E4flyby. Again, the primary induced dipole moment pointed closeto the Jupiter-facing meridian. The trajectory of the E14 passwas at higher altitude and latitude than the E4 pass. Closestapproach occurred at an altitude of 1647 km at 13:21:05 UT onMarch 29, 1998. The flyby was a mostly upstream pass whichended downstream.

As in the previous section, we discuss two different assumedocean thicknesses for the E14 flyby, 100 and 25 km. We startwith an assumed ocean thickness of 100 km. The measuredmagnetic field (red) along the E14 trajectory in the EPhiO coor-dinate system is shown in Fig. 7. In addition, the predicted fieldby neglecting induction (dashed black) and including inductionin a 100 km thick ocean for different assumed ocean conductiv-ities σoc is displayed. The color code used in Fig. 7 is the sameas in Fig. 5.

The Bx component of the measured magnetic field cannotbe explained by plasma interaction alone. We are able to fitthis component very well when using ocean conductivities of250 mS/m and higher. As mentioned above, an upper limitfor σoc cannot be assessed. The By component as well as theBz component and the magnetic field magnitude can almostbe explained by the plasma interaction. This demonstrates theimportance of including the plasma interaction for interpretingGalileo’s magnetic field data for a subsurface ocean. The en-

Fig. 7. Same as Fig. 5 for the E14 flyby. The ocean thickness is 100 km. Thepredicted field by including induction is shown for the ocean conductivities σoc:100 mS/m (blue), 250 mS/m (brown), 500 mS/m (green), and 5 S/m (black).

hancement of Bz and |B| occurs upstream of Europa where theplasma is slowed down and the magnetic field is compressed.Since the flyby was above the equator, the bending of the mag-netic field is visible as a positive perturbation of Bx .

Fig. 8 shows the simulation results when using a thicknessof the ocean of 25 km and a crust thickness of 25 km. In thiscase ocean conductivities of at least 1 S/m are necessary to re-produce the Bx component of the magnetic field. In reverse, wecan conclude that for the E14 flyby the ocean has to be thickerthan 25 km for σoc � 1 S/m. This is the same result as for theE4 flyby.

4.2.3. Europa flyby E26The E26 flyby was the crucial flyby to distinguish between

a permanent and an induced magnetic dipole moment as sourceof Europa’s internal magnetic field (Kivelson et al., 2000). Thispass occurred south of Jupiter’s magnetic equator in a regionwith low plasma density. Therefore, the induced magnetic fieldwas almost 180◦ out of phase with its value on the E4 andE14 pass. Closest approach occurred at an altitude of 346 kmat 17:59:43 UT on January 03, 2000. The flyby trajectory wasupstream of Europa, nearly radial toward Jupiter, and south ofEuropa’s equator.

Fig. 9 shows the magnetic field along the trajectory in theEPhiO coordinate system for an ocean thickness of 100 km


Fig. 8. Same as Fig. 5 for the E14 flyby. The ocean thickness is 25 km. Thepredicted field by including induction is shown for the ocean conductivitiesσoc: 100 mS/m (blue), 500 mS/m (green), 1 S/m (purple), and 5 S/m (black).

and a crust thickness of 25 km. We are able to fit the overallstructure of the magnetic field fairly well. Since this flyby wasvery close to Europa and off the equator, contributions fromthe plasma interaction can be found in every component of themagnetic field.

The bendback of the Alfvén wing leads to an enhancementof the Bx component around closest approach. While we can fitBx the overall form, we are not able to fit the double peak struc-ture of this component. This structure occurs around closestapproach at altitudes which are within or very close to Europa’sionosphere. As we use a simple model of Europa’s atmospherethe detailed structure of the magnetic field at these altitudes isbeyond the scope of our model. In the previous section we showthat the influence of the plasma interaction on the inductionprocess is weak. Hence, the details of the plasma interactiondo not influence our statement on the conductivity distributionin the interior of Europa. The By component contains contribu-tions from the plasma interaction as well as from the induction.Including induction with ocean conductivities of 250 mS/mand larger improves the fit of this component fairly well. The Bz

component also contains contributions from the plasma interac-tion. Including induction in an ocean with σoc � 250 mS/m andlarger leads to a better fit to the data. By using conductivities of250 mS/m and larger we can fit the magnetic field magnitudevery well. The enhancement of the field magnitude is due to the

Fig. 9. Same as Fig. 5 for the E26 flyby. The ocean thickness is 100 km. Thepredicted field by including induction is shown for the ocean conductivities σoc:100 mS/m (blue), 250 mS/m (brown), 500 mS/m (green), and 5 S/m (black).

compressing of the magnetic field upstream of Europa. Pleasenote, that our calibration parameters defined in Section 3.1.4 re-main the same for all three flybys. The good agreement of themagnetic field magnitude for all three flybys between the dataand the model supports our choice of parameters.

In Fig. 10 we show our simulation results when using anocean thickness of 25 km and a crust thickness of 25 km. Again,the Bx component is determined mainly by the plasma interac-tion. Using ocean conductivities larger than 1 S/m improvesthe fit of the By component. For Bz good fits are obtained whenusing σoc � 500 mS/m.

5. Conclusions

We have developed a self-consistent 3D Model of the time-dependent plasma interaction of Europa’s atmosphere and it’sinternal ocean with the jovian magnetosphere. The full in-clusion of the time-dependent induced magnetic fields in ourmodel, which is not straightforward, is the main advantage overprevious models of Europa’s interaction, which considered ei-ther only the plasma interaction or only the electromagneticinduction process inside Europa. This allows us to calculate theplasma induced magnetic fields. The induced magnetic fields ofthe plasma currents are complicated and contain higher-ordermultipoles. The dominating terms are the quadrupole terms.The plasma induced fields are strongest when Europa is located


Fig. 10. Same as Fig. 5 for the E26 flyby. The ocean thickness is 25 km. Thepredicted field by including induction is shown for the ocean conductivities σoc:100 mS/m (blue), 500 mS/m (green), 1 S/m (purple), and 5 S/m (black).

in the center of the plasma sheet and weakest when in-betweenthe two extreme conditions, i.e., the center of the plasma sheetand outside the plasma sheet. We find that the spherical har-monics coefficients of the plasma induced magnetic fields arean order of magnitude smaller than the spherical harmonicscoefficients of the background magnetic field induced dipole.Therefore, we conclude that the time-dependent plasma interac-tion generates only weak induction fields. However, the plasmainteraction has an important impact when interpreting the mag-netic field measurements for the induction within Europa. Thus,consideration of the plasma interaction of Europa’s atmosphereleads to significant improved analyses. We would also like topoint out that the plasma induced magnetic field may still influ-ence the lower part of Europa’s ionosphere.

When we compare our model results with the Galileo flybydata we focus on passes that occurred when Europa was lo-cated outside the current sheet, i.e., when the inductive responseis strongest. By modeling the time-dependent plasma interac-tion of Europa with the jovian magnetosphere we get the so farstrongest constraints on the conductivity and the thickness ofthe satellites subsurface ocean. We confirm the need for a highelectrical conducting shell below the surface of Europa and de-rive even larger values for the minimum electrical conductivitythan Zimmer et al. (2000).

The determination of the electrical conductivity of Europa’socean is ambiguous. Since the induced magnetic field dependsboth on the conductivity and the thickness of Europa’s inter-nal ocean we investigate two possible extreme cases discussedin the literature. First, a subsurface ocean with a thickness of100 km, and second a thin subsurface ocean with a thicknessof 25 km. We are not able to spatially resolve the thickness ofthe outer ice crust. Therefore, we use a constant thickness of25 km for this upper shell, which includes the elastic and theductile ice layer. However, an ice crust of 25 km represents theextreme case of a thick crust. Note that a thinner ice crust, e.g.,5 km, would lead to almost the same results, because the mag-netic field at the spacecraft altitude would differ only by a fewnT from that presented here.

Our results for the E14 and the E26 flyby show that if wetake into account the interaction of Europa with the ambientmagnetospheric plasma an even higher conductivity is neededfor the subsurface conducting shell. If the ocean thickness is100 km, the electrical conductivity has to be �500 mS/m. Us-ing the thin ocean model (25 km), we find that the conductivityof Europa’s ocean has to be larger than 1 S/m. In summary,we find the following relation in order to explain the measuredmagnetic field data: electrical conductivity × ocean thickness�50 S/m km, i.e., the conductance of the subsurface ocean hasto be larger than 50,000 S. With such high electrical conduc-tivities a subsurface ocean is even more likely than with theresults given by Zimmer et al. (2000). For an Earth-like ocean(σoc ≈ 5 S/m) a thickness of �10 km is needed. Note that notevery combination of ocean thickness and electrical conductiv-ity, which results in the same product, gives the same inductionresponse. This is particularly not true if the ocean thickness isclose to the skin depth.

It is important to point out that because of the different flybygeometries of the passes used, the lower limit of the suggestedocean conductivity may differ from pass to pass. Remarkably,the E4 flyby is most useful to determine the conductivity ofEuropa’s ocean. This flyby sets even closer constraints on theconductivity distribution inside Europa than the two other fly-bys. We point out that we are not able to set an upper limit onthe ocean conductivity since the induction is almost saturatedfor σoc larger than 5 S/m. Finally, we remind the reader thatat these high ocean conductivities no influence of the mantle orcore is visible in the data when using the synodical period ofJupiter.

The strong evidence for the presence of a deep water oceanbeneath the icy surface puts Europa among the most interestingtargets for planetary exploration in our Solar System. A Eu-ropa orbital mission could provide more extensive time andspace coverage of the magnetic field in the vicinity of Europa.This would allow for a more detailed investigation of the three-dimensional conductivity distribution inside the Moon and adetermination of the thickness of Europa’s ice crust. In addi-tion, periods other than the synodic rotation period of Jupiter,e.g., Europa’s orbital period (Khurana et al., 2002) or periodsdue to asymmetries of Jupiter’s magnetosphere, would be avail-able to an orbiting space craft, and would allow for a deepersounding of the Moons interior.


Acknowledgments

N.S. was partly supported by the Deutsche Forschungsge-meinschaft and partly by DLR.

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