Icarus 192 (2007) 41–55
www.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 Europa
and the jovian magnetosphere. The temporal variations are caused by periodic variations of the magnetospheric plasma and magnetic fields at
Europa. We develop a model which describes simultaneously the 3D plasma interaction of Europa’s atmosphere with Jupiter’s magnetosphere
and the electromagnetic induction in a subsurface conducting layer due to time-varying magnetic fields including their mutual feedbacks. We
find that inclusion of the magnetic field perturbations caused by the interaction with Jupiter’s magnetospheric plasma is important for interpreting
Galileo’s magnetic field measurements near Europa. This leads to improved constraints on the conductivity and thickness of Europa’s subsurface
ocean. 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 to
be 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 caused
by 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 and
modeling show Europa to be a differentiated satellite consisting of a metallic core, a silicate mantle and a water ice–liquid
outer shell. The minimum water ice–liquid outer shell thickness is about 80 km for plausible mantle densities (Anderson
et al., 1998). High resolution data obtained with the Solid
State Imaging (SSI) system show evidence of a young and
thin, 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 subsurface at the time of its recent geological deformation. Globalscale tectonic patterns can be explained by nonsynchronous
rotation and tidal flexing of a thin ice shell above a liquid water ocean (Geissler et al., 1998; Greenberg et al., 2000). The
* Corresponding author. Fax: +49 221 4705198.
E-mail address: schilli@geo.uni-koeln.de (N. Schilling).
0019-1035/$ – see front matter © 2007 Elsevier Inc. All rights reserved.
doi:10.1016/j.icarus.2007.06.024
water layer is likely comprised of three sub-layers: an outer,
brittle/elastic ice layer, an underlying ductile layer of potentially convecting ice, and a lower layer of liquid. Estimates
of the thickness of the ice layer (including the lower ductile
layer) range from a few km to 60 km (e.g., Schenk, 2002;
Hussmann et al., 2002). However, while the evidence for liquid
water in the past is favorable, there is no unambiguous indication from spacecraft imaging that such conditions exist today
(Pappalardo et al., 1999).
Thermal models indicate that a significant portion of the
outer water shell could be liquid today (e.g., Squyres et al.,
1983; Schubert et al., 1986; Spohn and Schubert, 2003). One
energy source for maintaining a liquid water ocean is tidal
heating caused by the three-body Laplace resonance with Io
and Ganymede. This process could offset the freezing of the
water 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 internal ocean depends on its composition, since the occurrence
of minor constituents in the ice and ocean such as hydrated salt
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 the
ice and the freezing temperature of the ocean.
The strongest indication for a liquid water ocean at present
time comes from the magnetic field observations (Kivelson
et al., 1999, 2000; Khurana et al., 1998). Europa has no detectable permanent internal magnetic field (Schilling et al.,
2004). The magnetic field data show evidence for electromagnetic induction taking place in the interior of Europa due to the
time-varying external magnetic field (Neubauer, 1998a, 1998b;
Khurana et al., 1998). Therefore, the observations support the
idea, that a global subsurface conducting layer may be present.
In addition, there are magnetic field perturbations owing to
the interaction of Europa’s atmosphere with the jovian magnetospheric plasma. These time-varying magnetic field perturbations are permanently there on top of the induction signature
compromising their interpretation. While induction signatures
are clearly visible in the data when Europa is well outside
Jupiter’s current sheet, the strong plasma interaction dominates
and hides the induction effect when Europa is close to the center 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-Resolution
Spectrograph. The observations imply molecular oxygen column densities in the range of ∼(2−14) × 1018 m−2 (Hall et al.,
1998). The atmosphere of Europa is produced by the interaction of energetic charged particles with Europa’s surface (e.g.,
Johnson et al., 2004). Shematovich et al. (2005) find by using a
collisional 1D Monte Carlo model of Europa’s atmosphere that
the atmosphere is more strongly structured in the radial direction than previously assumed and that the near-surface region of
the atmosphere is determined by both water and oxygen photochemistry. Recent images of Europa’s atomic oxygen emission
obtained with the HST Space Telescope Imaging Spectrograph
(STIS) indicate that the surface is not icy everywhere, but that
the 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 ratio
different 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 alkalis
is suggested (Johnson et al., 2002; Leblanc et al., 2002). A recent review on Europa’s atmosphere can be found in McGrath
et al. (2004).
Kliore et al. (1997) detected an ionosphere on Europa by
using Galileo radio occultation measurements. They derived
a maximum electron density of about 10,000 cm−3 with a
scale height of 240 km. By assuming a spherically symmetric
ionosphere for each derived electron density profile, Kliore et
al. (1997) found a strongly asymmetric ionosphere with maximum densities on the flanks and minimum densities downstream. The main source of Europa’s ionosphere is electron
impact ionization (Saur et al., 1998).
Zimmer et al. (2000) investigated the implications of the observed induced magnetic fields for the electrical structure of
Europa’s interior. By using a simple shell model they set bounds
on the characteristics of the current carrying layer. Zimmer et al.
(2000) find that the magnetic signature at Europa is consistent
with more than 70% of the induced dipole moment expected for
a perfectly conducting sphere. Therefore, currents are required
which flow in a shell with conductivity of at least 60 mS/m and
close to the surface (within a 200–300 km depth). Zimmer et
al. (2000) argue that solid ice, an ionosphere or a conducting
core cannot reproduce the amplitude of the observed magnetic
perturbation. In addition, Zimmer et al. (2000) argue that it
seems to be very unlikely that the magnetic signature can be explained 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 Europa
with the jovian magnetosphere, which we show to be important
for the interpretation of Galileo’s magnetic field measurements
near Europa. Instead they treat magnetic field perturbations due
to local plasma currents as a systematic error.
Previous models of Europa’s interaction with the jovian
magnetosphere considered either the plasma interaction only
(Saur et al., 1998), included an intrinsic dipole moment as a
free parameter but did not include the induced magnetic field
self-consistently (Kabin et al., 1999; Liu et al., 2000), or they
focused on the electromagnetic induction process inside Europa (Zimmer et al., 2000). For example, Kabin et al. (1999)
included an intrinsic dipole moment as a free parameter, where
they fitted their model results to the data of Galileo’s first Europa flyby. Depending on their model parameters they find different values for the internal dipole moment. When rotating the
plasma flow upstream of Europa by 20◦ , their intrinsic dipole
moment is close to the direction of an induced dipole moment,
but with a small magnitude of about 70% of that obtained by
Kivelson et al. (1999).
In this paper we combine the electromagnetic induction
process inside Europa and the interaction with the ambient
magnetospheric 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 conducting layer. The time-dependent interaction between Europa
and the jovian magnetosphere includes the local plasma interaction of Europa’s atmosphere and ionosphere as well as the
interaction of a potential internal ocean with the magnetosphere
of Jupiter. Due to Jupiter’s rotation with respect to Europa
and the inclination of Jupiter’s magnetic dipole moment, the
magnetospheric plasma density and the background magnetic
field vary at the position of Europa. The time-varying magnetic
fields induce currents in an electrically conducting ocean below
the Europan ice crust. These currents generate a time-varying
induced magnetic field which influences the plasma interaction. In addition, the periodic variations of the magnetospheric
plasma and the interaction with the atmosphere lead to a secondary induction effect. To study this time-dependent interaction, we develop, for the first time, a three-dimensional singlefluid MHD model which includes periodic magnetic fields from
the interior of the Moon.
In Section 2, we give a basic overview of the “classical”
electromagnetic induction problem applied to Europa. The induced magnetic fields derived in this section are part of our
interaction model. In Section 3, we introduce our model of the
Conductivity of Europa’s subsurface ocean
plasma interaction and describe how the induction is implemented self-consistently. We also describe the procedure used
to determine the plasma induced magnetic fields. In Section 4,
we then present and discuss the results of our full numerical
model. We investigate the influence of the plasma interaction
on the induction process. By comparing our model results with
the Galileo spacecraft in situ measurements we determine the so
far strongest constraints on the conductivity of Europa’s ocean.
In Section 5, we summarize our results.
Before proceeding we define the coordinate system used in
this work. We use the EPhiO coordinate system centered at Europa with its z-axis along Jupiter’s spin axis, the y-axis along
the radius vector towards Jupiter (positive inward) and the xaxis azimuthal with respect to Jupiter, i.e., in the ideal corotation direction.
2. The electromagnetic induction problem
We begin with a description of the electromagnetic induction process at Europa. In this section we describe the model
of Europa’s internal electrical structure used in this paper. We
derive the analytical solution for the internal induced magnetic
field which is part of our interaction model presented in the following section.
For studying the interior of the Galilean satellites electromagnetic induction sounding is well suited because Jupiter provides a strong primary signal. The orbit of Europa is located at
the outer edge of the Io plasma torus close to the transition to
the middle magnetosphere of Jupiter. At the position of Europa,
Jupiter’s magnetic field consists of a multipole field of internal origin and the magnetic field of an equatorial plasma sheet
(Connerney et al., 1981). The primary time-varying magnetic
field experienced by Europa is due to the rotation of Jupiter’s
tilted dipole. Together with the satellites orbital period this results in a uniform alternating magnetic field at the position
of Europa with a period of 11.1 h. The background magnetic
field varies mainly in the By - and to a minor degree in the
Bx -component at the position of Europa while Bz is almost constant (Khurana et al., 1998). If one assumes a conducting subsurface layer at Europa, the time-varying inducing field drives
currents in the conducting layer which generate a time-varying
induced magnetic dipole field. Simultaneously, magnetospheric
plasma and magnetic fields interact with Europa’s tenuous atmosphere and ionosphere and also with the time-varying magnetic field from the interior of the Moon. The induced magnetic
fields therefore influence the plasma interaction as it is discussed in Neubauer (1999). The plasma interaction generates
currents, e.g., Alfvénic and ionospheric currents, in the vicinity
of Europa. Due to the time-varying magnetospheric conditions
at the position of Europa during one synodical rotation, the
currents generated by the plasma interaction in the vicinity of
Europa are also time-dependent. This leads to a secondary induction effect, where the magnetic fields induced by the plasma
currents contain also higher order moments (Neubauer, 1999).
The depth of penetration of a time-varying magnetic field of
frequency ω in a conductor of electrical conductivity σ is given
1
by the skin depth δ = (σ μ0 ω/2)− 2 . Here μ0 is the permeabil-
43
ity of vacuum. While the skin depths for inducing fields with
Jupiter’s rotational period in a core similar to the Earth core
would be only ∼180 m, it would be ∼1000 km in the Earth
lithosphere (Stacey, 1992). In Sea water the skin depths would
be ∼45 km. A conducting layer with a thickness larger than
the skin depth in this layer can effectively shield an underlying layer even if this one has a large conductivity. For instance
a conducting subsurface layer at Europa which is thick enough
can effectively shield a conducting mantle or core from induction effects.
We model Europa as a sphere with four different conducting layers (Fig. 1) which are radially symmetric: core, mantle,
“ocean” and crust. Because of the low electrical conductivity
of solid ice at low temperatures, the solid ice crust is assumed
to be nonconducting. Similar problems were solved by Lahiri
and Price (1939) for an increase of conductivity with depth and
by Srivastava (1966) for a sphere made up of concentric shells,
each with uniform conductivity.
The fundamental equation of electromagnetic induction in
static conductors is the diffusion equation
1
∂B
∇ 2B
=
(1)
∂t
σ μ0
which can be obtained from Maxwell’s equations. Here we have
ignored the displacement currents which for our applications
are much smaller than the conduction currents. For the sake of
simplicity the permeability μ inside Europa will be taken equal
to the vacuum permeability μ0 .
Each component of B(t) can be expressed by a series of
superimposed sine waves of various frequencies. Because (1)
is linear in B we can examine one sine wave of a single frequency. The total field can then be determined by superposition.
The potential of the external inducing field can be expanded
in spherical harmonics. For a radially symmetric conductivity
distribution each surface spherical harmonic Snm of the inducing field gives rise to only the same Snm of the induced field
(Parkinson, 1983). Therefore we can deal with each harmonic
separately and superimpose the solutions.
Without loss of generality we can assume an inducing magnetic field with a potential:
n
r
Snm (θ, φ)e−iωt ,
Ue = bBe
(2)
b
where b is the radius of the outer shell (see Fig. 1), Be is a constant 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.
44
N. Schilling et al. / Icarus 192 (2007) 41–55
respectively, and r, θ, φ are spherical coordinates. The induction equation (1) then must be solved within each shell and in
the central region and the boundary conditions at each shell (Br
and Bθ must be continuous) have to be taken into account. The
resulting induced magnetic field outside Europa is then given
by
n+2
b
ind
Bi Snm e−iωt ,
Br = (n + 1)
(3)
r
n+2
∂S m
b
Bθind = −
(4)
Bi n e−iωt
r
∂θ
with Bi given through the boundary conditions at each shell by
F2 (bk1 ) D F2 (bk1 )
F1 (bk1 )
n
Bi
F1 (bk1 ) − (n + 1) + F1 (bk1 ) C F2 (bk1 ) − (n + 1)
=
F1 (bk1 )
F2 (bk1 ) D F2 (bk1 )
Be
n+1
F1 (bk1 ) + n + F1 (bk1 ) C F2 (bk1 ) + n
(5)
with
F1 (ak2 )
F1 (ak1 )
F1 (ak1 ) F2 (ak2 ) G F2 (ak2 )
D F1 (ak1 ) F1 (ak2 ) − F1 (ak1 ) + F1 (ak2 ) E F2 (ak2 ) − F1 (ak1 )
=
C
F2 (ak1 ) F2 (ak1 ) − F1 (ak2 ) + F2 (ak2 ) G F2 (ak1 ) − F2 (ak2 ) F2 (ak1 )
F1 (ak2 )
F1 (ak2 ) E F2 (ak1 )
F2 (ak2 )
(6)
and
G F1 (ck2 )
=
E
F2 (ck2 )
F1 (ck2 )
F1 (ck2 )
F1 (ck3 )
F1 (ck3 )
−
−
F1 (ck3 )
F1 (ck3 )
F2 (ck2 )
F2 (ck2 )
.
(7)
Here k1 , k2 and k3 are the wave numbers in the ocean, mantle
and core,
√ respectively. Thereby the wave number k is defined as
k = iσ ωμ0 . Further,
1
π
F1 (r) =
(8)
(rk)− 2 In+ 1 (rk),
2
2
1
π
(rk)− 2 Kn+ 1 (rk)
F2 (r) =
(9)
2
2
are the solutions of Bessel’s equation, where In+ 1 (rk) and
2
Kn+ 1 (rk) are the modified spherical Bessel functions of first
2
and third kind. The induced magnetic field outside a sphere
with multiple conducting shells can be related to that of a perfect conducting sphere by defining an amplitude A and a phase
lag Φ:
Φ
∞
.
Bind (t) = ABind t −
(10)
ω
The solution of the induction problem derived in this chapter
serves as a basis for our interaction model described in the next
section.
With the analytical solution derived above, we investigated
the influence of the core size, the mantle conductivity and the
thickness of a conducting outer layer on the induction signature
outside Europa depending on the conductivity of the subsurface layer (for more details, see Schilling, 2006). We find that
a conducting core of reasonable size is almost not detectable
outside Europa for conductivities of the subsurface layer larger
than 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) that
a conducting mantle alone cannot explain the induction signature found in the Galileo magnetic field data. An influence on
the induction signature, when using Jupiter’s synodical period
as the excitation period for the induction, is only expected for
an ocean conductivity σoc 100 mS/m. By varying the thickness and the conductivity of Europa’s subsurface layer, we find
that for ocean conductivities larger than 100 mS/m a resolution
of the oceans lower boundary is almost not possible if the ocean
thickness is larger than 100 km. In addition, the lower boundary of the ocean cannot be resolved if the ocean is thicker than
10 km and the ocean conductivity is larger than 1 S/m.
3. Satellite plasma interaction and the concept of a virtual
plasma
3.1. Description of the model
3.1.1. Concept
The basic idea of this paper is to study the time-dependent
plasma interaction of Europa’s atmosphere and its proposed internal ocean with the jovian magnetosphere. We describe the
plasma environment of Europa with a three-dimensional singlefluid MHD model and solve the MHD flow problem and the
internal induction problem simultaneously. Since both effects
interact, the MHD equations are rewritten such that the induced
magnetic field is explicitly included.
3.1.2. Mass balance
Magnetospheric plasma is convected into the atmosphere of
Europa where the density is modified by ionization and recombination processes. An ionospheric singly charged ion population with mO2 = 32 amu is produced in our model. Following
Kivelson et al. (2004), we use for the determination of the magnetospheric electron density an effective ion charge of the upstreaming plasma of z = 1.5. In our model the plasma consists
of one ion species. We describe the evolution of the bulk density
ρ by
∂ρ
(11)
+ ∇ · ρu = (P − L)mi ,
∂t
where P is the production rate, L the loss rate and u the bulk
velocity. Saur et al. (1998) find that electron impact ionization
is the dominant process to generate Europa’s ionosphere, being
over an order of magnitude larger than photoionization. Therefore, we only include electron impact ionization as a source
process for the ionospheric plasma. The ionization cross sections σj are taken from the NIST database (Kim et al., 1997).
Ionospheric electrons which are produced by electron impact are much cooler than the magnetospheric electrons and
therefore are not involved into the ionization process. While
the number of ionospheric electrons is determined by (11), we
assume a separate continuity equation for the magnetospheric
electrons. Electron impact ionization does not change the number of the magnetospheric electrons and the loss of the magnetospheric electrons due to dissociative recombination is negligible due to their low density and high temperature. We assume
Conductivity of Europa’s subsurface ocean
45
the magnetospheric electrons to move with the bulk velocity.
Thus, we solve the following continuity equation for the magnetospheric electrons
∂ne,ms
+ ∇ · ne,ms u = 0.
∂t
The loss rate L in (11) is given by
L = α D n2
(12)
(13)
with the number density n and a dissociative recombination coefficient (Torr, 1985)
300 0.7
m3 /s.
αD = 2 × 10−13
(14)
Te
3.1.3. Induction equation and momentum equation
The magnetic field at Europa can be subdivided into several portions. First, the time-varying background magnetic field
of Jupiter and the magnetospheric current sheet B0 . Second,
the magnetic field caused by the interaction of the magnetospheric plasma with Europa’s atmosphere BP . Finally, we
have the induced magnetic field from the interior due to the
time-varying background field and the time-varying plasma currents Bind = Bind (B0 ) + Bind ( jP ). We assume the background
magnetic field B0 to be a time-varying homogeneous magnetic
field at Europa. The periodicity of the background field is given
by the synodic rotation period of Jupiter. Doing so, we do not
account for other excitation periods, e.g., Europa’s orbital period.
The total magnetic field Btot at Europa can thus be written
Btot = B0 + BP + Bind (B0 ) + Bind ( jP ).
(15)
We assume the outer ice crust of Europa to be electrically
nonconducting, which causes the current systems outside and
inside Europa to be isolated against each other. Thus, B0 and
Bind (B0 ) are potential fields outside the Moon.
Generally, the total magnetic field has to fulfill the induction
equation, which can be written as
∂Btot
(16)
= ∇ × (u × Btot ) − ∇ × (η∇ × Btot ),
∂t
where η is the magnetic diffusivity. Inside Europa the dynamo
term (first term on the right-hand side) vanishes and (16) then
describes the diffusion of the magnetic field into the Moon.
For the variation of the Bx - and By -components of Jupiter’s
magnetospheric field at Europa’s location we fit an ellipse to
the data given in Kivelson et al. (2000) (Fig. 2). Thus, the timevarying inducing background magnetic field Bi0 is calculated
analytically by
Bi0 = −84nT sin (Ωt)x̂ − 210nT cos (Ωt)ŷ.
(17)
Ωt is the angle between the rotating dipole moment of Jupiter
and 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 rotating
dipole moment of Jupiter points towards (Ωt = 0◦ ) or away
from Europa (Ωt = 180◦ ). We use a constant B0,z = −410 nT.
Fig. 2. Background magnetic field at Europa. Values are obtained by fitting an
ellipse to the magnetic field data given by Kivelson et al. (2000).
Using (15) and (16) results in the following induction equation for the magnetic field caused by the interaction of the magnetospheric plasma with Europa’s atmosphere and the thereby
induced magnetic field which we solve with our model:
∂(BP + Bind ( jP ))
= ∇ × u × BP + Bind ( jP )
∂t
me
L
me
+
ν
+
∇
ν
−∇ ×
en
in
mi
n
ne2
× BP + Bind ( jP )
+ ∇ × u × B0 + Bind (B0 )
∂(B0 + Bind (B0 ))
(18)
.
∂t
Thereby we account for resistivity due to loss processes, ionneutral and electron-neutral collisions in the generalized Ohm’s
law. We use a constant initial plasma velocity with v0,x =
104 km/s. The collision frequency of the ions with the neutrals
is
α0 −1
νin = 2.6 × 10−9 nn
(19)
s ,
μa
−
where α0 is the polarizability of the neutral gas in units of
10−24 cm−3 , nn is the neutral gas density in cm−3 , and μa is
the reduced mass in amu. We use α0 = 1.59 for O2 (Banks
and Kockarts, 1973). For the electron-neutral collision frequency we use νen = 10−9 nn s−1 which we calculated by using
the momentum-transfer cross sections given by Itikawa et al.
(1989). Induction effects also have to be accounted for in the
momentum equation, which then reads
∂u
ρ
+ u · ∇u = ∇ × BP + Bind ( jP )
∂t
× BP + Bind ( jP ) − ∇p
P
me
νen + νin +
ρu
−
mi
n
46
N. Schilling et al. / Icarus 192 (2007) 41–55
+ ∇ × BP + Bind ( jP )
× B0 + Bind (B0 )
+ ∇ × B0 + Bind (B0 )
× B0 + BP + Bind (B0 ) + Bind ( jP ) .
the magnitude of the magnetic field is represented very well by
the model for all of the simulated flybys while using constant t0
and H , suggesting that this is a proper assumption.
(20)
3.1.4. Plasma temperature
The interaction of the plasma with the neutral gas (inelastic
as well as elastic processes) changes the internal energy e of the
plasma. Here, we deliberately use a simplified energy equation
∂e
(21)
= −∇ · eu.
∂t
Including additional terms in (21), such as frictional heating from the relative motion between plasma and neutral atmosphere or Joule heating, does not affect the results in this
paper as tested in multiple runs. The only process which is
strongly temperature dependent is the production of ionospheric
electrons by electron-impact ionization. For a proper description of this process, the temperature of the magnetospheric
electrons Te,ms is required. While the evolution of the density
of this population is given by (12), we use the following description for their temperature dependence: The plasma torus
around Europa represents an extensive energy reservoir, which
can provide energy via electron heat conduction along the magnetic field lines. This process is extremely effective at Europa
(Saur et al., 1998). Although this energy reservoir is strictly
speaking limited, it is legitimated to assume an unlimited energy reservoir for our purposes. Moreover, we assume infinite
heat conductivity along the field lines. Saur et al. (1998) show
that the electron temperature is reduced strongest close to the
surface and on the flanks where the electron density reaches
its maximum because of the longer transport time for a plasma
fluid element through the dense part of the ionosphere where
most of the inelastic collisions occur. By maintaining a con0
stant temperature Te,ms
for the magnetospheric electrons one
overestimates the production rate in these regions. Hence, the
plasma density is overestimated. In order to compensate for this
deficiency, we use a simple parameterization and implement
a calibration factor for the temperature Te,ms of the magnetospheric electrons. The temperature of the magnetospheric
electrons then has the following spatial dependence:
φ − Hh
0
T
Te,ms = Te,ms 1 − 1 − t0 cos
(22)
e
.
2
The azimuthal angle φ varies in the xy-plane from 0◦ (upstream) to 180◦ (downstream), h is the height above the surface. The factor t0 and the scale height HT are the calibration
parameters to be determined. We determine these parameters
by comparing our model with measurements of flybys during
which Europa was located in the middle of the current sheet
(e.g., E12), i.e., where induction effects are negligible. Doing
so, we find t0 = 0.1 and HT = 300 km. Once determined, the
calibration parameters are assumed to be the same for all model
scenarios. Thereby we assume that the atmosphere of Europa
does not have a strong asymmetry. As we will see in Section 4,
3.1.5. Atmosphere
We 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 observations by Hall et al. (1995). This is consistent with an O2
column density of Ncol = 5 × 1018 m−2 . Pospieszalska and
Johnson (1989) demonstrate that sputtering is not uniform over
the surface of Europa, but is decreasing from the trailing to the
leading hemisphere. We follow Saur et al. (1998) in assuming
that the surface density n0 (φ) varies in direct proportion to the
normalized flux variation calculated by Pospieszalska and Johnson (1989). The surface density then has the following spatial
dependence:
π
− φ cos φ
n0 (φ) = n0,0 1.08H
2
φ
+ 0.885 cos + 1.675 ,
(23)
2
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 Europa
The magnetospheric electrons at the location of Europa can
be divided basically in two populations, a thermal Maxwellian
and a suprathermal non-Maxwellian population. For the suprathermal population we use a density of nsth = 2 cm−3 at
Te,sth = 250 eV (Sittler and Strobel, 1987). For the thermal population we use Te,th = 20 eV (Sittler and Strobel, 1987), while
their density varies with the position of Europa in the plasma
sheet. For the upstream magnetospheric plasma we use an ion
mass mi = 18.5 amu (Kivelson et al., 2004). The variation of
the background plasma density stems from the tilt of the plasmasheet against Europa’s orbital plane by about 7◦ (Dessler,
1983). In addition, the rotational velocity of the plasma is larger
than the orbit velocity of Europa. Thus, Europa passes through
different plasma regimes during one synodic rotation of Jupiter.
We assume the plasma to be symmetric around the centrifugal equator and to vary periodically, with a minimum electron
number density of ne = 18 cm−3 when Europa is outside the
plasma sheet and a maximum value of ne = 250 cm−3 when
Europa is in the center of the plasma sheet. In between we assume the density to fall of with exp(−(z/H )2 ), where H is the
scale height of the plasma and z is the distance of Europa from
the center of the plasma sheet (Thomas et al., 2004). With the
values above, the scale height of the plasma in our model is
H = 0.7 RJ .
3.1.7. Numerics
The kernel of our model consists of Zeus3D (Stone and
Norman, 1992a, 1992b), a three-dimensional time-dependent
code which solves the ideal MHD equations. We extended the
Zeus3D code to nonideal MHD. We use a Cartesian grid divided into four regions with different spatial resolution: a very
Conductivity of Europa’s subsurface ocean
high resolution region from −1.5 RE to 1.5 RE in all three directions in space with a grid size of 80 km, a high resolution
region up to a distance of 3 RE in all three directions in space
with a grid size of 157 km, followed by a medium resolution
region with a grid size of 795 km and finally a low resolution
region 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 atmosphere
of Europa cannot completely shield the surface of Europa from
the streaming plasma flow. Plasma which hits the surface of Europa is absorbed by the surface and interacts with it. Thus, the
interior of Europa is free of plasma, while the magnetic field
can diffuse into the Moon. This fact can be implemented in
two ways: First, inner boundary conditions could be set at the
surface to connect the external regimes to the internal one described by (16). This solution is impractical, because it would
require a correct description of the diffusion into the Moon
through the boundaries. Secondly, the interior of Europa can
be part of the simulation. As the description of a solid body is
not included in the single-fluid equations a priori, we describe
the interior of Europa as a “virtual” plasma to mimic the real
relevant properties of a solid body. This leads to a consistent
description of the interior and the exterior of Europa. For numerical reasons the volume of the solid body cannot be free of
plasma 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 magnetic
Reynolds 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 to
/ρ. Hence, we control ηi by adjusting the neutral gas
nvirtual
n
density in every time step at every grid point in the interior
depending on the local plasma density. Simultaneously, loss
processes are implemented in order to reduce the plasma density in the interior of the Moon. For the plasma momentum
equation, (20) is applied in the interior. It must be controlled in
such a way that the velocity is small enough to guarantee RM
to be small.
47
approach (Neubauer, 1998b). The plasma interaction picture
adjusts to the slowly varying internal and external conditions in
a quasi-stationary way. Therefore, the plasma interaction picture as a function of time is given by consecutive stationary
states, where each steady state is given by the current upstream
conditions and the present internal magnetic field.
We then solve the external MHD problem and the internal
electrodynamic problem by using our 3D-MHD model in an iterative process. As a starting point we include the induction by a
homogeneous background field only. Thereby we use the timevarying components of the background magnetic field given
by (17). We then have the analytical solution of (1) as an initial condition for the internal magnetic field. Then we solve a
stationary problem for 8 different times ti which are equally distributed 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 considerably
at the different times ti .
In the second step the induced magnetic field Bind ( jP ) due to
the time-varying currents in the exterior is determined. Therefore, we first acquire the time variable part of the external field
Bp on the surface of the conducting sphere at a given time:
Bp = Bp + Bind ( jP ) − Bp + Bind ( jP )
(24)
where the constant part averaged over one synodic period is
calculated by
Bp + B
ind
1
( jP ) =
Tsyn
Tsyn
Bp (t) + Bind ( jP ) dt.
(25)
0
Subsequently, we determine the harmonic coefficients of the induced magnetic fields for each time ti . A Fourier expansion
of these harmonic coefficients determines the plasma induced
magnetic fields Bind ( jP ) for any time t . The analytical description of the plasma induced fields together with the induced field
Bind (B0 ) from the first iteration yields the new initial internal
magnetic field. We then repeat the above procedure until we
reach convergence for the determined harmonic coefficients.
4. Results and discussion
3.3. Procedure
4.1. Harmonic coefficients of the plasma induced magnetic
fields
In principle, the interaction problem could be solved by solving the set of single-fluid equations in the exterior and the
diffusion equation in the interior of the Moon simultaneously.
However, because of the different time scales of the processes
involved and the Courant–Friedrich–Lewy (CFL) criteria, the
simulation time needed for a stable periodic solution would
be unrealistically long. Therefore, we develop a different approach. For this approach two time scales are important: the
time it takes the plasma interaction to adjust to new external and
internal conditions, which is in the order of a few minutes and
the timescale of the electromagnetic induction, i.e., the diffusion time scale, which is on the order of the synodical period of
Jupiter (Neubauer, 1998b). Thus, we can use a quasi-stationary
After deriving the spherical harmonic coefficients (also
called Gauss coefficients) of the plasma induced magnetic fields
for each time ti , we do a Fourier expansion of these coefficients. This enables us to calculate the harmonic coefficients at
any time t . Hence, we are able to calculate the plasma induced
magnetic field at any point and at any time t . Fig. 3 shows the
harmonic dipole and quadrupole coefficients of the plasma induced magnetic fields during Jupiter’s synodical period for an
assumed ocean thickness of 100 km, an ocean conductivity of
5 S/m and a thickness of the ice crust of 25 km. The derivation of the spherical harmonic coefficients was done by using
the normalization according to the convention of Schmidt for
the associated Legendre polynomials (e.g., Blakely, 1995).
48
N. Schilling et al. / Icarus 192 (2007) 41–55
Table 1
Constant part of the Gauss coefficients of the external field (order 1 and 2)
generated by plasma currents. The multipole coefficients are in nT
Fig. 3. Harmonic coefficients of the plasma induced magnetic fields during
Jupiter’s synodical period. The upper panel shows the dipole, the lower panel
the quadrupole coefficients.
The Gauss coefficients gnm and hm
n for a given expansion can
be grouped under three headings; zonal, sectoral, and tesseral
harmonics. Zonal harmonics are those of order zero. Note that
none of the h coefficients can be of order zero. The zonal harmonics represent fields whose moments are aligned with the
z-axis. Therefore, the g10 term describes the moment of an axial Europa-centric dipole aligned with the z-axis, and the g20
term, an axial, Europa-centric, quadrupole. Sectoral harmonics
are those for which degree and order are equal. They represent fields which have their moments in the equatorial plane. In
the EPhiO coordinate system the g11 term describes a Europacentric dipole aligned with the y-axis while the dipole associated with the h11 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◦ and
180◦ . The g10 coefficient varies with a period of π but the
amplitude is not exactly symmetric. The h11 term also shows
an asymmetric behavior. The quadrupole coefficient with the
largest values is the g21 term. It is π -periodic with a maximum
value of ≈14 nT. The g20 can also reach values up to 10 nT and
is 2π -periodic.
None of the harmonic coefficients shows a purely π -periodic
behavior. This feature can be explained by the time-varying
G01
G11
H11
G02
G12
G22
H21
H22
12
−8
2
−2
−29
5
−2
2
magnetospheric background conditions in our model. Although
we use a plasma density model which is symmetric around
the centrifugal equator and a background magnetic field which
varies symmetric in time, the Bx component of our background
magnetic field breaks the symmetry.
Please note, that the expected differences in the measured
magnetic field near Europa when Europa is located in the center
of the plasma sheet compared to outside the plasma sheet is not
simply reflected in the difference of the harmonic coefficients.
The time-varying magnetic field caused by the interaction of
Europa’s atmosphere with the jovian magnetospheric plasma
depends on the external coefficients, which are related to the
internal Gauss coefficients by (5), and the location on the conducting surface given by (2). For example, the difference in
the g21 term between Europa in the center and outside of the
plasma sheet is 25 nT. Upstream of Europa on the surface of
the conducting layer this corresponds to a difference in Bz of
130 nT. Other coefficients give additional contributions. The
overall magnetic field due to the interaction of Europa’s atmosphere with the jovian magnetospheric plasma is given by
a Fourier synthesis of all coefficients. Since the source of the
plasma induced fields is outside Europa, the magnetic field
strength in the source region is even higher. The decrease in
field strengths between source region and surface of the conducting layer becomes even larger for higher order moments
(octupole, etc.) which therefore may be more important outside
of Europa.
We also show lower order moments of the constant part of
the Gauss coefficients of the external field calculated by (25)
in Table 1. The G12 term has the largest value and accounts for
the upstream enhancement and the downstream decrease of Bz
at Europa. Again the constant magnetic field of the plasma currents at Europa is then given by a Fourier synthesis of all the
Gauss coefficients and using (2) and (5). We remind the reader
that higher order moments become important outside Europa
close to the source region of those currents, and the extrapolation is only valid in the “source free” region below the external
current region. Therefore, a magnetic field calculated by using
only the extended coefficients given in Table 1 maybe inaccurate away from the surface.
The contribution of the single multipoles to the plasma induced magnetic field can be displayed by the spatial power
spectrum of the internal magnetic field Rn (Blakely, 1995),
where Rn is defined as the scalar product Bn · Bn averaged over
the spherical surface S(r):
1
Rn (r) =
4πr 2
2ππ
Bn · Bn r 2 sin θ dθ dφ
0 0
(26)
Conductivity of Europa’s subsurface ocean
with
Bn = −∇ Rc
Rc
r
+ hm
n sin mφ
n+1 n
49
gnm cos mφ
m=0
Pnm (θ )
(27)
and
B=
∞
(28)
Bn
n=1
where r = Rc represents the surface of the ocean. Using the
orthogonality property of spherical harmonics, Eq. (26) can be
reduced to (Backus et al., 1996):
2n+4
n
m 2 m 2 Rc
Rn (r) =
(29)
gn + hn .
(n + 1)
r
m=0
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 over
any sphere is the sum over the spectrum:
|B|2
=
S(r)
∞
|Bn |2
n=1
=
S(r)
∞
Rn (r).
(30)
n=1
Fig. 4 shows the spectral coefficients for dipole, quadrupole
and octupole terms at the surface of the ocean. It is obvious
that the main spectral power is in the quadrupole field. The
plasma induced fields are strongest when Europa is located in
the center of the plasma sheet. Smallest values are found when
Europa is between the two extreme conditions (inside and outside the plasma sheet), i.e., at Ωt = 45◦ , 135◦ , 225◦ , and 315◦ .
The power in the octupole terms is small compared to dipole
and quadrupole contributions. Higher-order multipoles (n > 3)
are even less important. The spectral coefficients for the background field induced dipole term varies between ∼14,000 and
∼88,000 nT2 (not shown).
We would like to point out that the external interaction fields
are mainly of order 2 and therefore induce mainly quadrupole
fields in a spherically symmetric conductor. This is in agreement with our results shown in Fig. 4. In a perfectly symmetric case, there would be no plasma induced dipole components. However, the following features break the symmetry:
The Alfvén wings are bend back and the background bulk velocity is not always perpendicular to the background magnetic
field. During a synodical rotation period of Jupiter, the strength
of the currents, which flow in the Alfvén wings, varies periodically. This also leads to time varying external uniform fields
that induce dipole components in the conductor. In addition,
differences between upstream and downstream ionospheric currents create an asymmetry. The asymmetric current system can
be illustrated by superposing the symmetric ionospheric currents with a closed current loop. While the symmetric currents
induce quadrupole fields, the closed loop gives a uniform magnetic field in the center of Europa and thus induces a dipole
magnetic 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 plasma
induced magnetic field.
Once we have calculated the plasma induced magnetic
fields, we include them as initial internal magnetic fields into
our model in addition to the background field induced dipole.
Coefficients of higher order are only important very close to
the surface since according to (29) Rn (r) ∼ r −(2n+4) . Therefore, we only consider dipole and quadrupole coefficients when
including the plasma induced fields in our model. In the next
iteration step, we then repeat the procedure described above
and calculate the plasma induced magnetic fields again. All coefficients derived after the second iteration differ from those
derived after the first iteration by less than 10%, which is good
enough for our calculations. Therefore, we stop the calculations
after the second iteration.
Comparing the dipole and quadrupole coefficients of the
plasma induced magnetic fields to the dipole coefficients of the
background magnetic field (Fig. 2), which are on the order of
100 nT, it is obvious that the plasma interaction has only a weak
impact on the induction process. However, the coefficients derived may have an influence on the lower part of the ionosphere
of Europa. Therefore, they may be important when modeling
this part of Europa in detail.
The results presented in this section were derived by assuming an almost saturated induction process. Note, that a smaller
ocean or an ocean which is less conductive leads to even smaller
harmonic coefficients.
4.2. New constraints on the conductivity of Europa’s
subsurface ocean
The comparison of our simulation results with the Galileo
magnetic field data along the trajectories of different flybys allows us to study the conductivity distribution inside Europa.
We concentrate on flybys that occurred when the inductive response is strongest, i.e., when Europa was well outside the
current sheet. This was the case for the E4, the E14, and the
E26 flyby [see Tables 1 in Schilling et al. (2004) and Kivelson
et al. (2000)]. We remind the reader that the model is parame-
50
N. Schilling et al. / Icarus 192 (2007) 41–55
terized such that for a flyby inside the plasma sheet (e.g., E12)
where induction effects play no role the agreement with magnetic field observations is very good.
4.2.1. Europa flyby E4
The first close Europa encounter of Galileo was the E4 flyby
on December 19, 1996. This pass occurred in Jupiter’s northern
magnetic hemisphere (Kivelson et al., 1997), i.e., the magnetic
background field at Europa pointed away from Jupiter. Thus,
the primary induced dipole moment pointed toward Jupiter. The
flyby was an equatorial pass, oblique through Europa’s wake.
Closest approach was at 06:52:58 universal time (UT) at an altitude of 695 km.
Fig. 5 displays the magnetic field along the trajectory in the
EPhiO coordinate system. The red curve shows the magnetic
field measured by the Galileo spacecraft (Kivelson et al., 1997).
Our model results for a pure plasma interaction without induction in the interior of the Moon are indicated by the dashed
black curve. In addition, the predicted fields including induction
into our model are shown. We choose different values for the
conductivity of Europa’s ocean and analyze what conductivity
Fig. 5. Observed and modeled magnetic field for the E4 flyby in the EPhiO coordinate system. From top to bottom: Bx , By , Bz , Bm . The red curve shows
the measured field (Kivelson et al., 1997). The dashed black curve shows the
predicted field when no induction is included in our model. The predicted 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). The assumed thickness of the crust is 25 km and the assumed thickness of the ocean
is 100 km.
fits the measured data best. We start with an ocean thickness of
100 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 conductivity of Sea water found on Earth. For larger values of the
σoc , the induction process is saturated, i.e., larger conductivities
yield effectively the same results.
The dashed black curve in Fig. 5 indicates that a small contribution from the plasma interaction can be found in the Bx and
By component of the measured data. Modeling the data without induction cannot explain these components. As the E4 flyby
was an equatorial pass, the main contribution in the Bx and By
component results from the induced magnetic field in the interior. With the used model of Europa’s interior, we are able to reproduce the Bx and By component for an ocean conductivity of
500 mS/m or larger. Note that by using a 100 km ocean thickness, 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 minimum
in the wake region in the Bx component (between UT 07:00
and 07:05) independent from the ocean conductivity used. This
suggests, that this feature is caused by the plasma interaction.
For the agreement between the data and our model, we see
no need for a deviation of the upstreaming plasma flow from
the nominal corotation direction as it was suggested by Kabin
et al. (1999). Both the Bz component and the magnetic field
magnitude can almost completely be explained by the plasma
interaction. Panel 3 and 4 of Fig. 5 show that our model reproduces the overall structure as well as the two local maxima.
The negative perturbation of Bz and |B| occurs in the downstream region where the plasma is accelerated and the magnetic
field magnitude is decreased. The agreement between the data
and the model for Bz and |B| can also be seen as a test for our
plasma model.
The induced magnetic field depends both on the conductivity and the thickness of the ocean. Thus, the determination of
σoc is ambiguous. To show this ambiguity, we vary the ocean
thickness. As an example Fig. 6 shows our simulation results
for 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, larger
ocean conductivities are necessary when using a thinner ocean.
Fig. 6 indicates that in this case ocean conductivities of 1 S/m
or less are insufficient to reproduce the Bx and By component
of the magnetic field when an ocean thickness of 25 km is assumed. A conductivity of, e.g., 5 S/m is needed. We remind the
reader that the induced magnetic field is saturated for conductivities larger than 5 S/m. Hence we can only set a lower limit
on the ocean conductivity. In reverse, we can conclude that if
the ocean conductivity is less than 1 S/m, the ocean has to be
thicker than 25 km. In summary, this results in the following relation: electrical conductivity × ocean thickness 50 S/m km.
4.2.2. Europa flyby E14
The Europa flyby, E14, occurred in the low plasma density
region above the current sheet (Kivelson et al., 1999). Thus,
Conductivity of Europa’s subsurface ocean
51
Fig. 6. Same as Fig. 5 for an ocean thickness of 25 km. The predicted 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).
Fig. 7. Same as Fig. 5 for the E14 flyby. The ocean thickness is 100 km. The
predicted 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).
the induced magnetic field is very similar to that during the E4
flyby. Again, the primary induced dipole moment pointed close
to the Jupiter-facing meridian. The trajectory of the E14 pass
was at higher altitude and latitude than the E4 pass. Closest
approach occurred at an altitude of 1647 km at 13:21:05 UT on
March 29, 1998. The flyby was a mostly upstream pass which
ended downstream.
As in the previous section, we discuss two different assumed
ocean thicknesses for the E14 flyby, 100 and 25 km. We start
with an assumed ocean thickness of 100 km. The measured
magnetic field (red) along the E14 trajectory in the EPhiO coordinate system is shown in Fig. 7. In addition, the predicted field
by neglecting induction (dashed black) and including induction
in a 100 km thick ocean for different assumed ocean conductivities σoc is displayed. The color code used in Fig. 7 is the same
as in Fig. 5.
The Bx component of the measured magnetic field cannot
be explained by plasma interaction alone. We are able to fit
this component very well when using ocean conductivities of
250 mS/m and higher. As mentioned above, an upper limit
for σoc cannot be assessed. The By component as well as the
Bz component and the magnetic field magnitude can almost
be explained by the plasma interaction. This demonstrates the
importance of including the plasma interaction for interpreting
Galileo’s magnetic field data for a subsurface ocean. The en-
hancement of Bz and |B| occurs upstream of Europa where the
plasma is slowed down and the magnetic field is compressed.
Since the flyby was above the equator, the bending of the magnetic field is visible as a positive perturbation of Bx .
Fig. 8 shows the simulation results when using a thickness
of the ocean of 25 km and a crust thickness of 25 km. In this
case ocean conductivities of at least 1 S/m are necessary to reproduce the Bx component of the magnetic field. In reverse, we
can conclude that for the E14 flyby the ocean has to be thicker
than 25 km for σoc 1 S/m. This is the same result as for the
E4 flyby.
4.2.3. Europa flyby E26
The E26 flyby was the crucial flyby to distinguish between
a permanent and an induced magnetic dipole moment as source
of Europa’s internal magnetic field (Kivelson et al., 2000). This
pass occurred south of Jupiter’s magnetic equator in a region
with low plasma density. Therefore, the induced magnetic field
was almost 180◦ out of phase with its value on the E4 and
E14 pass. Closest approach occurred at an altitude of 346 km
at 17:59:43 UT on January 03, 2000. The flyby trajectory was
upstream of Europa, nearly radial toward Jupiter, and south of
Europa’s equator.
Fig. 9 shows the magnetic field along the trajectory in the
EPhiO coordinate system for an ocean thickness of 100 km
52
N. Schilling et al. / Icarus 192 (2007) 41–55
Fig. 8. Same as Fig. 5 for the E14 flyby. The ocean thickness is 25 km. The
predicted 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).
Fig. 9. Same as Fig. 5 for the E26 flyby. The ocean thickness is 100 km. The
predicted 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).
and a crust thickness of 25 km. We are able to fit the overall
structure of the magnetic field fairly well. Since this flyby was
very close to Europa and off the equator, contributions from
the plasma interaction can be found in every component of the
magnetic field.
The bendback of the Alfvén wing leads to an enhancement
of the Bx component around closest approach. While we can fit
Bx the overall form, we are not able to fit the double peak structure of this component. This structure occurs around closest
approach at altitudes which are within or very close to Europa’s
ionosphere. As we use a simple model of Europa’s atmosphere
the detailed structure of the magnetic field at these altitudes is
beyond the scope of our model. In the previous section we show
that the influence of the plasma interaction on the induction
process is weak. Hence, the details of the plasma interaction
do not influence our statement on the conductivity distribution
in the interior of Europa. The By component contains contributions from the plasma interaction as well as from the induction.
Including induction with ocean conductivities of 250 mS/m
and larger improves the fit of this component fairly well. The Bz
component also contains contributions from the plasma interaction. Including induction in an ocean with σoc 250 mS/m and
larger leads to a better fit to the data. By using conductivities of
250 mS/m and larger we can fit the magnetic field magnitude
very well. The enhancement of the field magnitude is due to the
compressing of the magnetic field upstream of Europa. Please
note, that our calibration parameters defined in Section 3.1.4 remain the same for all three flybys. The good agreement of the
magnetic field magnitude for all three flybys between the data
and the model supports our choice of parameters.
In Fig. 10 we show our simulation results when using an
ocean thickness of 25 km and a crust thickness of 25 km. Again,
the Bx component is determined mainly by the plasma interaction. Using ocean conductivities larger than 1 S/m improves
the fit of the By component. For Bz good fits are obtained when
using σoc 500 mS/m.
5. Conclusions
We have developed a self-consistent 3D Model of the timedependent plasma interaction of Europa’s atmosphere and it’s
internal ocean with the jovian magnetosphere. The full inclusion of the time-dependent induced magnetic fields in our
model, which is not straightforward, is the main advantage over
previous models of Europa’s interaction, which considered either only the plasma interaction or only the electromagnetic
induction process inside Europa. This allows us to calculate the
plasma induced magnetic fields. The induced magnetic fields of
the plasma currents are complicated and contain higher-order
multipoles. The dominating terms are the quadrupole terms.
The plasma induced fields are strongest when Europa is located
Conductivity of Europa’s subsurface ocean
Fig. 10. Same as Fig. 5 for the E26 flyby. The ocean thickness is 25 km. The
predicted 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-between
the two extreme conditions, i.e., the center of the plasma sheet
and outside the plasma sheet. We find that the spherical harmonics coefficients of the plasma induced magnetic fields are
an order of magnitude smaller than the spherical harmonics
coefficients of the background magnetic field induced dipole.
Therefore, we conclude that the time-dependent plasma interaction generates only weak induction fields. However, the plasma
interaction has an important impact when interpreting the magnetic field measurements for the induction within Europa. Thus,
consideration of the plasma interaction of Europa’s atmosphere
leads to significant improved analyses. We would also like to
point out that the plasma induced magnetic field may still influence the lower part of Europa’s ionosphere.
When we compare our model results with the Galileo flyby
data we focus on passes that occurred when Europa was located outside the current sheet, i.e., when the inductive response
is strongest. By modeling the time-dependent plasma interaction of Europa with the jovian magnetosphere we get the so far
strongest constraints on the conductivity and the thickness of
the satellites subsurface ocean. We confirm the need for a high
electrical conducting shell below the surface of Europa and derive even larger values for the minimum electrical conductivity
than Zimmer et al. (2000).
53
The determination of the electrical conductivity of Europa’s
ocean is ambiguous. Since the induced magnetic field depends
both on the conductivity and the thickness of Europa’s internal ocean we investigate two possible extreme cases discussed
in the literature. First, a subsurface ocean with a thickness of
100 km, and second a thin subsurface ocean with a thickness
of 25 km. We are not able to spatially resolve the thickness of
the outer ice crust. Therefore, we use a constant thickness of
25 km for this upper shell, which includes the elastic and the
ductile ice layer. However, an ice crust of 25 km represents the
extreme case of a thick crust. Note that a thinner ice crust, e.g.,
5 km, would lead to almost the same results, because the magnetic field at the spacecraft altitude would differ only by a few
nT from that presented here.
Our results for the E14 and the E26 flyby show that if we
take into account the interaction of Europa with the ambient
magnetospheric plasma an even higher conductivity is needed
for the subsurface conducting shell. If the ocean thickness is
100 km, the electrical conductivity has to be 500 mS/m. Using the thin ocean model (25 km), we find that the conductivity
of Europa’s ocean has to be larger than 1 S/m. In summary,
we find the following relation in order to explain the measured
magnetic field data: electrical conductivity × ocean thickness
50 S/m km, i.e., the conductance of the subsurface ocean has
to be larger than 50,000 S. With such high electrical conductivities a subsurface ocean is even more likely than with the
results given by Zimmer et al. (2000). For an Earth-like ocean
(σoc ≈ 5 S/m) a thickness of 10 km is needed. Note that not
every combination of ocean thickness and electrical conductivity, which results in the same product, gives the same induction
response. This is particularly not true if the ocean thickness is
close to the skin depth.
It is important to point out that because of the different flyby
geometries of the passes used, the lower limit of the suggested
ocean conductivity may differ from pass to pass. Remarkably,
the E4 flyby is most useful to determine the conductivity of
Europa’s ocean. This flyby sets even closer constraints on the
conductivity distribution inside Europa than the two other flybys. We point out that we are not able to set an upper limit on
the ocean conductivity since the induction is almost saturated
for σoc larger than 5 S/m. Finally, we remind the reader that
at these high ocean conductivities no influence of the mantle or
core is visible in the data when using the synodical period of
Jupiter.
The strong evidence for the presence of a deep water ocean
beneath the icy surface puts Europa among the most interesting
targets for planetary exploration in our Solar System. A Europa orbital mission could provide more extensive time and
space coverage of the magnetic field in the vicinity of Europa.
This would allow for a more detailed investigation of the threedimensional conductivity distribution inside the Moon and a
determination of the thickness of Europa’s ice crust. In addition, periods other than the synodic rotation period of Jupiter,
e.g., Europa’s orbital period (Khurana et al., 2002) or periods
due to asymmetries of Jupiter’s magnetosphere, would be available to an orbiting space craft, and would allow for a deeper
sounding of the Moons interior.
54
N. Schilling et al. / Icarus 192 (2007) 41–55
Acknowledgments
N.S. was partly supported by the Deutsche Forschungsgemeinschaft and partly by DLR.
References
Anderson, J.D., Schubert, G., Jacobson, R.A., Lau, E.L., Moore, W.B., Sjogren,
W.L., 1998. Europa’s differentiated internal structure: Inferences from four
Galileo encounters. Science 281, 2019–2022.
Backus, G., Parker, R., Constable, C., 1996. Foundations of Geomagnetism.
Cambridge Univ. Press, New York.
Banks, P.M., Kockarts, G., 1973. Aeronomy, vol. A. Academic Press, San
Diego, CA.
Belton, M.J.S., and 33 colleagues, 1996. Galileo’s first images of Jupiter and
the Galilean satellites. Science 274 (5286), 377–385.
Blakely, R.J., 1995. Potential Theory in Gravity and Magnetic Applications.
Cambridge Univ. Press, New York.
Brown, M.E., 2001. Potassium in Europa’s atmosphere. Icarus 151, 190–195.
Brown, M.E., Hill, R.E., 1996. Discovery of an extended sodium atmosphere
around Europa. Nature 380, 229–231.
Carr, M.H., and 21 colleagues, 1998. Evidence for a subsurface ocean on Europa. Nature 391, 363–365.
Cassen, P., Reynolds, R.T., Peale, S.J., 1979. Is there liquid water on Europa.
Geophys. Res. Lett. 6 (9), 731–734.
Connerney, J.E.P., Acuña, M.H., Ness, N.F., 1981. Modeling the jovian current
sheet and inner magnetosphere. J. Geophys. Res. 86, 8370–8384.
Deschamps, F., Sotin, C., 2001. Thermal convection in the outer shell of large
icy satellites. J. Geophys. Res. 106 (E3), 5107–5121.
Dessler, A.J., 1983. Physics of the Jovian Magnetosphere. Cambridge Univ.
Press, New York.
Durham, W.B., Stern, L.A., 2001. Rheological properties of water iceapplications to satellites of the outer planets. Annu. Rev. Earth Planet.
Sci. 29, 295–330.
Geissler, P.E., and 14 colleagues, 1998. Evidence for non-synchronous rotation
of Europa. Nature 391, 368–370.
Greenberg, R., Geissler, P., Tufts, B., Hoppa, G., 2000. Habitability of Europa’s
crust: The role of tidal-tectonic processes. J. Geophys. Res. 105, 17551–
17562.
Hall, D.T., Strobel, D.F., Feldman, P.D., McGrath, M.A., Weaver, H.A., 1995.
Detection of an oxygen atmosphere on Jupiter’s moon Europa. Nature 373,
677–679.
Hall, D.T., Feldman, P.D., McGrath, M.A., Strobel, D.F., 1998. The farultraviolet oxygen airglow of Europa and Ganymede. Astrophys. J. 499 (5),
475–482.
Hussmann, H., Spohn, T., Wieczerkowski, K., 2002. Thermal equilibrium states
of Europa’s ice shell: Implications for internal ocean thickness and surface
heat flow. Icarus 156, 143–152.
Itikawa, Y., Ichimura, A., Onda, K., Sakimoto, K., Takayanagi, K., Hatano,
Y., Nishimura, H., Tsurubuchi, S., 1989. Cross sections for collisions of
electrons and photons with oxygen molecules. J. Phys. Chem. Ref. Data 18,
23–42.
Johnson, R.E., Leblanc, F., Yakshinskiy, B.V., Madey, T.E., 2002. Energy distributions for desorption of sodium and potassium from ice: The Na/K ratio
at Europa. Icarus 156, 136–142.
Johnson, R.E., Carlson, R.W., Cooper, J.F., Paranicas, C., Moore, M.H., Wong,
M.C., 2004. Radiation effects on the surfaces of the Galilean satellites. In:
Bagenal, F. (Ed.), Jupiter. The Planet, Satellites and Magnetosphere. Cambridge Univ. Press, Cambridge, UK, pp. 485–512.
Kabin, K., Combi, M.R., Gombosi, T.I., Nagy, A.F., DeZeeuw, D.L., Powell,
K.G., 1999. On Europa’s magnetospheric interaction: A MHD simulation
of the E4 flyby. J. Geophys. Res. 104 (A9), 19983–19992.
Kargel, J.S., Croft, S.K., Lunine, J.I., Lewis, J.S., 1991. Rheological properties of ammonia–water liquids and crystal–liquid slurries—Planetological
applications. Icarus 89, 93–112.
Khurana, K.K., Kivelson, M.G., Stevenson, D.J., Schubert, G., Russell, C.T.,
Walker, R.J., Polanskey, C., 1998. Induced magnetic fields as evidence for
subsurface oceans in Europa and Callisto. Nature 395, 777–780.
Khurana, K.K., Kivelson, M.G., Russell, C.T., 2002. Searching for liquid water
in Europa by using surface observatories. Astrobiol. J. 2, 93–103.
Kim, Y.K., and 12 colleagues, 1997. Electron-Impact Ionization Cross Section
for Ionization and Excitation Database (version 3.0) [online]. Available:
http://physics.nist.gov/ionxsec (2006, January 4).
Kivelson, M.G., Khurana, K.K., Joy, S., Russell, C.T., Southwood, D.J.,
Walker, R.J., Polanskey, C., 1997. Europa’s magnetic signature: Report
from Galileo’s first pass on 19 December 1996. Science 276, 1239–1241.
Kivelson, M.G., Khurana, K.K., Stevenson, D.J., Bennett, L., Joy, S., Russell,
C.T., Walker, R.J., Zimmer, C., Polanskey, C., 1999. Europa and Callisto:
Induced or intrinsic fields in a periodically varying plasma environment. J.
Geophys. Res. 104 (A3), 4609–4625.
Kivelson, M.G., Khurana, K.K., Russell, C.T., Volwerk, M., Walker, R.J., Zimmer, C., 2000. Galileo magnetometer measurements strengthen the case for
a subsurface ocean at Europa. Science 289, 1340–1343.
Kivelson, M.G., Bagenal, F., Kurth, W.S., Neubauer, F.M., Paranicas, C., Saur,
J., 2004. Magnetospheric interactions with satellites. In: Bagenal, F. (Ed.),
Jupiter. The Planet, Satellites and Magnetosphere. Cambridge Univ. Press,
Cambridge, UK, pp. 513–536.
Kliore, A.J., Hinson, D.P., Flasar, F.M., Nagy, A.F., Cravens, T.E., 1997. The
ionosphere of Europa from Galileo radio occultations. Science 277, 355–
358.
Lahiri, B.N., Price, A.T., 1939. Electromagnetic induction in non-uniform conductors. Philos. Trans. R. Soc. Ser. A 237 (784), 509–540.
Leblanc, F., Johnson, R.E., Brown, M.E., 2002. Europa’s sodium atmosphere:
An ocean source. Icarus 159, 132–144.
Liu, Y., Nagy, A.F., Kabin, K., Combi, M.R., Dezeeuw, D.L., Gombosi, T.I.,
Powell, K.G., 2000. Two-species, 3D, MHD simulation of Europa’s interaction with Jupiter’s magnetosphere. Geophys. Res. Lett. 27 (12), 1791–1794.
McCord, T.B., and 11 colleagues, 1998. Salts on Europa’s surface detected by
Galileo’s near infrared mapping spectrometer. Science 280, 1242–1245.
McGrath, M.A., Lellouch, E., Strobel, D.F., Feldman, P.D., Johnson, R.E.,
2004. Satellite atmospheres. In: Bagenal, F. (Ed.), Jupiter. The Planet, Satellites and Magnetosphere. Cambridge Univ. Press, Cambridge, UK, pp. 457–
483.
Neubauer, F.M., 1998a. Oceans inside Jupiter’s moons. Nature 395, 749–750.
Neubauer, F.M., 1998b. The sub-Alfvénic interaction of the Galilean satellites
with the jovian magnetosphere. J. Geophys. Res. 103 (E9), 19843–19866.
Neubauer, F.M., 1999. Alfvén wings and electromagnetic induction in the interiors: Europa and Callisto. J. Geophys. Res. 104 (A12), 28671–28684.
Pappalardo, R.T., and 31 colleagues, 1999. Does Europa have a subsurface
ocean. Evaluation of the geological evidence. J. Geophys. Res. 104 (E10),
24015–24056.
Parkinson, W., 1983. Introduction to Geomagnetism. Scottish Academic Press,
Edinburgh.
Pospieszalska, M.K., Johnson, R.E., 1989. Magnetospheric ion bombardment
profiles of satellites: Europa and Dione. Icarus 78, 1–13.
Saur, J., Strobel, D.F., Neubauer, F.M., 1998. Interaction of the jovian magnetosphere with Europa: Constraints on the neutral atmosphere. J. Geophys.
Res. 103 (E9), 19947–19962.
Schenk, P., 2002. Thickness constraints on the icy shells of the Galilean satellites from a comparison of crater shapes. Nature 417, 419–421.
Schubert, G., Spohn, T., Reynolds, R.T., 1986. Thermal histories, compositions and internal structures of the moons of the Solar System. In: Burns,
J.A., Mathews, M.S. (Eds.), Satellites. Univ. of Arizona Press, Tucson, AZ,
pp. 224–292.
Schilling, N., 2006. Time varying interaction of Europa’s atmosphere–
ionosphere and its conducting ocean with the jovian magnetosphere. Ph.D.
Dissertation, University of Cologne.
Schilling, N., Khurana, K.K., Kivelson, M.G., 2004. Limits on an intrinsic dipole moment in Europa. J. Geophys. Res. 109, doi:10.1029/2003JE002166,
E05006.
Shematovich, V.I., Johnson, R.E., Cooper, J.F., Wong, M.C., 2005. Surfacebounded atmosphere of Europa. Icarus 173, 480–498.
Sittler, E.C., Strobel, D.F., 1987. Io plasma torus electrons: Voyager 1. J. Geophys. Res. 92 (A6), 5741–5762.
Spohn, T., Schubert, G., 2003. Oceans in the icy Galilean satellites of Jupiter.
Icarus 161 (2), 456–467.
Conductivity of Europa’s subsurface ocean
Squyres, S.W., Reynolds, R.T., Cassen, P.M., Peale, S.J., 1983. Liquid water
and active resurfacing on Europa. Nature 301, 225–226.
Srivastava, S., 1966. Theory of the magnetotelluric method for a spherical conductor. Geophys. J. R. Astron. Soc. 11, 373–378.
Stacey, F., 1992. Physics of the Earth. Brookfield Press, Brisbane.
Stone, J.M., Norman, M.L., 1992a. Zeus-2D: A radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. I. The hydrodynamic algorithms and tests. Astrophys. J. Suppl. Ser. 80, 753–790.
Stone, J.M., Norman, M.L., 1992b. Zeus-2D: A radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. II. The mag-
55
netohydrodynamic algorithms and test. Astrophys. J. Suppl. Ser. 80, 791–
818.
Thomas, N., Bagenal, F., Hill, T.W., Wilson, J.K., 2004. The Io neutral clouds
and plasma torus. In: Bagenal, F. (Ed.), Jupiter. The Planet, Satellites and
Magnetosphere. Cambridge Univ. Press, Cambridge, UK, pp. 561–591.
Torr, D.G., 1985. The photochemistry of atmospheres. In: Levine, J.S. (Ed.),
Upper Atmosphere. Academic Press, San Diego, CA, pp. 165–278.
Zimmer, C., Khurana, K.K., Kivelson, M.G., 2000. Subsurface oceans on Europa and Callisto: Constraints from Galileo magnetometer observations.
Icarus 147, 329–347.