Induced magnetic fields on planetary bodies

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Universidad de los Andes

Final Degree Project

Induced magnetic fields on planetary

bodies

Author: FelipePe˜na

Supervisor:

Natalia G´omez

A thesis submitted in fulfilment of the requirements for the degree of Physicist

in the

Geophysic’s Group

Deparment of Physics

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UNIVERSIDAD DE LOS ANDES

Abstract

Faculty of Sciences

Deparment of Physics

Physicist

Induced magnetic fields on planetary bodies

by Felipe Pe˜na

Induced magnetic fields on a planetary scale have not been studied extensively but a few

bodies in the solar system may present this phenomena (Mercury, Ganymede, Callisto

and Europa). It is know that the interaction between a conducting shell and an external

magnetic field produces an induced magnetic field. Also, it has been suggested that this

phenomena only occur in two satellites of the Jovian system.

This problem is interesting because an ocean (salty-water), it’s the best candidate as

the layer that cause a magnetic field for Europa and Callisto. Recently, life’s search in

other places in the universe has become important. Relations between life and liquid

water have been proposed. If water is a universal solvent that allows the development

of life, Europa is an important satellite for astrobiology studies.

We study the magnetic response of a conducting shell with a variable external magnetic

field using numerical models featuring various electrical conductivities. Our results

show that the induced field can have a quadrupole or octupole configuration. Analitical

solutions published in the literature are in good agreement with our numerical results.

Galileo spacecraft has done about thirty flybys Europa after orbiting the Jovian system

for seven years. Data from these flybys are in good agreement with our results from

the numerical models. However, plasma effects in the jovian magnetosphere may be

important for the ambient magnetic field and having a better understanding of these

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List of Figures

1.1 Basic geometry of the internal and external radius. . . 4

2.1 Response of a conductive sphere (with and without rotation) in a constant magnetic field . . . 8

2.2 Geometry used in fluid within a sphere . . . 8

3.1 Magnetic field for differents passes around Europa . . . 15

3.2 Model of layers of Europa . . . 17

4.1 Magnetic energy stored in the water for a constant field . . . 21

4.2 Magnetic energy stored in the water . . . 21

4.3 r-component of the magnetic field on the surface of Europa Pm = 0.1−0.5 23 4.4 r-component of the magnetic field on the surface of Europa forPm= 0.6−1.0 24 4.5 θcomponent of the magnetic field on the surface of Europa forPm= 0.1−0.5 25 4.6 θcomponent of the magnetic field on the surface of Europa forPm= 0.6−1.0 26 4.7 φcomponent of the magnetic field on the surface of EuropaPm= 0.1−0.5 27 4.8 φcomponent of the magnetic field on the surface of EuropaPm= 0.6−1.0 28 4.9 Azimuthal and equatorial cut of the φ component of the magnetic field for differents magnetic Prandtl (0.1-0.5) . . . 29

4.10 Azimuthal and equatorial cut of the φ component of the magnetic field for differents magnetic Prandtl (0.6-1.0) . . . 30

4.11 Azimuthal and equatorial cut of the r component of the magnetic field forPm= 0.2. . . 31

4.12 Azimuthal and equatorial cut of the θ component of the magnetic field forPm= 0.2. . . 31

4.13 Azimuthal and equatorial cut of the r component of the magnetic field forPm= 1.0. . . 31

4.14 Azimuthal and equatorial cut of the θ component of the magnetic field forPm= 1.0. . . 32

4.15 Geometry of the magnetic field inside and outside Europa. Pm= 0.1−0.5 33 4.16 Geometry of the magnetic field inside and outside EuropaPm = 0.5−1.0 34 4.17 By(nT) component of the magnetic field forPm= 0.1−0.3 . . . 36

4.18 By(nT) component of the magnetic field forPm= 0.4−0.6 . . . 37

4.19 By(nT) component of the magnetic field forPm= 0.7−0.9 . . . 38

4.20 By(nT) component of the magnetic field forPm= 1.0 . . . 39

C.1 Azimuthal and equatorial cut of the r component of the magnetic field for differents magnetic Prandtl (0.1-0.5) . . . 49

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List of Figures v

C.2 Azimuthal and equatorial cut of the r component of the magnetic field for differents magnetic Prandtl (0.6-1.0) . . . 50 C.3 Azimuthal and equatorial cut of the θ component of the magnetic field

for differents magnetic Prandtl (0.1-0.5) . . . 51 C.4 Azimuthal and equatorial cut of the θ component of the magnetic field

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List of Tables

3.1 Thickness of each European layer. . . 16 3.2 Density, viscosity and kinematic viscosity for differents temperatures at

1 atm. From [39] . . . 18

B.1 Normalization of magnetic field and magnetic energy, and ratio model. . . 48

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Symbols

Symbol Meaning Units

α Coefficient of thermal expansion K−1

B Magnetic induction vector T

χ Radius ratio = ro

ri

D ro−ri m

E Ekman number

η Dynamic viscosity Pa s

g Gravity acceleration m s−2

go Gravity at the outer boundary m s−2

κ Thermal diffusivity m2 s−1

λ Magnetic diffusivity m2 s−1

µ Magnetic permeability H m−1

µ0 Vacuum permeability H m−1

ν Kinematic viscosity m2 s−1

P Pressure Pa

Pm Magnetic Prandtl number

Pr Prandtl number

Ra Rayleigh number

ri Radius of the internal boundary m

ro Radius of the external boundary m

ρ Mass density kg m−3

σ Electrical conductivity S m−1

T Temperature K

t Time s

u Velocity vector m s−1

Ω Angular frequency rad s−1

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Dedicated to my Parents and brothers. . .

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Chapter 1

Introduction

For centuries, planets and stars have been studied by humanity. Relations between

planets and gods have been proposed since the ancient greeks. Besides, life is being

searched in other worlds. The presence of water seems to be the key for life in our

own planet. Recently, it has been discovered that some planets and moons in the solar

system have water. Mercury, Callisto and Europa may be suitable places where to look

for life. Extreme temperature conditions may render Mercury inhabitable. It is likely

that Callisto and Europa have the right conditions to foster life.

Electrical and magnetic properties inside a planet may play an important role for

protec-tion to radiaprotec-tion due to charged-particles, and more relevant to us, for the generaprotec-tion of

a magnetic field. This project will focus on induced magnetic fields in planetary bodies,

and particularly in Europa. We identify suitable parameters for the model, obtain the

average electrical conductivity of the European ocean and analyze the magnetic field of

Europa. Given the resemblance between the Galilean moons; Callisto and Europa have

similar internal compositions, Ganymede may have an induced magnetic field produced

by plasma, and it has an internal self-sustained dynamo. This similarity allow us to

interpret that our results can be applied to the other Galilean moons introducing some

modifications. For this reason we include a brief summary of each moon in section1.1.

In section1.2, a brief description of the inductive model is presented.

1.1 Jovian System

Jupiter is by far the largest planet in the Solar System, with a radii of 11.2 Rearth. In

mythology, Jupiter was the main god of the Romans; the greeks referred to Jupiter as

Zeus. Jupiter, with Saturn, are called the gas giants. It was discovered very early in

history along with other planets.

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Chapter 1. Introduction 2

Jupiter’s size and other features like fast rotation and composition create a huge magnetic

field. Its magnetic field is tipped by 10 degrees with respect to its rotation axis. It was

not until Pioneer 10 and 11 that the existence of a magnetosphere was confirmed; these

spacecrafts found a strong non-dipolar field component [2].

Jupiter has a complex system of multiple satellites, somewhat resembling a small

plane-tary system. Galilean satellites, those discovered by Galileo, are the largest moons in the

Jovian system. They can be found from smaller to larger orbital distance: Io, Europa,

Ganymede and Callisto.

1.1.1 Io

Io is the innermost moon. In Greek religion, Zeus was in love with Io, but he turned

Io into a white heifer to disguise her from his wife. Io is one of the most remarkable

objects in the solar system. It is only 2 % larger than our moon. It is known by being an

endogenically active solid body. Io has a dry surface because of the high rate of volcanic

activity. Eruptions from Io can affect plasma environment of Europa and Ganymede.

Jupiter’s auroras are created mainly by Io’s particles erupted from volcanoes. Tidal

heating is the main source of internal heat of Io, this is produced by the orbital resonance

of Io with Europa and Ganymede.

A potential explanation for the magnetic field of Io is an instric magnetic field generated

by dynamo action [30]. The data is consistent with a superposition of an Io-centered

dipole and antiparallel to Jupiter’s dipole moment.

1.1.2 Europa

This moon is distinguished from others because of its internal ocean, maybe larger than

Earth’s ocean beneath its outer shell, a water-ice layer, Callisto has a similar structure.

Europa’s geology is dominated by smooth plains and mottled terrain.

European’s magnetic field data revealed an induced dipole moment. This is how we

know that an internal layer is an ocean made-up with salt and water. More information

about this moon will be presented in chapter3

1.1.3 Ganymede

Ganymede is the solar system’s largest satellite. In Greek mythology, Ganymede was

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Ganymede has an interesting geology, about 40% of its surface is covered by dark,

heavily cratered terrain, and the rest consists of heavily tectonized bright terrain [33].

In addition, Ganymede is covered by water ice.

Three-layer models suggest that Ganymede is differentiated into an outermost ice layer,

a silicate mantle and a central core of Fe or FeS. The magnetic field data support the

idea of an internal dynamo and a metallic core [32].

1.1.4 Callisto

This is the outermost Galilean moon, it is approximately of the same size of Mercury.

Callisto is named after a woman from a Greek myth. Callisto was Zeus (Jupiter) lover

as well as the other moons.

Callisto is a heavily cratered body with a lacking of endogenic volcanic acivity of

tec-tonic landforms. Callisto does not have an internal magnetic field. The magnetic field

data, obtained from Galileo spacecraft, fits well by the expected response of a conductor

in Jupiter’s time variable magnetic field [19]. Measurements suggest that Callisto

con-tains an internal conducting layer. Callisto’s surface is made of an icy shell with other

components (darker material) because of it is low albedo [33].

Three-layer and four-layer interior models have been proposed for Callisto, an iron core

is improbable given its mean density (ρ ∼ 1800 kg m−3) [5] [4]. Callisto may have a

central metallic core, an intermediate rock and an ice layer.

1.2 Description of the model

Equations 1.1 to1.8 describe the computational model used to find the electrical

con-ductivity of european ocean, which consists of an electrically conductive fluid, bounded

by two spherical shells of internal and external radii ri and ro, respectively. We take

the inner sphere to be electrically conductive and solid. The radius ratio is defined as

χ=ri/ro ; distance by the shell gap with, D=ro−ri (See Figure 1.1). The system

ro-tates with an angular velocity Ω, with the axis of rotation parallel to the unitary vector

ˆ

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ri

ro

D

Figure 1.1: Sketch of the geometry used on the model. Inner circle is the internal

radii, outer circle is the outer radii andD is the shell gap.

We use non-dimensional equations are derived using scaled quantities. Temperature is

scaled by the temperature difference between the inner and outer shells, ∆T; time by

the viscous diffusion time, τν = D2ν−1, where ν is the kinematic viscosity; velocity by

νD−1; pressure by ρνΩ; and magnetic induction by √ρµλΩ, where ρ is the density, µ

the magnetic permeability andλis the magnetic diffusivity of the fluid.

The non-dimensional equations describing velocity and magnetic field induction, in the

rotating frame of reference, and under the MHD and Boussinesq approximations are:

E

∂u

∂t + (u· ∇)u− ∇ 2_u

+ 2ˆz×u=−∇P +RaE

Pr g g0

ˆ

rT + 1

Pm

(∇ ×B)×B (1.1)

∇ ·u= 0 (1.2)

∇ ·B= 0 (1.3)

∂B

∂t =∇ ×(u×B) +

1

Pm

∇2B (1.4)

where u and B are the velocity and magnetic induction vectors; t is the time; T and

P are the temperature and pressure scalars, respectively; g is the radially dependent

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Equations1.1to1.2are expressed in terms of the following non-dimensional parameters:

The Rayleigh number,

Ra=

αgo∆T D3

κν (1.5)

where α is the thermal expansion coefficient and κ is the thermal diffusivity. Ra is

associated with the thermal balance within the fluid; when the Rayleigh number is below

the critical value for that fluid, heat transfer is exclusively in the form of conduction;

when it exceeds the critical value, heat transfer is in the form of convection. In our

model we useRa= 0, this means that there is no heat transfer of any kind. The Ekman

number,

E = ν

ΩD2 (1.6)

which is the ratio between viscous and Coriolis force in the system. The Prandtl number,

Pr = ν

κ (1.7)

which is the ratio between the viscous and the thermal diffusivites. And the magnetic

Prandtl number,

Pm= ν

λ (1.8)

which is the ratio between the viscous and the magnetic diffusivities.

In this project we include inductive models for differents magnetic Prandtl numbers. In

chapter2, we present analitical models of an uncharged sphere immersed in an external

field. Chapter 3 includes important information about Europa and how it will be used

in our model. In chapter4we present results of the numerical simulations. A theoretical

model is important to our simulations because we can compare both solutions and the

results should be in agreement (chapter 4). We find that, quadrupoles and octupoles

are plausible solutions with a dependency of the magnetic Prandtl number.

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————————————————————-Chapter 2

Theoretical Model

2.1 Equations

Maxwell’s equations are the basis of electromagnetic theory, and so we will use them to

find a solution to our problem: an homogeneous conducting sphere (or shell) rotates at

a constant angular velocity in a non-uniform magnetic field.

In general Maxwell’s equation are:

∇ ×E~ =−∂ ~B

∂t (2.1)

∇ ×B~ =µ0~J+

1

c2 ∂ ~E

∂t (2.2)

∇ ·B~ = 0 (2.3)

∇ ·E~ = ρ

0

(2.4)

Here E is the electric field, B the magnetic field, J is the current density, µ0 is the

permeability in free space, c is the speed of light, ρ is the charge density and is the

dielectric constant.

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Chapter II. Theoretical Model 7

With the assumption of non-relativistic speeds: i.e. v2 c2, this can be justified because

the fluid or sphere velocity is small compared to the speed of light1. This allow us to

discard the term _c12∂_∂tE. ([17])

We will present an overview of different solutions under certain approximations. In

section 2.2 we present possible solutions for 3 cases with a constant external magnetic

field. In section 2.3the external magnetic field has an arbitrary frequency.

2.2 Constant External Magnetic Field

The problem of moving conductors have been studied by several authors, specially

Lor-rain. Results from [24] and [28] will be shown in this section.

In these papers, they assume steady state conditions. With respect to a stationary

reference frame S, the field variables E and B, the electric current density J, and the elecstrostatic volume and surface charge densities, are all independent of time. Bind is

the induced magnetic field.

∇ ×~E= 0 (2.5)

∇ ×B~ =µ0~J (2.6)

∇ ·B~ = 0 (2.7)

∇ ·E~ = ρ

0

(2.8)

The first case is: Rigid object, no rotation, J= 0 , Bind = 0.

The second case is: Rotating sphere,J= 0 , Bind = 0

The only problem in the second case is how can there be no induced current when there

is av×B field ? The answer is that the electric field cancels v×B at every point. In both cases the magnetic field is unaffected by the rotating conducting sphere.

1

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Figure 2.1: Representation of a conductive sphere inmersed in a constant magnetic

field (arrows represent the magnetic field)

The third case is: Fluid within a sphere,J6= 0 ,v×Bind = 0 , Bind 6= 0

The conducting fluid inside a sphere of radius R has an azimuthal velocity v that is a function of ρ andz:

v=ωρ1−K(z2/R2)φˆ (2.9)

r

Ρ

Θ

R

B Ω

z

Figure 2.2: Geometry used in 3rd case (Fluid within a sphere). Figure adapted from [24]

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If K is positive, the equator rotates faster than the poles. The sphere rotates in a

uniform axial magnetic fieldBextzˆ

With ω, Bext, K positive,Bind points in the −φˆ direction in the northern hemisphere,

and in the + ˆφdirection in the southern hemisphere: magnetic field lines are distorted

downstream, but they are stationary. The sign of the direction changes ifK is negative,

and the magnetic field lines are distorted upstream.

2.3 Induced field on a conducting sphere under an

inho-mogeneous ambient magnetic field

In the final solution, the main equation is 2.1, and also we have that (based on [34]):

E=−dA

dt (2.10)

Combining the equations we obtain:

µ τ

dA

dt =∇

2_A _(2.11)

Equation 2.11has the form of the well-known equation of the diffusion of heat, but the

dependent variable is a vector instead of a scalar. Now, our problem has spherical

sym-metry, let us assume that the magnetic field producing the eddy currents is independent

of φ and has no φ-component. Then the vector potential has only a φ-component and

may be written2

A=Aφ(r, θ, t) ˆφ (2.12)

The solution of the vector potential is:

Ao =

1

2B(r+D 1

r2) sinθφ,ˆ fora≤r≤ ∞ (2.13)

Ai =

1 2

BC

r1/2I3/2[(ip)

1/2_r_{] sin(}_θ_{) ˆ}_φ, _{for 0}_≤_r _≤_a _(2.14)

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where a is the radius of the sphere,C and D are constants, B is the amplitude of the

external magnetic field and I_3/2 comes from the modified Bessel equation term. With

adequate boundary conditions when r =a, constant C and D can be found. Now the

magnetic field at any point outside is:

Boθ =−

1

r ∂

∂r(rAo)ˆθ=−B

1− D

2r3

sinθθˆ (2.15)

Bor=

1

rsinθ ∂

∂θ(sinθAo)ˆr=B

1 + D

r3

ˆ

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Chapter 3

The Second Galilean moon:

Europa

3.1 Introduction

Several terms are unknowns variables in equations 1.1, 1.2,1.3, 1.4. This chapter will

focus on various methods for finding some of the values that are needed to solve the

equations developed in Chapter1.

3.2 Interior stratification

3.2.1 Core

Europa’s radius metallic core has been estimated to be 460-840 km. Different approaches

were used by different authors to estimate the size of this layer. Estimates for the core

size can be attributed to Anderson and his group [3]; they used data from Galileo’s

spacecraft radio signal during two Europa encounters (E4 and E6). These passes allowed

a measurement of the Europa’s gravitational field. From the gravity coefficients it is

possible to find some properties of a planetary body; the gravitational potential can be

expressed in the form of:

V(r, φ, λ) = GM

r "

1 + ∞

X

n=2 n X

m=0

(R/r)n(Cnmcosmλ+Snmsinmλ)Pnm(sinφ) #

(3.1)

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Chapter III Europa 12

The coefficients C21 and S21 can be interpreted as corrections to the orientation of the

polar axis for example the coefficientS22 measures a rotation of the x and y axes about

the polar axis z.

The coefficientC22can be used to infer the moon’s internal structure. C22can be related

with a parameter α (C22 = 3αq₄r ), where qr is the ratio of the centrifugal force to the

gravitational force at Europa’s equator andα is a dimensionless quantity that depends

on the distribution of density with depth inside the satellite. With α, Anderson and

his group can find depths for each layer; solving Clairaut’s equation for the adequate

α is possible to build different models: the best is a three-layer model of Europa, and

assuming that the core has different densities for a Fe-core and Fe-FeS core, they found

that a Fe core would have a radius of 0.3−0.4RE, whereas that a Fe-FeS core would

have a radius of 0.4−0.6RE.

Anderson and his group, with data from two subsequent encounters (E11 and E12),

refined models of Europa’s interior with the same approach of gravity coefficients [6].

With the new data, they deduced that the core could be as large as about 0.5RE.

Furthermore, they find a solution of 2 layer model(without a metallic core). This solution

is possible but improbable because it is inferred from the magnetic signature that Europa

has a core [20] [21].

Kuskov and Kronrod [22] found that for a Fe core the radius of this could be 420-510 km

and for a Fe-FeS core 610-710 km. They used numerical models to introduce physical

constraints from geophysical (the mass and moment of inertia from Galileo gravity

mea-surements), geochemical (the chemical composition of meteorites) and thermodynamic

(modeling of phase relations and physical properties in theN a2O−T iO2−CaO−F eO−

M gO−Al2O3−SiO2−F e−F eS system). Kuskov and Kunrod included phase diagram

of H2O and equations of state of high-pressure ices and meteoritic matter [23]. With

these refinements they found a more realistic estimated of the core size which is: 470-640

km.

3.2.2 Mantle

Depth of the mantle is unknown, even its composition is uncertain. A silicate mantle is a

probable composition with perhaps hydrated silicates [6]. If we assume an intermediate

value between 0.4−0.5RE for the core and 200 km for the water-ice layer [3] [6], the

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3.2.3 Water Shell

The depth of this layer is 80-300 km, depending on the size of the core, density of

innermost layers (mantle and core) and composition of the core (Fe or Fe-S) [3], [6].

Kuskov and Kronrod found values between the range mentioned; 120-140 km for their

first calculation [22], 115± 10 km for a differentiated mantle and 135 ±10 km for an

undifferentiated mantle [23] (See section3.2.1, for an explanation of the methods). It is

important to bear in mind that the gravity experiment described in section3.2.1can not

distinguish between liquid water and ice because their densities are similar [6]. Other

approach using planetary models [29], gives a width of 150 km for this layer.

3.2.4 Thick ice shell

This is the outer layer of Europa, and the most studied because more data is available.

Thickness of the ice shell can be determined with a comparison between flexural models

and observations, simulations of impact craters or thermodynamic analyses (See table 1

in [8] for a complete reference of all method developed). From impact crater constraints

the estimate is≥19 km [31]; convective tidal dissipation models find thicknesses between

25 and 50 km [15]; flexural models applied to craters 15+20₋₉ km [10] and from impact

crater simulations≥3−4 km [38].

Reflectance spectra studies justify the presence of others components like magnesium

sulfates and sodium carbonates or mixtures of these minerals on the surface [26] [25].

Additionally, alternative brines, such as MgSO4 and Na2SO4, exposed to Europa’s

sur-face conditions exhibit spectral features similar to those measured for Europa’s sursur-face

[27] [37]. Theoretical models based on the composition, relative abundances and

stabil-ity of hydrated salts predicted the predominance of Mg and Na sulfates in the european

surface [41].

3.3 Electrical conductivity of water

Pure water by itself is an insultadorσ = 4×10−6 S/m. Good conductors, such as iron,

have several orders of magnitude larger valuesσF e= 1×107S/m. At room temperature

and pressure the conductivity of saturated salt-water may beσSW = 22.7 S/m [13].

The magnetic signature attained by Galileo spacecraft shows an induced field caused

by a layer with a significant electrical conductivity. These conducting layers may be

explained, for example, by salty liquid-water oceans [19]. Conductivity of salt-water at

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Ocean’s composition have been estimated with different techniques. Results from

ex-periments with a meteorite sample (CM) expound that water with cations: Na∼Mg>

Ca, K > Fe and anions: SO4 Cl are more feasible as components of the ocean [11].

The origin of this salty ocean is unknown, but some authors justify that the non-ice

component of the ice shell must be endogenous [41], with this assumption, all elements

or brines from section 3.2.4should be the salt that we are looking for.

3.4 Magnetic field

Through out each flyby of Europa by Galileo spacecraft new information was revealed.

The first pass was on 19 December 1996, the magnetic signal of this pass was modeled

as an internal field, but could not confirm the existence of a dynamo [20]. Amplitude’s

magnetic field ranges from 240 nT to 120 nT if Europa has a centered internal dipole.

But other processes can produce the same perturbation to the magnetic field observed

by the Galileo, for example, plasma currents can generate similar magnetic signatures.

Perturbations caused by plasma could become as large as∼100 nT [20].

With more flyby of Galileo, new data was acquired and there is evidence now that the

magnetic field was a field induced by a deeper electrically conductive layer. [19]. Data

obtained on 3 January 2000 allowed to distinguish between and induced and an internal

magnetic dipole moment; because and induced equatorial dipole moment changes

ori-entation and amplitude in a predictable way, with a southern hemisphere pass, it was

possible to evidence this difference [21] (See Figure3.1).

The external magnetic field near Europa has a peak amplitude of ∼ 220 nT and a

frequency of 11.2 hours. If the conductivity of the ocean is 1 S m−1 _{the skin depth is}

S ≈95 km [19] this means that the magnetic field can penetrate a maximum distance

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Figure 3.1: The heavy line represents a model of a highly conducting sphere. The

narrow line is a least squares fit. The measure of E26 would be in the triangle if the magnetic field of Europa was internal. Taken from [21].

The coordinate system in the figure 3.1 correspond to: ˆx pointing to the corotating

plasma flow, ˆy radially toward Jupiter and ˆz is parallel to the spin axis. The My

component is plotted because is the ‘least modified’ by plasma effect for the low latitude

passes (E4 and E14). Although this is an assumption, a correction of ∼100 nT can be

used for all latitudes [20]. The pass E26 was in the south hemisphere, E4 and E14 were

near the equatorial plane, and E19 happened in the north hemisphere.

Plasma and other effects can increase the uncertainty in figure3.1, but an induced dipole

preserve the diagonal of figure3.1. This means that all points must fall in right-up and

left-down sector. Flyby E19 looks like an exception to this simple test, but the measured

My is negative and modeled is positive, in a plot of measured vs measured, E19 would

be in the left-down quadrant. The E26 is an exceptional data that can differentiate

between an internal dipole or an induced one. If the E26 was internal the data may be

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3.5 Real vs Model Non-dimensional parameters

In the previous sections (3.2,3.4,3.3), we reported values for layers thicknesses, among

other properties and the techniques used to find them. Now, it is possible to find values

for radius ratio and shell gap (section 1.2).

It is worth noticing the layer’s values; we have an extensive range for the thickness of each

one (core, mantle, water and ice). Accordingly to section 3.2.1, a value of 0.5RE ∼780

km or less is acceptable1, mantle has a thickness of 500-700 km (section 3.2.2), water depth is 125 km (section3.2.3 and ice thickness is larger than 4 km (section 3.2.4). In

the table3.1the values that we choose for each layer are shown

Shell Thickness (km)

Core 700

Mantle 600

Water 125

Ice 120

rE 1545

Table 3.1: Thickness of each European layer.

Others numbers can be adequate due to the uncertainty of the method of each layer, but

these are in good agreement with section 3.2. Ice layer is larger in our case because we

need a considerable size to produce eddy currents on the outer shell. A representation

of Europa with the values from table 3.1is in figure3.2

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Core

Mantle

H20

Ice

ri

ro

Figure 3.2: This figure represents a scale model of layers of Europa from table3.1

The inner boundary layer is defined as:

ri =rcore+Dmantle= 1300 km (3.2)

whereriis the inner radii,rcoreis the radius of the metallic core,Dmantleis the thickness

of the mantle. Also:

ro =ri+DH2O= 1425 km (3.3)

wherero is the outer radii andDH2O is the thickness of water; please noticeDH2O =D

from the figure1.1;

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Now, we will find values for equations1.5to 1.8. Rayleigh number will be examined in

section 3.6. All numbers need kinematic viscosity of water, this parameter depends on

temperature and pressure (see table 3.2).

T, ◦C ρ, kg/m3 µ, N·s/m2 ν, m2/s

0 1000 1.788 E-3 1.788 E-6

10 1000 1.307 E-3 1.307 E-6

20 998 1.003 E-3 1.005 E-6

30 996 0.799 E-3 0.802 E-6

40 992 0.657 E-3 0.662 E-6

50 988 0.548 E-3 0.555 E-6

60 983 0.467 E-3 0.475 E-6

70 978 0.405 E-3 0.414 E-6

80 972 0.355 E-3 0.365 E-6

90 965 0.316 E-3 0.327 E-6

100 958 0.283 E-3 0.295 E-6

Table 3.2: Density, viscosity and kinematic viscosity for differents temperatures at 1

atm. From [39]

.

The rotation rate of Europa is ∼2×10−5 s−1[1]. Given this, the Ekman number is:

E = ν

ΩD2 = 5.7×10

−12_∼₆_×₁₀−12 _(3.5)

The thermal diffusivity of water at 0◦C is 1.35 ×10−7 m2/s [16]. The Prandtl number

is therefore computed as:

Pr= ν

κ = 13.24∼13 (3.6)

In addition, we need the magnetic diffusivity of salty water, defined as λ = _µ1

0σ, in

section3.3, we know the value ofσ ∼10 S/m, the conductivity for the Earth’s ocean is

2.75 S/m, the magnetic Prandtl number will be:

Pm= ν

λ = 6.17×10

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These values, probably are not adequate because some of the constants, such as

ther-mal diffusivity, kinematic viscosity and density, are uncertain for conditions of Europa

(temperature ∼270 K and pressure ∼2×108 Pa [36]).

Unfortunately, even if these values are adequate, the resolution of the simulations is not

high enough to resolve a model with those parameters. Instead we use adimensional

numbers that may be appropiate for Europa:

E = 1.0×10−4 (3.8)

Pr = 1 (3.9)

Pm = 0.1−1.0 (3.10)

3.6 Rayleigh number

The Rayleigh number is defined in equation 1.5. Thermal convection in Europa is still

under debate. Oceanographic measurement are not available. Numerical models have

been studying a convective ocean [9], finding values for Rayleigh as a function of ocean

thickness: Ra = 106−1010. A convective ocean may explain the chaotic terrain on the

surface of Europa, hiper-convection is required in this caseRa= 1020−1023[18]. Similar

values Ra∼1020 are found with temperature models of the subsurface ocean [14].

We choose a Rayleigh number∼0 given the uncertainty of the value. This will simplify

our model and it means that the ocean is adiabatic (dQ = 0). This will simplify our

model and reduce simulation’s time.

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Chapter 4

Results and Discussion

4.1 Introduction

In this chapter, results of the different simulations with the parameters estimated in

sections3.5and3.6are presented. Magnetic energy of the shell in section4.2, magnetic

field on the surface in4.3and inside Europa in4.4, lines of the magnetic field outside and

inside Europa are showed and analyzed 4.5according with models explained in chapter

2. We used a modified version of MagIC by [40], this numerical code implements a

pseudo-spectral algorithm which solves the equation presented in section1.2

4.2 Magnetic Energy

The magnetic energy stored in the shell can be written as:

E = 1 2µ0

Z

B2dV (4.1)

with the respective adimensionalization (see appendix B). In figures 4.1 and 4.2, the

magnetic energy is represented as a function of diffusion time

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Chapter IVResults and Discussion 21

Figure 4.1: Magnetic energy stored in the water for different values of Magnetic Prandtl as a function of viscous diffusion time for a constant magnetic field

Figure 4.2: Magnetic energy stored in the water for different values of Magnetic

Prandtl as a function of viscous diffusion time for a time dependent magnetic field

The energy stored saturate for different values depending on the conductivity of the

liquid, if the conductivity decreases the energy stored increases. This is true for both

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The energy stored in both models differ because with a constant magnetic field the

energy stored is only caused by the magnetic lines enclosed by the shell, in contrast in

the time dependent magnetic field an induced field is created an the energy stored must

increase.

4.3 Magnetic field on the surface of Europa

We use an external magnetic field with a frequency of 11 days (section3.4). The solution

shown in section 2.3 does not represent the problem due to the high frequency of the

external field, this means that the second term in equation B.4 is zero and we have an

approximately constant field.

The solution that better represents our problem is the one assuming a fluid within a

sphere in a constant magnetic field (section2.2), from it we expected only aφcomponent

for the induced field and no r and θ component. Figures 4.3, 4.4, 4.5,4.6, 4.7 and 4.8

reveal that the theoretical model and our models are in a good agreement. Notice the

colour-scale forBr and Bθ: they are very small compared to Bφ.

Particularly, for aPm = 1.0 the magnetic field for the θand r are small, of the order of

0.15 nT, but the representation of the magnetic field on surface are different (see figures

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(a) Pm= 0.1 (b)Pm= 0.2

(c)Pm= 0.3 (d)Pm= 0.4

(e)Pm= 0.5

Figure 4.3: Radial component of the magnetic field on the surface of EuropaPm=

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(a) Pm= 0.6 (b)Pm= 0.7

(c)Pm= 0.8 (d)Pm= 0.9

(e)Pm= 1.0

Figure 4.4: Radial component of the magnetic field on the surface of Europa for

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(a) Pm= 0.1 (b)Pm= 0.2

(c)Pm= 0.3 (d)Pm= 0.4

(e)Pm= 0.5

Figure 4.5: Polar component of the magnetic field on the surface of Europa for

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(a) Pm= 0.6 (b)Pm= 0.7

(c)Pm= 0.8 (d)Pm= 0.9

(e)Pm= 1.0

Figure 4.6: Polar component of the magnetic field on the surface of Europa for

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(a) Pm= 0.1

(b)Pm= 0.2 (c)Pm= 0.3

(d)Pm= 0.4 (e)Pm= 0.5

Figure 4.7: Azimuthal component of the magnetic field on the surface of Europa

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(a) Pm= 0.6 (b)Pm= 0.7

(c)Pm= 0.8 (d) Pm= 0.9

(e)Pm= 1.0

Figure 4.8: Azimuthal component of the magnetic field on the surface of Europa

Pm= 0.6−1.0

At the poles, the induced field on Europa should cancel the external field [19]. Therefore

the magnetic field is zero in these region. Figures 4.3, 4.4, 4.5, 4.6, 4.7 and 4.8 show

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4.4 Magnetic field inside Europa

As in section 4.3, inside Europa we only predicted an azimuthal component of the

magnetic field. Figures 4.9 and 4.10 reveal that at the center of Europa the magnetic

field decreases its magnitude until it vanishes. Skin depth for the external field is∼100

km (see sectionB.1and [19]) induction of magnetic field on the dip core is not expected.

This is in agreement with our models, see figures4.9and 4.10.

(a)Pm= 0.1 (b)Pm= 0.2

(c)Pm= 0.3 (d)Pm= 0.4

(e)Pm= 0.5

Figure 4.9: Azimuthal and equatorial cut of the φcomponent of the magnetic field for different magnetic Prandtl numbers (0.1-0.5)

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(a)Pm= 0.6 (b)Pm= 0.7

(c)Pm= 0.8 (d)Pm= 0.9

(e)Pm= 1.0

Figure 4.10: Azimuthal and equatorial cut of theφcomponent of the magnetic field

for different magnetic Prandtl numbers (0.6-1.0)

Here we show one of the plots for radial and polar components (figures4.11and4.12), but

for the others magnetic Prandtl the plot is almost the same, changing only in magnitude

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Figure 4.11: Azimuthal and equatorial cut of thercomponent of the magnetic field forPm= 0.2

Figure 4.12: Azimuthal and equatorial cut of theθ component of the magnetic field forPm= 0.2

When Pm = 1.0, the internal magnetic field changes (figure 4.13 and 4.14). The same

may be observed in figures4.6(e) and4.4(e) in section4.3.

Figure 4.13: Azimuthal and equatorial cut of thercomponent of the magnetic field forPm= 1.0

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Figure 4.14: Azimuthal and equatorial cut of theθ component of the magnetic field forPm= 1.0

4.5 Geometry of the magnetic field

According to section 3.4an induced dipole can fit the data that the Galileo spacecraft

measured. Figures4.15 and 4.16revealed that higher order terms like quadrupoles and

octupoles are solutions to our models. Pm = 0.1 (figure 4.15(a)) and Pm = 0.5−1.0

(figures4.15(e)-4.16(e)) show a quadrupole structure andPm = 0.2−0.4 (figures

4.15(b)-4.15(d)) show an octupole structure.

These configurations may be applicable to Callisto, if we assume that layer thickness

and compositions are similar, but lower electrical conductivity of the fluid is required. It

is known that the ice giants (Uranus and Neptune) have strong non-dipolar components,

it may be possible that Uranus and Neptune have a conductive outer layer as Europa,

interacting with an internal field ([12] [35] )

Our results show that magnetic field lines can reach the mantle and even the surface of

the core, Figure4.16(e). All figures in this section were computed when the system had

reach a steady state. Transitions between quadrupoles and octupoles were observed,

before the steady state was reached, but further studies are needed to conclude that this

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(a) Pm= 0.1 (b)Pm= 0.2

(c)Pm= 0.3 (d)Pm= 0.4

(e)Pm= 0.5

Figure 4.15: This figures is the representation of the magnetic field obtained by

our models. Black semi-circle is the core-mantle boundary, dark blue semi-circle is mantle-water boundary and outer semi-circle is water-ice boundary. Blue and red lines surrounding the semi-circles is the representation of the magnetic field streamlines.

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(a) Pm= 0.6 (b)Pm= 0.7

(c)Pm= 0.8 (d)Pm= 0.9

(e)Pm= 1.0

Figure 4.16: This figures is the representation of the magnetic field obtained by

our models. Black semi-circle is the core-mantle boundary, dark blue semi-circle is mantle-water boundary and outer semi-circle is water-ice boundary. Blue and red lines

surrounding the semi-circles is the representation of magnetic field streamlines.

4.6 A comparison with Galileo data

In this section we present comparisons between Galileo data (figure 3.1) and our

sim-ulations. We plot values of the same component of the magnetic field (By) as in [21].

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We do not plot our values in the figure 3.1 because the order of magnitude from our

results are lower. Despite the fact that the magnitude is not the same, we can appreciate

a similar behavior in our results and spacecraft measurements, figure3.1[21]. All figures

(4.17,4.18,4.19 and 4.20) are well fitted according with section 3.4because all data are

in the up-right or down-left quadrants. If Europa had an internal field all measurements

would be in a single quadrant; for example in an internal dipole regardless in the

hemi-sphere in which the measurement is made the sign for the magnetic still the same. This

do not happen for an induced field.

Low magnitude obtained for the induced fields is a result of neglecting the plasma

contribution in our models of the external magnetic field . Plasma provides a higher

magnitude of the magnetic field with a maximum of∼100 nT. With these results we can

support that all configurations of quadrupoles and octupoles showed in section 4.5can

represent the European magnetic field. The ratio model is a number that quantify how

the numerical model approximate to the real problem (it is better if this is 1) (SeeB).

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−6 −5 −4 −3 −2 −1 0 1

x 10−12 −6 −5 −4 −3 −2 −1 0 1x 10

−12 By (nT) By (nT) E26 E14 E12 E19 E4

(a) Pm= 0.1

−5 −4 −3 −2 −1 0 1

x 10−12

−5 −4 −3 −2 −1 0 1x 10

−12 By (nT) By (nT) E26 E14 E12 E19 E4

(b)Pm= 0.2

−2 −1 0 1 2 3 4 5 6 7 8 x 10−12 −1 0 1 2 3 4 5 6 7x 10

−12 By (nT) By (nT) E26 E14 E12 E19 E4

(c)Pm= 0.3

Figure 4.17: By(nT) component of the magnetic field forPm= 0.1−0.3 (Coordinate

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−5 0 5 10 15

x 10−14

−2 0 2 4 6 8 10 12 14x 10

−14 By (nT) By (nT) E26 E14 E12 E19 E4

(a) Pm= 0.4

−5 0 5 10 15

x 10−14 −4 −2 0 2 4 6 8 10 12 14 16x 10

−14 By (nT) By (nT) E26 E14 E12 E19 E4

(b)Pm= 0.5

−5 0 5 10 15

x 10−14

−4 −2 0 2 4 6 8 10 12 14 16x 10

−14 By (nT) By (nT) E26 E14 E12 E19 E4

(c)Pm= 0.6

Figure 4.18: By(nT) component of the magnetic field forPm= 0.4−0.6 (Coordinate

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−10 −8 −6 −4 −2 0 2 4

x 10−13 −10 −8 −6 −4 −2 0 2 4x 10

−13 E26 E14 E12 E19 E4

(a) Pm= 0.7

0 1 2 3 4 5 6

x 10−13 0 1 2 3 4 5 6x 10

−13 By (nT) By (nT) E26 E14 E12 E19 E4

(b)Pm= 0.8

−10 −8 −6 −4 −2 0 2 4

x 10−13 −10 −8 −6 −4 −2 0 2 4x 10

−13 By (nT) By (nT) E26 E14 E12 E19 E4

(c)Pm= 0.9

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−20 −15 −10 −5 0 5

−14 −12 −10 −8 −6 −4 −2 0 2 4 E26 E14 E12 E19 E4

(a) Pm= 1.0

Figure 4.20: By(nT) component of the magnetic field forPm= 1.0

4.7 Conclusions

Models for an induced magnetic field in planetary bodies were made extensively. We

choose appropriate parameters to simulate a moon with a conductive external shell

immersed in a magnetic field with a frequency of 11 hours. Different electrical

con-ductivities (Pm = 0.1−1.0) were used to test how the geometry of the magnetic lines

change.

This models can be applied to other systems like Callisto or Ganymede, with some

modifications: different compositions or an internal magnetic field. Ice giants have

strong non-dipolar component like the results presented in section4.5, this may indicate

that a conductive outer layer is present in Uranus and Neptune, but the internal dynamic

of the ice giants is different; because they have an internal self sustained dynamo.

Theoretical models explained in section 2.2 support figures from sections 4.3 and 4.4.

Bφ is dominant, the other components of the magnetic field are negligible for all cases

of Pm inside and on the surface of Europa.

Magnetic induced quadrupole or octupole are posible configurations for the magnetic

field. An induced dipole is in agreement with measurements of Galileo spacecraft but

results in sections4.2and4.5showed that quadrupoles or octupoles can match the same

measurements polarity. Plasma effects may modify the ambient field significantly and

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Further work is necessary to reproduce satisfactory Galileo’s data measurements,

in-cluding an ocean with convection and the effects on the external field due to plasma

environment. These refinements may lead to a better understanding of the interior

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Appendix A

Complementary calculations of

induced magnetic field

(Based on[34]) Faraday’s law of induction states that, if the magnetic induction Bin a conductor, is changing, an electric fieldE is produced which is given in magnitude and direction by :

∇ ×E=−dB

dt (A.1)

In terms of the magnetic vector potentialA:

E=−dA

dt (A.2)

Since this electric field is produced in a conductor, a current will flow according to Ohm’s

law. Ifτ is the resistivity andi is the current density, we get:

i= ∇E

τ (A.3)

So :

τ(∇ ×i) =−dB

dt (A.4)

and

τi=−dA

dt (A.5)

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Appendix A.Complementary calculations of induced magnetic field 42

These currents, flowing in the conductor of permeabilityµ, will produce magnetic fields

which, are given by :

∇ ×B=µi (A.6)

∇2A=−µi (A.7)

The equations that must be satisfied by i ,B, andA in a conductor when a changing field is present derive from the 4 equations before. With some algebra:

d

dt(∇ ×B) = d dt(µi)

∇ ×dB

dt =µ di

dt

∇ ×[−τ(∇ ×i)] =µdi

dt → ∇ ×(∇ ×i) =−

µ τ

di

dt

∇2i− ∇(∇ ·i) =∇2i

Finally :

∇2i= µ

τ di

dt (A.8)

In a similar way, it’s possible to get:

∇2B= µ

τ dB

dt (A.9)

∇2A= µ

τ dA

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Equations (A.8),(A.9) and (A.10), all have the form of the well-know equation of the

conduction of heat, but the dependent variable is a vector instead of a scalar. Now, our

problem has spherical symmetry, let us assume that the magnetic field producing the

eddy current is independent of φ and has no φ-component. Then the vector potential

has only a φ-component and may be written:

A=Aφ(r, θ, t) ˆφ (A.11)

where ˆφ is a unit vector in theφ-direction, as usual, given by:

φ=−sinφˆi+ cosφˆj (A.12)

Now with some algebra :

µ τ

∂Aφ ∂t =∇

2_A₌_∇2_A

φφˆ+Aφ∇2φˆ (A.13)

Writing out∇2 _{in polar coordinates:}

µ τ ∂Aφ ∂t = 1 r2 ∂ ∂r

r2∂Aφ ∂r

+ 1

r2_sin_θ ∂ ∂θ

sinθ∂Aφ ∂θ

(A.14)

Making the substitutionu= cosθ, we get after some cumbersome algebra:

µ τ ∂Aφ ∂t = 1 r2 ∂ ∂r

r2∂Aφ ∂r

+(1−u

2₎1/2

r2

∂2(1−u2)1/2Aφ

∂u2 (A.15)

Considering the steady-state eddy currents when the magnetic field oscillates with an

angular frequency ω. We shall seek a solution that is the product of a function of θ by

a function of r. Then we write:

Aφ= Θr−1/2Reiωt (A.16)

Substituing in A.15, multiplying through by r2, and dividing through by Θr−1/2Reiωt

give:

r2 R

d2R dr2 +

rdR

Rdr −1/4−ipr

2₊(1−u2)1/2

Θ

d2 du2

h

(53)

wherep=τ−1µω. Setting the termis in A.17 involvingθ equal to−n(n+ 1) and those

involving r equal ton(n+ 1), expanding the derivatives,

(1−u2)d

2_Θ n du2 −2u

dΘn du −

Θn

1−u2 +n(n+ 1)Θn= 0 (A.18)

d2Rn dr2 +

1

r dRn

dr −

ip+n(n+ 1) + 1/4

r2

Rn= 0 (A.19)

These equations look familiar (like Legendre and Bessel) but maybe we need some

mod-ifications. From [7], the associated Legendre Functions:

(1−x2) d

2

dx2P m

n (x)−2x d dxP

m n (x) +

n(n+ 1)− m

1−x2

P_nm(x) = 0 (A.20)

Comparing A.18 and A.20, look the same but with m = 1 forA.20. Now the Modified

Bessel Functions:

d2_R dv2 +

1

v dR

dv −

1 +n

2

v2

R= 0 (A.21)

Rearranging equation and fromA.16

Aφ=r−1/2

AnPn1(u) +BnQ1n

CnIn+1/2[(ip)1/2r] +DnKn+1/2[(ip)1/2r]

eiωt (A.22)

Now, in a region where the conductivity is zero (for example, outside of the sphere), the

left side of equation A.15 is zero and if we let Aφeiωt = RΘeiωt. We obtain A.18 but

instead ofA.19 we get:

d dr

r2dR

dr

−n(n+ 1)R= 0 (A.23)

whose solution is:

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Consider now the specific example of a sphere of resistivityτ, permeabilityµ, and radius

a placed in a uniform alternatingz-directed magnetic fieldBeiωt. The vector potential

of this field is, when eiωt is divided out:

A= 1

2Brsinθ ˆ

φ= 1 2BrP

1

1(cosθ) ˆφ (A.25)

Thus n=1 inA.24 and in A.22. Outside the sphere:

Ao =

1

2B(r+Dr

−2_{) sin(}_θ_{) ˆ}_φ, _for _a_≤_r_{≤ ∞} _(A.26)

Atr= 0,Ai is finite so only I3/2[(ip)1/2r] can occur inside the sphere.Thus setting n =

1 , we have

Ai=

1 2BCr

−1/2_I

3/2[(ip)1/2r] sin(θ) ˆφ, for 0≤r≤a (A.27)

The boundary conditions whenr =aare :

Ao=Ai

and

µv ∂

∂r(rsinθAi) =µ ∂

∂r(rsinθAo)

Puttingr =a, and writingIn forIn[(ip)1/2a] and v for (jp)1/2a

a3+D= a3/2CI_3/2

(2a3−D)µ= µva3/2[1₂I3/2+vI3/2]

Solving for C and D gives :

C= 3µva

3/2

(µ−µ)v)vI−1/2+ [µv(1 +v2)−µ]I1/2

(A.28)

D= (2µ+µv)vI−1/2−[µv(1 +v

2_{) + 2}_µ_]_I 1/2

(µ−µv)vI−1/2+ [µv(1 +v2)−µ]I1/2

(55)

With these, the magnetic field at any point outside is:

Boθ =−

1

r ∂

∂r(rAo)ˆθ=−B

1− D

2r3

sinθθˆ (A.30)

Bor=

1

rsinθ ∂

∂θ(sinθAo)ˆr=B

1 + D

r3

ˆ

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Appendix B

Normalization

In this appendix there are the full calculations of some of the normalization values of

magnetic field, magnetic energy, kinetic energy, and others.

The first value that is explained is magnetic field, normalization is given by:

B=pρµλΩBˆ (B.1) In the section 3.5are the values of the non-dimensional number that we choose. With

the numbers we can find the conversion value to give the real magnetic field. Remember

that: Pm = ν_λ = 0.1−1.0. Assumingν= 1.788×10−6m2/s. It’s possible to findλ:

λ= ν

0.1−1.0 = 1.788×(10

−5₋₁₀−6₎ _(B.2)

Please notice that ’-’ indicate the range of the value. With a density ofρ= 1000kg/m3,

a frequency ∼11h and a magnetic constant ofµ= 4π×10−7N/A2.

Now, the normalization of the magnetic energy is given by:

EB= B2V

2µo

ˆ

E (B.3)

where V is the volume of the shell∼2.9×1018m3.

The real external magnetic field has a a magnitude of ∼220 nT. The magnetic field in

MagIC is modeled with the equation:

B(t) =A+Ccos(ωt) (B.4)

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Appendix BNormalization 48

We can get defined a quantity ratio model (A/A’), which is defined as the ratio between

the normalization and the real value, if this quantity approach to 1, the model reflects

a better approximation to the real problem.

Pm B (nT) EB(J)×1029 _AA0

0.1 750 6.52 0.29

0.2 530 3.26 0.42

0.3 433 2.17 0.51

0.4 375 1.63 0.59

0.5 335 1.30 0.66

0.6 306 1.09 0.72

0.7 283 0.93 0.78

0.8 265 0.81 0.83

0.9 250 0.72 0.88

1.0 237 0.65 0.93

Table B.1: Normalization of magnetic field and magnetic energy, and ratio model.

B.1 Skin depth

Skin depth is defined as the penetration of a periodically varying magnetic field acting

on an electrically conducting object:

S = (ωµσ 2 )

−1

2 (B.5)

(58)

Appendix C

Additional results

Complete results of section 4.4are showed in this appendix.

(a) Pm= 0.1 (b)Pm= 0.2

(c)Pm= 0.3 (d)Pm= 0.4

(e)Pm= 0.5

Figure C.1: Azimuthal and equatorial cut of the r component of the magnetic field

for differents magnetic Prandtl (0.1-0.5)

(59)

Appendix C. Additional results 50

(a) Pm= 0.6 (b)Pm= 0.7

(c)Pm= 0.8 (d)Pm= 0.9

(e)Pm= 1.0

Figure C.2: Azimuthal and equatorial cut of the r component of the magnetic field

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(a) Pm= 0.1 (b)Pm= 0.2

(c)Pm= 0.3 (d)Pm= 0.4

(e)Pm= 0.5

Figure C.3: Azimuthal and equatorial cut of the θ component of the magnetic field

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(a) Pm= 0.6 (b)Pm= 0.7

(c)Pm= 0.8 (d)Pm= 0.9

(e)Pm= 1.0

Figure C.4: Azimuthal and equatorial cut of the θ component of the magnetic field

(62)

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Induced magnetic fields on planetary bodies

Universidad de los Andes

Final Degree Project

Induced magnetic fields on planetary

bodies

Abstract

Contents

List of Figures

List of Tables

Symbols

Dedicated to my Parents and brothers. . .

Chapter 1

Introduction

1.1

Jovian System

1.2

Description of the model

————————————————————-Chapter 2

Theoretical Model

2.1

Equations

2.2

Constant External Magnetic Field

2.3

Induced field on a conducting sphere under an

inho-mogeneous ambient magnetic field

Chapter 3

The Second Galilean moon:

Europa

3.1

Introduction

3.2

Interior stratification

3.3

Electrical conductivity of water

3.4

Magnetic field

3.5

Real vs Model Non-dimensional parameters

3.6

Rayleigh number

Chapter 4

Results and Discussion

4.1

Introduction

4.2

Magnetic Energy

4.3

Magnetic field on the surface of Europa

4.4

Magnetic field inside Europa

4.5

Geometry of the magnetic field

4.6

A comparison with Galileo data

4.7

Conclusions

Appendix A

Complementary calculations of

induced magnetic field

Appendix B

Normalization

B.1

Skin depth

Appendix C

Additional results

Bibliography