Tidal deformation of Europa and Phobos: implications on their structure and history

M.H.P. Kleuskens

Delft, 12 October 2006

Delft University of Technology Faculty of Aerospace Engineering Astrodynamics & Satellite Systems

Tidal deformation of Europa and Phobos: implications on their structure and history

M.Sc. Thesis Report

M.H.P. Kleuskens

Delft, 12 October 2006

This document was typeset with LATEX 2ε. The layout was designed by Remco Scharroo c 1993.

Astrodynamics & Satellite Systems Faculty of Aerospace Engineering Delft University of Technology Kluyverweg 1, 2629 HS Delft The Netherlands

Preface

In September 2005, I finished a “sabbatical” year of working for a student choir. From the world of arranging concert halls and rehearsal rooms, I entered back in the world of another passion: science.

This M.Sc. thesis work was carried out at the section Astrodynamics and Satellite Systems of the Aerospace Faculty of Delft University of Technology. This chair works mainly on precise orbit determination, Earth observation and Space instrumentation. Recently, planetary science has become a more important subject at this chair. This work is a continuation of previous M.Sc. theses researches about Jupiter’s moon Europa.

This report is set up for M.Sc. students or higher on the same subject. For people who are interested in planetary sciences but do not have a technical back- ground, I advice to read the background information of Europa and Phobos, in sections 4.1 and 5.1, respectively. The results in sections 4.6 and 5.5 can be fairly understood without mathematical background.

This work would have been impossible to accomplish without the help of a lot of people. The people on the ninth floor of the Aerospace Faculty building created a pleasant atmosphere to work in. I could not have completed this work without the help of fellow students and staff members, who did not only provide help, but moreover encouraged my interest in planetary sciences. In particular, I want to thank my tutor Bert Vermeersen, who always kept the door open for questions, and Luuk van Barneveld, who shared his knowledge about Europa.

Next, I want to thank Tony Dobrovolskis for sending me his numerical model of Phobos to make my graphics look more realistic and Ren´ePischel for keeping me up to date about the encounters of Phobos by Mars Express. Furthermore, I thank Terry Hurford for sharing his methods and results on Europa.

A special gratitude goes to friends, family and the One above for hearing my stories about tidal deformation over and over again.

Marco Kleuskens

v

Summary

Tidal forces are very important in various processes in our solar system. This thesis shows the effects of tidal deformation on the Jovian moon Europa and Martian moon Phobos. It makes use of the normal mode analysis, which was initially developed to model the deformation of the Earth.

Grooves and ridges on the surface of Europa show a possibility of a liquid ocean under the icy surface, caused by tidal heating as a result of Europa’s eccentric or- bit around Jupiter. By using an altimeter on an orbital mission to Europa the daily tidal deformation could be measured. The normal mode model is adapted to include internal fluid or low-viscosity layers. The viscoelastic response that is modelled is used to determine the amplitude and phase of the deformation as a function of rigidity, viscosity and thickness of the icy surface. For all realistic sit- uations, the relaxation times of the mantle and the core are too long to contribute to a phase lag with the surface ice layer. Based on surface features, it is suggested that Europa’s rotation is faster than its revolution. In this case, Europa would complete a rotation counter clockwise with respect to Jupiter on the timescale of 104 years. Since the relaxation times of the mantle and the core are on this order of magnitude, it is possible that the tidal bulge of the mantle and core is not aligned with the subjovian point. An altimetry mission could observe the presence of such an offset. This would make it possible to constrain the material and rheo- logical properties of those layers. Furthermore, it could be predicted whether the surface is co-rotating with the mantle or not. In the latter case, only the surface would rotate with respect to Jupiter, while the mantle and core would be locked to Jupiter.

Compared to Europa, Phobos is an irregular moon that is about one hundred times smaller, an its orbit is very close to Mars. Moreover, the moon is in a spiral motion that will end on the surface of Mars within 32 million years. But before that will happen, it will have broken up already by tidal forces. Although it is likely to be a captured asteroid, the orbit is almost circular. Furthermore, it has a large crater and a remarkable pattern of grooves. The origin of Phobos and its grooves lead to many paradoxes, and have driven a discussion between scientists for decades. This study sheds a new light on various arguments by adopting the normal mode analysis. Although the technique is linear and not accurate for an ellipsoidal figure, it gives new insight in the history of the moon. By simulating the orbital decay, an average viscosity of 4 × 1019P a would deform Phobos to its present shape. It is therefore plausible that the shape of Phobos is mainly caused by tidal deformation. Unfortunately, the impact that caused the main crater on Phobos and the tidal forces cannot be excluded not confirmed as the cause for the grooves.

vii

Contents

1 Introduction 1

2 Potential theory 3

2.1 Gravity ...... 3

2.2 Centrifugal potential ...... 5

2.3 Tides ...... 6

2.4 Variation in tidal potential due to eccentricity ...... 8

3 Normal mode analysis 13

3.1 Rheology ...... 14

3.2 Matrix propagation technique ...... 15

3.3 Convolution ...... 24

3.4 Stress ...... 27

4 Europa 31

4.1 Introduction ...... 31

4.2 Tidal potential of Jupiter, Io and Ganymede ...... 35

4.3 Normal mode analysis for fluid layers ...... 39

4.4 Validation of the method ...... 47

4.5 Sensitivity to changes in Europa’s rheology ...... 54

4.6 Results ...... 57

5 Phobos 63

5.1 Introduction ...... 63

5.2 Theories about Phobos ...... 68

ix x Contents

5.3 Normal mode analysis for a homogeneous sphere ...... 71

5.4 Determining material properties by the deformation ...... 75

5.5 The causes of the grooves ...... 81

6 Conclusions and recommendations 87

A Data of moons 89

A.1 Europa ...... 89

A.2 Phobos ...... 90

B Mathematical tools 91

B.1 Spherical coordinates ...... 91

B.2 Spherical harmonics ...... 91

B.3 Fourier series ...... 94

Bibliography 99 Chapter 1 Introduction

The Dutch interest in planetary exploration is growing rapidly. Industry, techno- logy institutes and scientists cooperate in the Netherlands Platform for Planetary research to ensure a good position in future planetary missions. Different groups of scientists are involved in this research. While geologists and biologists try to connect detailed processes on planets to the processes on Earth, geophysicists deal with the structure of planets on global scales and provide a valuable contribution to the knowledge about planets.

This M.Sc. thesis deals with tidal deformations of moons in our solar system, in particular the Martian moon Phobos and the Jovian moon Europa. These moons differ in many aspects: first, Phobos is a tiny, irregular rocky moon while the shape of Europa is more similar to our Moon. Second, their parental planets are an earthly planet and a gas giant, respectively. Their connection is the fact that their their surface features are most probably the result of extreme tidal forces.

On Europa, tidal heating by its eccentric orbit could melt the icy upper layers and could form an ocean in the subsurface. The most important clue is the fact that the Europan surface is covered with ridges and cracks. The presence of a liquid or slush water layer, together with the tidally driven reopening of cracks and water convection make Europa one of the most plausible places outside the Earth for sustaining life. The goal of this thesis is to assess whether it is possible to predict the inner structure of Europa by observing the tidal deformation. In particular, the presence of an ocean in the subsurface and the rheology of the surface are of interest. The effect by tides of other Jovian moons and the possible non-synchronous rotation of Europa is taken into account. All these relationships can put constraints on the instruments and the orbit of a future mission to Europa. A study to a Europa mission itself is not covered in this thesis, but is described in detail in e.g. [Weimar, 2005].

Although many UFO believers think life exists on or even inside Phobos, the opposite is more plausible. The small, irregular rock has not sufficient gravity to sustain an atmosphere. Surface features show evidence of a tumultuous history. Phobos has many craters of which the diameter of the largest one, Stickney, is almost half the size of the total diameter of Phobos. Furthermore, the surface is covered with a fine structure of grooves that follow a remarkable pattern. Not only the past was severe, the future brings an inevitable, disastrous end. Due to tidal friction the orbit of Phobos decays, which increases the tidal forces even

1 2 Introduction

more and will eventually cause a collapse on the surface of Mars. Before this will happen, the increasing tidal forces will disrupt Phobos. Learning more about the background and origin of Phobos can indirectly provide new information about the creation of our solar system. This research about Phobos is therefore rather focussed on the history than the present situation. The main goal of this part of the thesis is to summarise all possible causes for the grooves and assess them by use of the normal mode analysis. Two most favourable causes are compared, namely an almost destructive impact that also caused the giant crater Stickney, or tidal stress. The secondary goal is to relate the tidal deformation and rheology to the present, ellipsoidal shape of Phobos. Although the adopted method gives basic insight in the historical background, it is not very accurate for ellipsoidal figures like Phobos. For detailed modelling in the future, the method should be adapted for an ellipsoidal shape, or using finite element methods.

The report is built up in the following way. Before going into details on the particular moons, some general information concerning tidal deformation is pro- vided. In chapter 2, the potential theory is clarified. In chapter 3, the normal mode theory is explained. In these chapters, the theory is kept as general as possi- ble, in order it apply it on different bodies of the solar system. The research about Europa and Phobos will be shown in the chapters 4 and 5, respectively. The struc- ture in both chapters is similar. First, a general introduction is given. Since this report is focused on how knowledge can be gathered from observations together with mathematical methods, the information about the moons will be presented in the order of which the observations are made. Next, the normal mode analysis is adapted for the particular moon. For Europa, the normal mode analysis is adapted for internal fluid layers using the approach of [Van Barneveld, 2005]. However for Phobos, there is no proof that it has a layered structure. In this case, the tidal and load response are derived completely analytically. The next sections of these chapters show the validity of the method and the results. The report ends with conclusions and recommendations in chapter 6. Chapter 2 Potential theory

The main force that drives the deformation of celestial bodies is gravity. In order to understand this mechanism, first the general theory of gravity, centrifugal and tidal potential will be explained.

The forces that will be discussed are ordered by magnitude. First, the gravita- tional potential is derived using Newton’s law. In general, the gravitational force acts on all pairs of point masses and is therefore a fundamental force for astrody- namics. Next, the differences in gravity potential are shown more in detail. When two bodies are in an orbit, these local differences on the surface cause a resultant force, which is called tidal force. Finally, the temporal differences of tidal potential by an eccentric orbit are discussed, and cast in a mathematical description.

Various notations are commonly used in potential theory. The main confusion occurs in defining a spherical coordinate system, where terms as colatitude, lati- tude, θ and ϕ are often mixed. The spherical coordinate systems that is adopted here is defined in appendix B.1.

2.1 Gravity

According to Newton’s law of gravitation [Newton, 1687], two point masses m1 and m2 have a mutual attraction F~ that is proportional to the masses of the points and inversely proportional to the squared distance r

m m ~r F~ = −G 1 2 , (2.1) r2 r where G is the gravitational constant. This constant relates mass of a body with the gravitational force. It is the scale and distance in which the gravitational force differs from molecular forces: while molecular forces are dominant for small dis- tances, the gravitational force is a governing force at astronomical scale. Another method way to measure the mass is by observing its inertia. This method is used in human space flight. While gravitational force starts to become a substantial force for only large masses, inertia can only be measured for relatively small bodies. Therefore, this constant is most difficult to measure of all physical constants. The

3 4 Potential theory

Committee on Data for Science and Technology, recommends the value [CODATA, 2006]

G = (6.6742 ± 0.0001) × 10−11m3s−2kg−1.

According to Newton’s law of inertia, the resultant force equals the rate of change of momentum, or mass times acceleration. Therefore, the acceleration by the gravitational force of an external point mass m is m ~r ~g = −G , (2.2) r2 r

where ~g is a vector field for the mass to be considered. Because gravity acts radially towards a point mass, the field is irrotational:

∇ × ~g = 0.

As gravity is described as a vector, it is less convenient to work with for astrody- namical problems than a scalar. Therefore, a sphere with surface S is considered around the mass. The flux of the gravitational field yields (e.g. [Ditmar, 2006])

~ Φ = x (~g · dS). (2.3) S

A convenient property of this scalar value is that fluxes can be summed up: X ~ X ~ X Φ = x (~g · dS) = x (~g · dS) = Φi. (2.4) S i i S i

In astrodynamics, the flux of the gravitational field is called the gravity potential. Consider again the gravitational field of a point mass, equation 2.2. The flux in this case becomes m ~r  Φ = −G · dS~ . (2.5) r2 x r S

~r ~ Since r is a unit vector and is parallel with S, the part inside of the integral can be simplified to m m Φ = −G dS = −G S = −4πGm. (2.6) r2 x r2 S

It can easily be shown that this is also valid for a deformed spherical surface. Using the summation property, equation 2.6 is valid for a surface that encloses a group of point masses

Φ = −4πGM, (2.7)

where M is the total mass of a group os point masses. The potential can also be related to the density ρ of the body with volume V . Substituting M = R dm = R ρdvin equation 2.7yields 2.2 Centrifugal potential 5

Φ = −4πGρ y dV, (2.8) V

and using the divergence theory

~ Φ = x (~g · dS) = y ∇ · dV, (2.9) S V

one obtains the property

4Φ = ∇ · g = −4πGρ, (2.10)

which is called Poisson’s differential equation. For the potential of a vacuum ρ = 0 and consequently 4Φ = 0. The latter is called Laplace’s differential equation. The solution of this differential equation in spherical coordinates is

∞ l l GM X X R Φ = (C cos mϕ + S sin mϕ)P (cos θ), (2.11) r r lm lm lm l=0 m=0

which is derived in appendix B.2. R stands for the radius of the body, Plm the Legendre function and Clm and Slm the Legendre coefficients. The most important advantage of this notation is the fact that the potential is split up into different degrees l and orders m. In this way, the potential can be approximated up to the required accuracy for the designated purpose. The most important terms are C00 that accounts for the radial symmetric potential. Next, the terms C20, C22 and S22 are primary components for the tidal and centrifugal potential, which will be discussed in the next sections. Since most celestial bodies are close to an ellipsoid, the expansion up to order and degree 2 can already give much insight in astrodynamics.

2.2 Centrifugal potential

When a rotating reference frame is chosen around a celestial body, an observer fixed to the rotating reference frame experiences an extra centrifugal acceleration of [Torge, 1991]

2 ~c =ω ˙ ~r⊥, (2.12)

whereω ˙ is the rotational speed in rad/s and ~r⊥ is the vector perpendicular to the rotational axis. With

∇Φc = ~c, (2.13)

the centrifugal potential becomes

ω˙ 2 Φ = ~r2 , (2.14) c 2 ⊥ 6 Potential theory

and in spherical coordinates ω˙ 2 Φ = r2 sin2 θ. (2.15) c 2

Using table B.2, the centrifugal potential can be expanded in spherical harmonics 1 Φ = ω˙ 2r2 (P (cos θ) − P (cos θ)) . (2.16) c 3 00 20

The first term is small compared to the primary component of the gravity potential and is often neglected.

2.3 Tides

The effect of tides is mostly known for the sealevel change at coastal areas on Earth. Not often is it realised that these tides are the engine for various cycles on Earth. Without these cycles, it is doubtful that life could evolve on Earth.

Generally, all celestial objects that have gravitational interaction show tidal forces. They cause various processes in astrodynamics. In this section, the tidal force is derived mathematically. Next, the process of tidal locking is discussed in a qualitative manner. Other processes like tidal heating and orbital decay are less general concepts and are therefore discussed in chapters 4 and 5, about Europa and Phobos respectively.

Tidal potential

Although Kepler published his laws of planetary motion in 1609, it was Newton who connected this motion to the mutual gravity of celestial bodies [Newton, 1687]. When two bodies orbit around each other, the gravitational force and centrifugal force are in equilibrium in the centre for both bodies, see figure 2.1. At other loca- tions, the distance and direction of the secondary body is different. Therefore, the two forces are not in equilibrium. The equality of the centrifugal and gravitational potential in the centre of the body requires GM Φ = . (2.17) c r

The tidal potential is the difference between the centrifugal potential and the gravitational potential of the secondary body, 1 1 Φ = GM( − ). (2.18) T l r

The Legendre polynomial, which is defined in appendix B.2, has not only an ap- plication in spherical coordinates, but can also be used as an approximation in geometry. Combining the property

∞ l X R r Pl(cos ψ) = , (2.19) r p 2 2 l=0 r + R − 2Rr cos ψ 2.3 Tides 7

Figure 2.1 A system of two celestial bodies. Two external forces are acting on each point of mass: the centrifugal force and the gravitational force of the other body. For the centre of spherical bodies, the forces are in equilibrium.

with the cosine law for triangles l2 = r2 + R2 − 2rR cos ψ, (2.20)

the tidal potential yields

∞ l ! 1 X R 1 Φ = GM P (cos ψ) − . (2.21) T r r l r l=0

The contribution of the functions P0(cos ψ) and P1(cos ψ) will be compensated by the centrifugal force. A short and elegant notation of the tidal potential is therefore ∞ l GM X R Φ = P (cos ψ). (2.22) T r r l l=2

Note that the term (R/r)l rapidly converges to zero for relatively small radii com- pared to large distances between the bodies, which is the case for most objects in space. For most situations, terms up to only the third degree are taken into account.

A disadvantage of this tidal potential is that it cannot be added to the grav- ity potential and centrifugal potential, because different coordinates are adopted. Fixating the secondary body on coordinate [r, π/2, 0] and using spherical trigonom- etry, the transformation is given by cos ψ = cos ϕ sin θ. (2.23)

the tidal potential becomes (e.g. [Kaula, 1964])

∞ l l GM X R X (l − m)! Φ = (2 − δ ) · P (cos θ)P (0) cos(mφ). (2.24) T r r (l + m)! 0m lm lm l=2 m=0

The tidal potential up to the third degree is

GMR2  1 1  Φ = − P (cos θ) + P (cos θ) cos(2ϕ) + T r3 2 20 4 22 R1 1  P (cos θ) cos ϕ − P (cos θ) cos(3ϕ) . (2.25) r 4 31 24 33 8 Potential theory

Figure 2.2 A schematic overview of the mechanism that locks a body to a neighbouring body. When it rotates counter clockwise, the tidal bulges will lag behind as shown in the figure. The tidal force on the tidal bulge is not perpendicular to the surface, but has a component tangential to the surface of the body. This component causes a clockwise torque.

As mentioned before, the terms of the third degree can be neglected in most cases.

Tidal locking

When two bodies have a large mutual interaction, the tidal force can dominate the envelope of the system’s history. The force can be substantially dominant, that one body forces the other to slow down its rotation. This has happened for almost all moons in our solar system. In the Pluto-Charon system, the mutual force was large enough to lock both bodies in a configuration where they both show the same side to each other.

Consider a moon around a planet with a rotational speed faster than its revo- lution speed. By tidal force, the moon tends to deform to an ellipsoid, see chapter 3. In general, there will be a delay in the deformation. While the moon rotates further, the tidal bulge will have an offset from the planet, see figure 2.2. The tidal bulge that is attracted by the planet creates a torque acting clockwise and thus decelerating the counter clockwise rotation of the moon.

Although the tidal lock stabilises the rotation of a moon, it is possible that the tidal potential forces Europa to rotate slowly with respect to Jupiter. This mechanism is described more in detail in section 4.6.

2.4 Variation in tidal potential due to eccentricity

Ever since an object is tidally locked, the tidal force acts on the same side. Con- sidering the age of our solar system, most of these bodies had enough time to fully deform to the disturbed potential field. In such cases, other temporal variations of the tidal force become important. The main cause is the eccentricity of the orbit. According to figure 2.3, the distance between a planet and a moon shows a periodical variation. Furthermore, the rotation speed of the moon is constant, while the revolution speed has a periodical variation. From a moon’s point of view, the planet seems to rotate clockwise around an imaginary point.

In order to find the time dependent tidal potential, first consider Kepler’s law for an elliptical orbit (e.g. [Wakker, 2002])

a(1 − e)2 r = , (2.26) 1 + e cos Ω 2.4 Variation in tidal potential due to eccentricity 9

Figure 2.3 The movement of a moon in an eccentric orbit; left in a planet-fixed reference frame, and right in a moon-fixed reference frame [Greenberg, 2005].

with a the half long axis of the orbit, Ω the angle between pericentre and the actual position of the body, and e is the eccentricity, defined by

 e = 0, circular orbits  r − r  0 < e < 1, elliptic orbits e = a p , (2.27) r + r e = 1, parabolic orbits a p   e > 1, hyperbolic orbits

For most planets and moons, the eccentricity varies between e = 0.0 and e = 0.2 (e.g. [Solarviews, 2006]. Since the angular momentum by the revolution of a body

H = r2Ω˙ , (2.28)

must remain constant, it implies that Ω˙ is not constant for e > 0. The relation between time and Ω is given in an implicit form as r !  1 − e Ω p sin Ω t = Ω¯ 2 tan−1 tan − e 1 − e2 , (2.29) 1 + e 2 1 + e cos Ω

where Ω¯ is the average orbital speed. Since this function gives no physical insight in the problem, a different approach is generally used. Using the eccentric anomaly E

θ r1 + e E tan = tan , (2.30) 2 1 − e 2

as defined in figure 2.4, the relation becomes:

E − e sin E = Ω¯t, (2.31)

which is known as Kepler’s equation. To find the position of a moon at a certain moment, an easy iterative method like Newton-Rhapson can be used. But to get more insight in the characteristics of the orbit, it is preferable to find an approximation in terms of a series. First define:

E − Ω¯t = x. (2.32) 10 Potential theory

Figure 2.4 Definition of the eccentric anomaly [Wakker, 2002].

Substituting this into Kepler’s equation and using series expansion of the gonio- metric elements:

x = e sin(Ω¯t + x) = e sin(Ω¯t) cos(Ω¯t) + e cos(Ω¯t) sin x 1 1 1 = e sin(Ω¯t)(1 − x2 + x4 − ...) + e cos(Ω)¯ t(x − x3 + ...).. (2.33) 2! 4! 3!

Since −e < x < e and e < 1, x can be approximated by:

2 3 4 x = x1e + x2e + x3e + O(e ),

and Kepler’s equation becomes:

 1  x e+x e2+x e3+O(e4) = e sin(Ω¯t)+e2x cos(Ω¯t)+e3 x cos(Ω¯t) − x2 sin(Ω¯t) +O(e4). 1 2 3 1 2 2 1

Grouping the x-terms yields:

x1 = sin(Ω¯t) 1 x = x cos(Ω¯t) = sin(2Ω¯t) 2 1 2 1 3 1 x = − x2 sin(Ω¯t) + x cos(Ω¯t) = sin(3Ω¯t) − sin(Ω¯t). (2.34) 3 2 1 2 8 8

Substituting back into E of equation 2.32 yields:

 e2  e2 3e3 E = Ω¯t + e 1 − sin Ω¯t + sin 2Ω¯t + sin 3Ω¯t + O(e4). (2.35) 8 2 8

With the same method, an expression for higher accuracies can be obtained. Using equation 2.30 the real orbital angle Ω can be found. 2.4 Variation in tidal potential due to eccentricity 11

When no inclination is assumed (The inclinations of Europa and Phobos are less than 1◦) and the orbital period is the same as the rotational period, the time dependent tidal potential up to the second harmonic degree can be approximated using s binomial expansion [Kaula, 1964]

∞ 1 X Φ (t) =r2ω˙ 2 − P A cos(qωt˙ ) T 2 20 q q=−∞ (2.36) ∞ ! 1 X   + P B cos(2ϕ) cos(qωt˙ ) − sin(2ϕ) sin(qωt˙ ) , 4 22 q q=−∞

where, according to Kepler’s third law (e.g. [Wakker, 2002])

GM ω˙ 2 = . (2.37) r3

The eccentricity polynomials Aq and Bq are only a function of e. The terms with index q = 0 are not regarded here since they account for the constant part of the tidal potential. When assuming only first and second order in the eccentricity, the terms with q = ±1, 2 are sufficient. The values for these polynomials are q Aq Bq -2 9/4e2 0 -1 3/2e -1/2e 1 3/2e 7/2e 2 9/4e2 17/2e2

Substituting these coefficients into equation 2.36, the tidal potential function yields

 1   Φ (t) = r2ω˙ 2e − P (cos θ) 3 cos(ωt ˙ ) + 17e cos(2ωt ˙ ) T 2 20 1  17 + P (cos θ) cos(2ϕ)(3 cos(ωt ˙ ) + e cos(2ωt ˙ )) (2.38) 4 22 2 17  + sin(2ϕ)(4 sin(ωt ˙ ) + e sin(2ωt ˙ )) . 2

The second line of equation 2.38 can be split into two effects. The term with cos(2ϕ) accounts for the variation in orbital distance and the term sin(2ϕ) dom- inates the libration. When neglecting the second order term in eccentricity, one obtains  3  Φ (t) = r2ω˙ 2e − P (cos θ) cos(ωt ˙ ) T 2 20 (2.39) 1   + P (cos θ) 3 cos(2ϕ) cos(ωt ˙ ) + 4 sin(2ϕ) sin(ωt ˙ ) . 4 22

Since the normal mode analysis, which is described in the next chapter, assumes lateral homogeneity, equation 2.39 should be transformed to a form with only lat- eral terms. The first term of equation 2.39 has no lateral components. The second term can be expressed as only lateral components when a spherical coordinate sys- tem is adopted with the pole at the subplanetal point. The third term is analogous 12 Potential theory

to a laterally homogeneous potential with a pole in point [θ = π/2, ϕ = π/4] in the original coordinate system. Using these transformations, the tidal potential becomes   2 2   ΦT (t) = r ω˙ e 3 cos(ωt ˙ )P20(cos ψ)+4 sin(ωt ˙ ) P20(cos γ)+P20(cos θ) , (2.40)

where θ is the angle with respect to the geographical pole, ψ the angle with respect to the subplanetal point, and γ the angle with respect to the point [θ = π/2, ϕ = π/4].

When two or more moons orbit a planet, they also have a mutual tidal inter- action. Although this effect may be small, the relative distance and angle between the moons varies enormously over one epicycle. This is shown for the Galilean moons in section 4.2. Because the mutual distance and angle vary periodically, an expansion into series converges very slowly. In order to illustrate this, fig- ure 4.6 shows the mutual distance between Io and Europa during one orbit. A mathematical description up to the sufficient order would be too laborious to use.

Therefore, a numerical approach is adopted. First, the tidal potential on every point on the surface of the moon at every moment can be calculated using

∞ l GM X  R    Φ (t) = P cos(ψ(t)) , (2.41) T r(t) r(t) l l=2

where r(t) is the distance between the two moons and ψ(t) the angle between the location on the moon and the vector between the two moons, both as function of time. These values can be derived geometrically. When the solution is found for a grid on the sphere, the solution can be expanded into normalised spherical harmonics according to appendix B.2. Furthermore, when this is done in regular time intervals over a complete epicycle, it can be expanded into a Fourier series, see appendix B.3,

∞ l X X X ΦT (θ, ϕ, t) = Glmf Ylm(θ, ϕ) cos(fωt˙ ) + Hlmf Ylm(θ, ϕ) sin(fωt˙ ). f l=0 m=−1 (2.42) Chapter 3 Normal mode analysis

In this chapter it is shown how the deformation of a celestial body can be related to an external potential or load by normal mode analysis.

First, it is shown how stress, strain and strain rate are related with one another. It is found that the Maxwell model forms the most basic model to describe the viscoelastic behaviour sufficiently.

Next, the deformation of the body as a whole will be addressed. In principle, the deformation, stress field and gravity field can be solved by means of numerical integration techniques with appropriate initial, boundary and continuity condi- tions. By this method, virtually all possible shapes and loads can be modelled, however it gives no insight in the problem.

By using three mathematical transformations, the problem can be solved al- most analytically. First, the problem is transformed to the Laplace domain in order to simplify the Maxwell rheology. Next, when lateral homogeneity is as- sumed, spherical coordinates simplify the calculations. Finally, when expanding a load in spherical harmonics, the solution is decoupled for all harmonic orders and degrees. The way to find the deformations is briefly the following: between two subsequent layers, six properties are propagated: radial deformation, tangen- tial deformation, radial stress, tangential stress, change in potential and potential stress. At the surface, constraining the radial stress, the tangential stress and the potential stress to the external load sets the boundary condition. These conditions are propagated inward and linked with the core conditions. These conditions are propagated outward again to find the three unknowns at the surface, namely the radial deformation, the tangential deformation and change in potential.

This approach has several advantages: since the normal mode analysis is almost completely analytical, it needs less CPU time to solve it, thus many runs can be performed in a short time. Furthermore, it helps in gaining more insight in geophysics. But the most important advantage is the fact that the deformation can be split into several contributions: different harmonic degrees, different layers of the body, and different frequencies of tidal variation. As will be shown in chapter 4 and 5, only some of these contributions are substantial; the others can be easily neglected for a first approximation.

Here, only the basic outline of the Normal Mode Analysis is given. A more

13 14 Normal mode analysis

elaborate description can be found in e.g. [Sabadini and Vermeersen, 2004] and [Vermeersen, 2002].

3.1 Rheology

In geology often the macroscopic properties of materials are considered rather than focussing on interaction on molecular scale. A lot of research is spent on the rheo- logical behaviour of material. Although most materials deform non-linearly, it will be assumed that the rheology for small grain sized materials is linear, which sim- plifies the calculations substantially. In this section the different linear rheologies are discussed.

Hooke’s law relates stress σ linear proportionally to the strain ε. This approx- imates the behaviour of a material that is not loaded enough to cause irreversible deformation. In reality, the material will show creep dependent on the viscosity. For now only the elastic behaviour is shown, which represents a short lasted stress. By assuming an isotropic material, Hooke’s law simplifies to

σij = λεkkδij + 2µεij, (3.1)

with δij the Kronecker delta, and λ and µ Lam´econstants. The behaviour can best be described by a spring. The strain ε is defined as   1 ∂ui ∂uj εij = + . (3.2) 2 ∂xj ∂xi

The strain can be interpreted as the relative displacement: when the indices are equal, the strain represents elongation or shortening, whereas it represents defor- mation when the indices are unequal.

Now consider the purely viscous response. When a material is subjected to a constant force for an infinite time, the elastic deformation has already occurred and material can be assumed as pre-stressed. It will now only deform by viscous behaviour. The linear relation between stress and strain-rate is then

0 σij = λ ε˙kkδij + 2νε˙ij, (3.3)

in which ν is the Newtonian viscosity of the material. In geological processes it is generally assumed thatε ˙ = 0, the so-called Stokes-condition. Purely viscous behaviour can best be interpreted as the behaviour of a dashpot. Analogous to the strain, the strain-rate is defined by   1 ∂u˙ i ∂u˙ j ε˙ij = + . (3.4) 2 ∂xj ∂xi

Real materials are neither completely elastic or viscous. Even the hardest stone will show creep on geological time scales, and almost liquid materials will still lag behind fast oscillations of a force. Elastic and viscous behaviour can be combined in a linear manner to describe viscoelastic behaviour. In one dimensional analogy, the 3.2 Matrix propagation technique 15

viscoelastic behaviour is often represented as a combination of springs (σ = 2µε) and dashpots (σ = 2νε˙). The three basic combinations (parallel, serial, parallel followed by serial) of these elements are depicted in figure 3.1. The simplest rheology that describes viscoelastic behaviour is the Maxwell model. For planetary sciences where no laboratory experiments of materials have been conducted yet it is sufficient to adopt this model. The total strain-rate is the sum of the strain-rates of the spring and dashpot σ˙ σ ε˙ = + . (3.5) 2µ 2ν

For a constant applied stress σ0, the strain becomes σ σ ε(t) = 0 t + 0 . (3.6) 2ν 2µ

Figure 3.1b shows the strain for a constant load that is applied at the interval [t0 − t1]. The jump at t = t0 represents the elastic response, while the slope at [t0 − t1] represents the viscous behaviour. When the stress is removed, only the viscous deformation remains.

By taking the strain as constant rather than the stress, the material behaves quite differently. First when the strain is applied, the strain consists only of elastic deformation. Therefore, the stress is large, causing a large viscous respond. As the material deforms viscously, the elastic deformation becomes less and subsequently also the stress. By solving equation 3.5 for constant strain, the solution becomes µ − t −t/τM σ(t) = σ0e ν = σ0e , (3.7)

where τM is the Maxwell relaxation time, defined as: ν τ = (3.8) M µ

This characteristic time indicates the transition between the dominance of elastic and viscous behaviour. The complete three dimensional form of the Maxwell model is [Peltier, 1974]:

3 ! 3 µ 1 X X σ˙ + σ − σ δ = 2µε˙ + λ εδ˙ . (3.9) η ij 3 kk ij ij k=1 k=1

3.2 Matrix propagation technique

In order to determine the deformation of a celestial body, it is preferable to consider the body first without external loads. By its own gravity, it is already hydrosta- tically pre-stressed. The density and potential are split into a constant part by self-gravity and a perturbed part by an external load,

Φ = φ0 + φ1,

ρ = ρ0 + ρ1. (3.10) 16 Normal mode analysis

Figure 3.1 The strain as function of time for different linear viscoelastic models: a) Kelvin model, b) Maxwell Model, c) Burgers Model [Ranalli, 1995].

This leads to the linearised equation of momentum

∇ · σ − ∇(ρ0g~u · ~er) − ρ0∇φ1 − ρ1g0~er = 0. (3.11)

The first term describes the contribution by the induced stress, the second term the advection of the pre-stress, the third term the changed self-gravity and the fourth term the changed density by compression. The perturbed gravity potential satisfies Poisson’s equation, analogous to equation 2.10

4φ1 = −4πGρ1. (3.12)

For incompressible problems, ρ1 will become zero. Using the appropriate boundary conditions, equations 3.9, 3.11 and 3.12 can be solved numerically. Several software packages exist where standard rheologies are implemented. A major disadvantage of these software packages is that the viscoelastic behaviour is difficult to decouple from the external loads in numerical results. By transforming the problem to another domain, it is possible to solve it almost completely analytically. In this case, the Laplace transform will be used. Furthermore, the problem can be solved up to a certain degree of accuracy. For planetary science, low degree solutions help already much in the understanding of the global picture. This will be shown in chapters 4 and 5.

Correspondence Principle

First, the viscoelastic behaviour of the material can be described in a much simpler form when transforming it to the Laplace domain. Consider the Laplace transform 3.2 Matrix propagation technique 17

of a function, defined by (e.g. [Boyce and DiPrima, 1997]):

∞ Z L[f(t)] = f˜(s) = f(t)estdt, (3.13) 0

This transformation is often used for solving partial differential equations. By transforming to another domain, a complicated mathematical structure can be- come substantially easier. By using the property

L[f 0(t)] = sL[f(t)] − f(0) (3.14)

the derivatives of stress and strain can be easily transformed to the Laplace domain. The complete Laplace transformation of equation 3.9 yields:

3 3  µ 1 µ X X s + σ˜ (s) − σ˜ (s)δ = 2µsε˜ (s) + λs ε˜ (s)δ . (3.15) ν ij 3 ν kk ij ij kk ij k=1 k=1

The equation can now be simplified by

3 ˜ X σ˜ij(s) = λ(s) ε˜kk(s)δij + 2˜µ(s)˜εij(s), (3.16) k=1

where the the so-called compliances λ˜(s) andµ ˜(s) are given by

µ 2 sλ+ ν (λ+ 3 µ) λ˜(s) = µ , s+ ν sµ µ˜(s) = µ . (3.17) s+ ν

The Maxwell rheology has become a Hookean rheology in the Laplace domain. The so-called correspondence principle states that by calculating the associated elas- tic solutions in the Laplace-transformed domain, the time dependent viscoelastic solutions can be found by Laplace inversion in a unique way. Besides simplifying the response of the body it has another advantage. Since the Laplace transformed stress is linearly dependent on the strain, the stress can be derived from the de- formation. This makes it possible to assess the direction of cracks in the surface. This will be discussed in detail in section 3.4. Besides the Maxwell rheology it can be shown that all linear rheologies (Kelvin, Burgers) will behave as Hookean in the Laplace domain.

Expansion in spherical harmonics

By using the Laplace transform, the problem is simplified in the time dimension. Now the spatial dimensions are assessed to simplify. Since most celestial bodies can be approximated by a sphere, spherical coordinates are adopted. In chapter 2, the potential was already expanded into spherical harmonics, simplifying the description of the potential substantially. Similarly, the equations describing the interior of a body can be expanded in a similar manner. 18 Normal mode analysis

Before rewriting the equations, it is convenient to adopt another assumption. In most cases, the external load acts only radially on the body. In this way, the spherical coordinates can be chosen such that the problem is rotationally symmet- ric. Therefore, the longitudinal component drops out, and the solution has to be only expanded in Legendre polynomials Pl(cos θ),

∞ X u˜(r, θ, s) = U˜l(r, s)Pl(cos θ) l=0 ∞ X ∂Pl(cos θ, s) v˜(r, θ, s) = V˜ (r, s) l ∂θ l=0 σ˜rr(r, θ, s) =σ ˜rr(r, s)Pl(cos θ)

σ˜rθ(r, θ, s) =σ ˜rθ(r, s)Pl(cos θ) ∞ ˜ X ˜ φ1(r, θ, s) = φl(r, s)Pl(cos θ). (3.18) l=0

Note that the introduced θ is not necessarily similar to the angular distance to the geometrical north pole. Toroidal components only occur when the body is loaded to oblique loads, like friction of the atmosphere or for models with lateral variations. For the case of Phobos the main crater could be caused by an oblique impact. In this thesis this possible cause is not covered, but could be an interesting subject for a study.

Matrix propagation

The equations 3.11, 3.12 and 3.15 can now be solved for all spherical degrees l. To this extent first consider a layer in a spherical body. For an incompressible Maxwell viscoelastic case, the equations 3.11, 3.12 and 3.15 can be written as d~y˜ = A˜ · ~y,˜ (3.19) dr l

With ˜ T ~y˜ = [U˜l, V˜l, σ˜rrl, σ˜rθl, φl, Q˜l] (3.20)

and  2 l(l+1)  − r r 0 0 0 0  − 1 1 0 1 0 0   r r µ˜   4  3˜µ  l(l+1)  6˜µ  l(l+1) ρ(l+1)   − ρg − − ρg 0 −ρ0 ˜ r r r r r r Al =  2  . (3.21)  1  6˜µ  2(2l +2l−11)˜µ 1 3 ρ  − − ρg 2 − − − 0   r r r r r r   l+1   −4πGρ0 0 0 0 − r 1  l+1 l(l+1) l−1 −4πGρ0 r 4πGρ r 0 0 0 r

The last term in equation 3.20 is sometimes nicknamed potential stress, and is defined by dφ˜ l + 1 Q˜ = − l − φ˜ + 4πGρU˜ . (3.22) l dr r l l 3.2 Matrix propagation technique 19

The parameters are chosen such that they are continuous at internal boundaries between two subsequent layers. Furthermore, if no external load is applied, the third, fourth and sixth term must be zero in order to match the external boundary conditions at the surface.

Equation 3.19 is a system of six linear first order differential equations and can therefore be solved as (see e.g. [Boyce and DiPrima, 1997])

~y˜l(r, s) = Y˜ l(r, s) · C~l(r), (3.23)

with

 lrl+1 l−1 2(2l+3) r 0  (l+3)rl+1 rl−1  2(2l+3)(l+1) l 0  2 l  (lρ0g0r+2(l −l−3)˜µ)r (ρ g r + (2l − 1)˜µ)rl−2 −ρ rl Y¯˜ (r) =  2(2l+3) 0 0 0 ... l  l l−2  l(l+2)˜µr 2(l−1)˜µr 0  (2l+3)(l+1) l  l  0 0 −r l+1 2πGρ0lr l−1 l−1 2l+3 4πGρ0r −(2l + 1)r (l+1)r−l −l−2  2(2l−1) r 0 (2−l)r−l r−l−2  2l(2l−1) − l+1 0  2  (l+1)ρ0g0r−2(l +3l−1)˜µ ρ0g0r−2(l+2)˜µ ρ0  l+1 l+3 − l+1  2(2l−1)r r r  (3.24) (l2−1)˜µ 2(l+2)˜µ  l+1 l+3 0  l(2l−1)r (l+1)r  0 0 − 1  rl  2πGρ0(l+1) 4πGρ0 (2l−1)rl rl+2 0

The solutions of two subsequent layers can be connected in the following way

˜ (i) ~ (i) ˜ (i+1) ~ (i+1) Yl (ri+1, s) · Cl (ri) = Yl (ri+1, s) · Cl (ri+1). (3.25)

Substituting the unknowns from the previous equation by extending the result for all layers gives a relation between the vector at the surface and the unknowns in the core

N−1 ! ~ Y ˜ (i) ˜ (i)−1 ˜ (N) ~ (N) y˜l(R, s) = Yl (ri, s)Yl (ri+1, s) Yl (rc, s)Cl (rc), (3.26) i=1

where

˜ −1 ˜ ¯˜ Yl (r, s) = Dl(r) · Yl(r, s), (3.27)

˜ with Dl(r) being a diagonal matrix with the elements

1 l + 1 l(l + 1) 1 l(l + 1)  diag(D (r)) = , , − , lrl, rl+2, −rl+1 , l 2l + 1 rl+1 2(2l − 1)rl−1 rl−1 2(2l + 3) (3.28) 20 Normal mode analysis

and  ρgr r lr ρr  µ˜ − 2(l + 2) 2l(l + 2) − µ˜ µ˜ µ˜ 0  ρgr 2(l2+3l−1) 2 r (2−l)r ρr   − µ˜ + l+1 −2(l − 1) µ˜ µ˜ − µ˜ 0    ¯˜  4πGρ 0 0 0 0 −1  Yl(r, s) =   (3.29)  ρgr + 2(l − 1) 2(l2 − 1) − r − (l+1)r ρr 0   µ˜ µ˜ µ˜ µ˜   ρgr 2(l2−l−3) r (l+3)r ρr   − µ˜ − l −2l(l + 2) µ˜ µ˜ − µ˜ 0  4πGρ 0 0 0 2l + 1 −r

For the core the right half of the propagation matrix becomes singular because it contains factors of r with a negative power. For a solid core, only the left half of the propagation matrix is used. For a fluid core, it is assumed that:

(N) The radial displacement is close to an equipotential surface,y ˜1,l (rc) = y˜(N)(r ) 5,l c + K . gc 3 (N) The tangential displacement is undefined:y ˜2,l (rc) = K2. The radial stress is dependent on the difference between the equipotential sur- (N) face and the radial displacement:y ˜4,l (rc) = gcρcK3. The tangential stress is zero, because the fluid is assumed to be frictionless. The gravity is proportional to rl. The potential stress is continuous.

In matrix form:  l−1  −rc /Ac 0 1    0 1 0    K1   K1 ˜ (N) ~ (N) ˜  0 0 ρcAcrc Y C (rc) = Ic,l · K2 =   · K2 (3.30) l l  0 0 0  K3  l  K3  rc 0 0  l−1 2(l − 1)rc 0 3Ac

4 with Ac = 3 πGρc. For a liquid core, only the density and radius need to be specified.

The elastic response

For the surface, it was already mentioned that the third, fourth and sixth element are zero if no forcing is applied. When projecting equation 3.26 on these elements, the system becomes,     N−1 ! K1 K1 ~ Y ˜ (i) ˜ (i)−1 b = P1 Yl (ri, s)Yl (ri+1, s)Ic,l(rc) ·K2 = P1BIc,l ·K2 (3.31) i=1 K3 K3

With P1 the projection matrix on the third, fourth and sixth component  0 0 1 0 0 0  P1 =  0 0 0 1 0 0  . (3.32) 0 0 0 0 0 1 3.2 Matrix propagation technique 21

The vector ~b is dependent on the kind of external forcing. When a tidal force is applied, the stresses at the surface remain zero in order to fit the continuity condition. Only the potential stress changes, by the external gravity potential [Takeuchi et al., 1962]

 2l + 1T ~b = 0 0 − . (3.33) T R

In the case of a unit point mass, the radial stress at the surface is perturbed. The boundary condition becomes

 1 2l + 1 2l + 1T ~b = − g(R) 0 − G . (3.34) L 4π R2 R2

The unknowns parameters are collected when projecting the solution on the first, second and fifth element  ˜    Ul K1 ˜  Vl  (R, s) = P2BIc,l · K2 (3.35) −Φ˜ l K3

With P2 the corresponding projection matrix  1 0 0 0 0 0  P2 =  0 1 0 0 0 0  . (3.36) 0 0 0 0 1 0

By substituting equation 3.35 into equation 3.31, the unknowns at the core can be eliminated, ending up with an equation that relates the constrained parameters (stress, potential stress) to the unconstrained parameters (displacement, poten- tial),   U˜l ˜ −1 ~  Vl  (R, s) = (P2BIc,l) · (P1BIc,l) · b. (3.37) −Φ˜ l

For the elastic case s → −∞ and the compliances are similar to the Lam´epara- meters for a Hookean rheology. The stress strain relationship is simply linear,   U˜l ˜ −1 ~ ~ e  Vl  (R) = (P2BIc,l) · (P1BIc,l) · b = Kl (R). (3.38) −Φ˜ l

Viscoelastic response

For the viscous response, the solution is much more complex. First, consider the homogeneous version of equation 3.36,   K1 ~ 0 = P1BIc,l · K2 . (3.39) K3 22 Normal mode analysis

This equation has only non trivial solutions if the determinant

|P1BIc,l(rc)| = 0

This expression is called the secular equation. The solutions s = sj are the inverse relaxation times of different modes of the body. It can be solved for each harmonic degree l. The number of modes M is dependent on the structure of the body. Using numerical codes, it was found that the following modes occur [Han and Wahr, 1995]:

M0 represents the mode contributed by the surface,

L0 exists if there is an elastic lithosphere on top of a viscoelastic mantle,

Mi, i = 1, 2, 3,... are triggered by every change in density between two subse- quent layers, and are also called buoyancy modes. Every jump in Maxwell relaxation time triggers a pair of transitions modes Ti, i = 1, 2, 3,.... A change from fluid to solid or from solid to fluid represents one mode. The transition from the lowermost viscoelastic mantle to the inviscid core is labelled C0.

If the density of the body is nowhere increasing over the radius then all inverse relaxation times are negative for an incompressible model.

The search for modes is the only part in the described procedure that has to be done numerically. First, the determinant is calculated for values of s on a regular interval. An interval contains an odd number of solutions if the determinant switches sign, and an even number of solutions if the sign does not change. The grid spacing has to be reduced if not all modes are found. If the values of all roots are roughly known, a bisectional algorithm can narrow the particular intervals. This process is repeated iteratively to obtain the required accuracy.

Obtaining the total displacement vector caused by viscosity is another problem. Equation 3.37 becomes singular for all values of sj. Therefore, it can be shown that the viscoelastic response can be written as

 U˜  M l K~ j(R) ˜ X l  Vl  (R, s) = . (3.40) s − sj −Φ˜ l j=1

In this case, the vector K~ is solved in an alternative way:

† ! ~ j (P2BIc,l(rc)) · (P1BIc,l(rc)) ~ Kl (R) = d · b, (3.41) |P1BIc,l(rc)| ds s=sj 3.3 Convolution 23

where the daggered matrix denotes the matrix of the complementary minors (e.g. [Scienceworld, 2006]):   a22 a23 a21 a23 a21 a22 −  a32 a33 a31 a33 a31 a32    −1 1  a12 a13 a11 a13 a11 a12  1 † A = ·  − −  = · A |A|  a32 a33 a31 a33 a31 a32  |A|    a12 a13 a11 a13 a11 a12  − a22 a23 a21 a23 a21 a22

The derivative of the determinant can be solved either analytically or numerically. The numerical way is most straightforward:

d (P1BIc,l(rc))s=s +h − (P1BIc,l(rc))s=s −h D = |P BI (r )| = j j ds 1 c,l c 2h

Only the choice of h can be a problem. By choosing h too large, the maximum accuracy is not obtained. However, if h is too small, numerical errors will occur. In order to find the optimum result, the derivatives Dn are calculated for a range of hn, where every next h is somewhat smaller than the previous one. Initially, the accuracy will improve, which is visible by the fact that the subsequent derivatives Dn and Dn+1 differ less and less. After a certain hoptimum the numerical problems become visible, because D starts to fluctuate. Therefore the best approximation of the derivative can be found where the difference between two subsequent deriv- atives is minimum. The algorithm takes the form:

hn+1 = αhn, 0 < α < 1

(P1BIc,l(rc))s=sj +hn − (P1BIc,l(rc))s=sj −hn Dn = 2hn d |P BI (r )| = min(|D − D |) (3.42) ds 1 c,l c n n−1

The analytical alternative is found in the following way. The derivative of the ˜ fundamental matrix Yl is composed by setting all elements that are not a function of s to zero, and by replacingµ ˜ by: d d  µs  µ2 µ˜ = = . (3.43) ds ds s + µ/ν ν(s + µ/ν)2

˜ −1 For the derivative of the fundamental matrix inverse, Yl , a similar approach is applied. Here, the factor 1/µ˜ is replaced by: d  1  d s + µ/ν  1 = = . (3.44) ds µ˜ ds µs −s2 ∗ ν

Since |P1BIc,l(rc)| is a series of products, the derivative is found by using the product rule of differentiation multiple times, d d d AB = A B + B A (3.45) ds ds ds

For the applications of Phobos and Europa both methods provided the same re- sults. 24 Normal mode analysis

Figure 3.2 A schematic of a response to a Heaviside load (left) and a periodical load (right).

3.3 Convolution

The last step is transforming back from the Laplace domain to the time domain. Following the residue theorem, the viscoelastic response becomes   Ul M ~ e X ~ j (sj t)  Vl  (R, t) = Kl (R)δ(t) + Kl (r)e . (3.46) −Φl j=1

Finally, this response function must be convolved with the external load. Below the convolution for a Heaviside load and a periodical load are described more in detail.

Heaviside load

A constant load is most basic and is used as starting point. When this load is applied instantly, this represents a Heaviside load, defined as:  0, t < th H(th) = (3.47) 1, t ≥ th

By adding a series of Heaviside functions with different start times, every random function can be approximated. The case of the spiral motion of Phobos is de- picted in figure 5.8. The convolution with a Heaviside function can be found using standard tables (e.g. [Boyce and DiPrima, 1997]) and yields

 U  M j l K~ (r)   ~ e X l (sj th)  Vl  (R, t) = Kl (R)H(th) + 1 − e H(th). (3.48) −sj −Φl j=1

The response is depicted schematically in the left of figure 3.2 and can be inter- preted in the following way: the first term represents the elastic response, which works instantly after th. The viscous response also starts at this time, but the factor 1 − e(sj th)
starts at zero and builds up to the asymptote of 1. The inverse relaxation time sj decides the speed of this process, which explains its name. By 3.3 Convolution 25

using a logarithmic timescale it is obvious to see when a certain response becomes important. The Heaviside load is also used to define dimensionless properties de- scribing the rheology of the body. The dimensionless numbers are called Love numbers, after A.E.H. Love [Love, 1927].

Love numbers

Consider the elastic deformation of a body by a tidal potential. The elastic Love numbers for each harmonic degree are then defined as

U eg he = l T,l Φ V eg le = l T,l Φ Φe ke = l , (3.49) T,l Φ

where g is the average gravity at the surface and the capital index T denotes the tidal deformation. As an example, the tidal Love numbers can be interpreted as follows: consider a disturbance in the potential field of a body by tides ΦT . The surface of the body will move upwards by hT ΦT /g. Since the mass of the body is redistributed due Love number hT , the potential changes with an additional factor e kT ΦT . In total the potential becomes Φ + kT ΦT . The value of hT can be larger than unity, as will be seen in chapter 4. This seems contradictory: the surface tries to reshape itself to the geoid, but with hT > 1 the surface deforms more than the change in potential demands. The explanation is a secondary effect: when the potential changes, the surface changes. By the changing of the surface height the mass of the body is redistributed, causing the extra change in potential kT . This extra on its turn will deform the surface even more. In chapter 5, it will be shown that hT,2 = 2.5 is the maximum value. This can occur for a fluid, homogeneous body. A body with high-density mantle and low density core can theoretically have higher Love numbers, but entropy will make these bodies unstable and are therefore unrealistic.

The inverse relaxation times sj are converted to the time domain in order to make them better interpretable, 1 τ = − . (3.50) sj

The viscous response is expressed in modal strength. The number of modal strengths is equal to the number of modes per harmonic degree times the har- monic degree for which it is calculated,

j j Ul g hl = − sj j j Vl g ll = − sj j j Φl kl = − . (3.51) sj 26 Normal mode analysis

After an infinite elapse of time the total deformation will be represented by the sum of the modal strengths together with the elastic Love number. For the case of a mass load, the dimensionless Love numbers are

U eM he = l L,l R V eM le = l L,l R ΦeM ke = − l , (3.52) L,l gR

in which M is the total mass of the body with radius R and surface gravity g. The index L stands for a load.

Periodical load

Libration and the eccentricity of an orbit generate a periodical variation in tidal force. According to equation 2.42, these variations can be expanded in Fourier series. This forcing has to be convolved with the viscoelastic response in the time domain. The elastic response poses no problem, analogous to the problem of a Heaviside load. The convolution for the viscoelastic response is less straightfor- ward. By using the mathematical properties (e.g. [Scienceworld, 2006]):

L[f(t) ∗ g(t)] = L[f(t)] ·L[g(t)] L−1[F (s) · G(s)] = L−1[F (s)] ∗ L−1[G(s)]

it is shown that the Laplace transform of a convolution is simply a multiplication. Mathematically it is easier to first transform the tidal forcing to the Laplace do- main, multiply with the response and then transform back to the time domain. The Laplace transform for the tidal potential (see equation 2.42) yields   ˜ X ΦT,l(R, s) = L  Glmf cos(fωt˙ ) + Hlmf sin(fωt˙ ) f (3.53) X  s fω˙  = G + H . lmf s2 + (fω˙ )2 lmf s2 + (fω˙ )2 f

The viscous response in the Laplace domain becomes     U˜l M ˜ X ~ j X (fωH˙ lmf + sGlmf )  Vl  (R, s) = Kl (R)  2 2  . (3.54) (s − sj) · ((fω˙ ) + s ) −Φ˜ l j=1 f 3.4 Stress 27

Using standard Laplace tables, this solution can be transformed back to the time domain  ˜  Ul M  sjGlmf + fωH˙ lmf ˜ X ~ j X sj t  Vl  (R, t) = Kl (R) 2 2 e − ˜ (fω˙ ) + sj −Φl j=1 f (3.55) ! sjGlmf + knHlmf fωG˙ lmf − sjHlmf 2 2 cos(fωt˙ ) + 2 2 sin(fωt˙ ) (fω˙ ) + sj (fω˙ ) + sj

The first term is exponential and describes the deformation when a body would be subjected to this periodical load suddenly. Taking into account geological timescales the effect of the initialisation of the periodical function can be neglected. The next terms are periodical and describe phase lag of the response in time. This is visible in the fact that Glmf does not only appear in the cosine term, but also in the sine term. The same counts for Hlmf together with the cosine term. The viscous response of a sinusoidal load is depicted in the right of figure 3.2. The total deformation by a periodical tidal load is:

    U˜l M ˜ ~ e X ~ j X −sj cos(fωt˙ ) + fω˙ sin(fωt˙ )  Vl  (R, t) = K cos(fωt˙ ) + K (R)  Glmf + l l (fω˙ )2 + s2 −Φ˜ j=1 f j l .  M  ~ e X ~ j X −sj sin(fωt˙ ) − fω˙ sin(kωt˙ ) K sin(fωt˙ ) + K (R)  Hlmf l l (fω˙ )2 + s2 j=1 f j (3.56)

Using simple geometric relations the tidal bulge and phase lag can be calculated.

3.4 Stress

In previous sections the deformation was calculated by propagating the stress from the centre to the surface. Here the stress is derived from the displacement.

The strain tensor components are directly related to the displacements [Saba- dini and Vermeersen, 2004]

∂u ε = rr ∂r 1 ∂v  ε = + u θθ r ∂θ 1 ε = (v cot θ + u) ϕϕ r 1 ∂v 1 1 ∂u ε = ε = − v + . (3.57) rθ θr 2 ∂r r r ∂θ

All other shear components are zero because of the axial symmetry. Using the spherical expansions of u and v in equation 3.18, the strain through the whole 28 Normal mode analysis

sphere can be obtained. The expansion to spherical harmonics separates the three coordinate axes of the polar coordinate system: Ul and Vl are only dependent on r, the Legendre function itself is only dependent on θ and since lateral homogeneity is assumed, the solution is axial symmetric and the expansion is independent on ϕ. The strain in spherical harmonics yields

∞ X ∂Ul ε (θ) = P (cos θ) rr ∂r l l=0 ∞  2  1 X ∂ Pl(cos θ) ε (θ) = V + U P (cos θ) θθ r l ∂θ2 l l l=0 ∞   1 X ∂Pl(cos θ) ε (θ) = V cot θ + U P (cos θ) ϕϕ r l ∂θ l l l=0 ∞   1 X ∂Vl 1 Ul ∂Pl(cos θ) ε (θ) = ε (θ) = − V + . (3.58) rθ θr 2 ∂r r l r ∂θ l=0

When incompressibility is assumed, the stress in the Laplace domain can be related directly to the strain:

σii = 2˜µεii (3.59)

Note that the radial components of the stress at the free surface are zero by definition. Consider the stress for a periodical load. In order to keep the equations simple, only the stress in the radial direction for a sinusoidal load is discussed. The Laplace transformed tidal potential is, according to equation 3.53 ω˙ Φ˜ (s) = L (H sinωt ˙ ) = H . (3.60) l l l s2 +ω ˙ 2

The viscous radial deformation in the Laplace domain becomes, according to equa- tion 3.54   ∞ M ~ j X X ∂Kl,1(r) ωH˙ l ε˜rr(θ, s) =  2 2  Pl(cos θ). (3.61) ∂r (s − sj)(ω ˙ + s ) l=0 j=1

With use of equation 3.59, the radial stress becomes   ∞ M ~ j X X ∂Kl,1(r) µs ωH˙ l σ˜rr(θ, s) = 2  2 2  Pl(cos θ). (3.62) ∂r s + µ/ν (s − sj)(ω ˙ + s ) l=0 j=1

The transformation back to the time domain can be performed by a mathematical software package like Mathematica

  2 ! −1 µs 1 ω˙ ω˙ (µ − sjν) sin(ωt ˙ ) − (sjµ +ω ˙ ν) cos(ωt ˙ ) L 2 2 = µνω˙ 2 2 2 2 2 . s + µ/ν s − sj s +ω ˙ (sj +ω ˙ )(µ +ω ˙ ν ) (3.63) 3.4 Stress 29

In this transformation, the exponential terms are left out since they die out rapidly and do not influence the stress strain relation for a periodical load, which acts in geological timescales. For the case of the radial stress this yields  ∞ M ~ j X X ∂Kl,1(r) σ˜ (θ, t) =2 µνω˙ rr  ∂r l=0 j=1 (3.64) 2 !! ω˙ (µ − sjν) sin(ωt ˙ ) − (sjµ +ω ˙ ν) cos(ωt ˙ ) · 2 2 2 2 2 Pl(cos θ). (sj +ω ˙ )(µ +ω ˙ ν )

A similar approach can be used to calculate the other stress components. In case of a delta load or a periodical load of which the period is many orders shorter than the relaxation times, the material can be approximated by an elastic material, and µ˜ of equation 3.59 can be replaced by µ, which simplifies the solution substantially.

Chapter 4 Europa

Europa has some features that makes it unlike other moons. The tidal forces on the moon dominate the processes on the moon because it is located close to the largest planet in our solar system and the orbit has a substantial eccentricity. Furthermore, under the icy surface there might be an ocean that governs the tidal deformation of the upper layer, creating a rich structure of cracks, grooves and ridges.

The goal of this chapter is to establish a connection between the internal struc- ture of Europa and the tidal deformation. The analysis to calculate the tidal de- formation has to be adapted for internal fluid layers. This was already done by numerical shooting methods or by assessing the limit of low rigidity and viscos- ity. [Van Barneveld, 2005] provided a nearly complete analytical method to solve the elastic response for an ocean in the subsurface, but the viscoelastic response imposed some unsolved problems. In this thesis, the problems for the viscoelastic response are solved. Furthermore, minor errors are corrected. Next, the results are validated by comparing the outcome of this model with other literature, and by comparing the model of a liquid ocean with a low-viscous layer. Once the vis- coelastic response can be calculated, it can be assessed whether the phase lag of the tidal bulge gives extra information about the structure of Europa.

The outline of this chapter is the following: first a brief introduction on Europa is given in section 4.1. Next, the potential theory that is given in chapter 2 and 3 is adapted for the case of Europa in section 4.2. The normal mode analysis is modified in section 4.3 and validated in section 4.4 by comparing with other literature and by interpreting the results in a qualitative way. After the model is validated, the sensitivity of the model to variations of parameters is assessed in section 4.5. As a result, it can be seen which celestial bodies cause which deformations in section 4.6.

4.1 Introduction

When the Italian physicist Galileo Galilei pointed one of the first telescopes to- wards Jupiter in 1610, he saw that the planet was accompanied by four spots. He interpreted them as moons and assigned them as JI, JII, JIII and JIV . The Ger- man astronomer Simon Marius, who discovered the moons independently, but only

31 32 Europa

Figure 4.1 Left: the trailing hemisphere of Europa, seen by the Galileo spacecraft. Right: an impression of the composition of Europa [NASA, 2006].

some days later than Galilei, gave the moons the names Io, Europa, Ganymede and Callisto, from Greek mythology. Even before anything of the moons themselves was known, an interesting property could already be distinguished: the revolution time of the three inner Galilean moons follow a striking pattern: they are in an orbital resonance of 1:2:4. During the time that Ganymede completes one orbit around Jupiter, Europa has completed two orbits, while Io completed four orbits. After every cycle, Io and Europa are on one side of Jupiter while Ganymede is on the opposite side. This pattern was already discovered in the 17th century, but only Laplace could explain the cause of this effect in the 19th century, which was named the Laplace resonance. These complex dynamics show how moons can change each other’s orbital radius and eccentricities when they conjugate always at the same place, and eventually get locked in the resonance. This resonance keeps the eccentricities of the moons to a certain value, while Jupiter would oth- erwise damp them out. Until this discovery, it was assumed that the moons had no eccentricity. If that would have been the case, they would be cold and dead worlds like our moon. But even before Voyager, the first mission to Jupiter, it was predicted that the present eccentricity pumps tidal energy into the moons, causing interesting features. The Voyager missions in the seventies confirmed this: volcanic activities are abundant on Io, and for the case of Europa a complex structure of grooves and ridges over the icy surface showed the consequences of tides (see left figure of 4.1).

Besides observing surface features, the Voyagers could measure the potential field of Europa. This was done by measuring the Doppler shift in the radio carrier wave that is generated by the Deep Space Network. This resulted for Europa in a mean density of 3018±35 kg/m3 [Campbell and Synnott, 1985]. Four close encoun- ters by the Galileo spacecraft in 1996 and 1997 gave the possibility to predict the internal structure of Europa. During the encounters it was possible to measure the gravitation field perturbations C2 and C22 using the same Doppler shift technique. Based on the differences of gravity by tides and the rotation of Europa, the nor- 4.1 Introduction 33

. Figure 4.2 Some details of Europa’s surface and the cycle of opening and closing of cracks, pumping up new material to the ridges (Copyright Pearson Education)

malised axial moment of inertia was found to be I/MR2 = 0.346±0.005 [Anderson et al., 1998]. This is less than 0.4, which corresponds to a homogeneous sphere. Therefore the density must vary over the radius of the moon. When this radial difference would be caused only by the presence of the light ice layer, the internal material should be about 3800 km/m3. The most plausible constituents for this high density are a mixture of iron compounds and silicates. The tidal heating and radiogenic decay is sufficient to differentiate the components. A model of a liquid Fe or Fe-FeS core, covered by a rocky mantle and an icy surface are presently most accepted (see right figure of 4.1).

For numerous reasons, it is believed that a liquid layer is present under the icy surface. The ridges and grooves in the surface reveal the first reason. If an ocean is present, the surface is more susceptible to tidal loads. Therefore, the icy shell can be easily broken. In some cases the cracks close again with a substantial vertical displacement. Blocks of crustal ice seem to have floated and drifted over a liquid sea. The ridges that are often formed along a crack are up to 100 metres high [Greenberg, 1998]. The process in figure 4.2 shows how ridges can be formed when an ocean is present: cracks can open and close periodically by tidal deformation. When the crack reopens, the open ocean is filled with debris falling from the shaft and contaminations in the ocean that are lighter than water. When the crack closes, the material is crushed downwards, but a part is pushed to the surface. A quantitative estimate of this process shows that a ridge of 100 metres can be formed on an timescale of 104 years [Greenberg, 2005].

Furthermore, young, large craters like the crater Pwyll are remarkably shallow compared to craters on the moon. If an asteroid would protrude the ice layer, the water would fill the crater and heal the deep wound partially. Also the amount of craters is not as predicted. Europa should be heavily bombarded by asteroids, be- cause it has nearly no atmosphere and it catches many asteroids that are attracted by Jupiter. Nevertheless, impact features are very rare. Based on crater density 34 Europa

and a statistical prediction of the frequency of asteroids, the current surface must be less than 50 million years [Greenberg, 2005]. Images from Galileo show sepa- ration zones where new ice is created and subduction zones where ice disappears under other layers. Europa is a vivid body where the surface shows continental drift and continuously renews its surface.

The dimensions of the ice and ocean were predicted in several ways. By ex- amining how the ice bends under other layers in a subduction zone and assuming material properties of the ice, it was calculated that the ice near Conamara Chaos is about 300 metres thick [Tufts et al., 1997]. This estimate is disputable, since the thickness near cracks can be different than the average thickness. Another estimate was done using the heat dissipation of the moon. The inside of Europa is heated by tidal friction and nuclear degradation. For a thermal equilibrium, this heat is radiated at the surface. With the surface temperature of 100 Kelvins, and an estimate of the temperature gradient, the ice thickness can be predicted. With an average heat dissipation of 20 mW/m2 the thickness of the ice-layer is around 30 km [Hussmann et al., 2002].

Considering the thickness of the possible ice layer, the pressure below the ice can be high enough that water close to the melting point behaves like a slushy material. The layer can also be contaminated by material of the mantle, creating a muddy mixture.

The option for a local ocean is unlikely. When a local ocean is present and it will grow slightly, the tidal deformation will become larger and so is the tidal heating. The tidal heating will melt more ice, causing a cascade towards a global ocean. Similarly, when the ocean would freeze partially, the tidal deformation would become smaller and Europa would slowly cool down to an icy moon. So either a global ocean exists (and has already existed for a long time), or no liquid water is present. Only local processes like volcanic activity could create a local ocean in the subsurface, but such an ocean would be too small to cause extra tidal deformation.

Since the Voyager missions and the Galileo mission observed Europa only dur- ing fly-bys, limited information about the composition of the moon could be gath- ered. With future missions new ways can be addressed to study the icy surface more in detail. A straightforward way is to send a lander to Europa with drilling capacities. In this way material properties can be examined, the thickness can be measured very accurately, and it will even be possible to send a submarine robot under the ice. The disadvantage of such mission is that one lander can study only one spot in detail. Furthermore, landing and drilling on a moon that is barely understood is very risky. For example, it is difficult to predict the depth at which the driller has to dig. A global survey of the surface from a Europa orbiter gives is more interesting for the first mission dedicated specially to Europa. Using radar sounding, the reflection of inner layers can be received. This method is applied on the Mars Express and Venus Express, returning a lot of information about the subsurface. On Europa, contamination in the ice and the cracks will interfere the signal dramatically. This method will learn us a lot about the composition of the subsurface, but not about the rheology of the ice. It is likely that the contami- nation in the ice is too much for the radar to reach the ice-ocean boundary. The method that is described in the next sections assumes that the moon is composed 4.2 Tidal potential of Jupiter, Io and Ganymede 35

Figure 4.3 The tidal bulge on Europa and the relative motion of Jupiter if the tidal bulge had a phase lag of one quarter of an orbit (adapted figure of [Greenberg, 2005])

of homogeneous layers. In this case the relation between the composition of the different layers and the tidal deformation can be modelled.

4.2 Tidal potential of Jupiter, Io and Ganymede

In this section, the potential theory as explained in chapter 2 is used to deal with the potential in the Jupiter system.

Tidal force by Jupiter

As Europa is located close to the largest planet in our solar system, the tidal forces are substantial. The consequences of these tidal forces were and still are dominant for Europa. The torque created by phase lag of the tidal bulge decelerated the rotation of Europa. Within the order of magnitude of 104 years, Europa’s rotation was tidally locked to Jupiter [Greenberg, 1998]. If the orbit had no eccentricity, the moon would deform until a hydrostatic equilibrium is reached. However, as explained in section 4.1, orbital resonances of the other Galilean moons pull Europa into an eccentric orbit. Consequently, the tidal forces of Jupiter vary both in magnitude and direction each orbit. When a reference frame fixed to a moon is chosen, Jupiter seems to revolve counter clockwise around a point at average orbital distance, as shown in figure 2.3.

While Europa is always assumed to be tidally locked, some arguments show that Europa could be rotating slightly faster than its revolution speed. This means that it rotates very slowly with respect to Jupiter. This movement will be ad- dressed as “non-synchronous rotation”. Examining the ridges more carefully, it seems that some patterns are repeated over the same latitude. It looks as if Jupiter leaves the same footprint while Europa moves. The physical mechanism that could drive the non-synchronous rotation is the following: consider the tidal bulge that is formed by eccentricity of the orbit. Viscoelastic response of Europa implies that the tidal bulge has a phase lag. This is shown schematically in figure 4.3. When Jupiter is nearest, the tidal bulge is in a position corresponding to the old location of Jupiter. This makes Jupiter to drag Europa counter clockwise. The same happens when Europa is in apojovium, however, this effect is smaller due to the larger distance then between Europa en Jupiter. The force that pulls counter 36 Europa

Figure 4.4 The variation of the Galilean constellation in two orbits of Europa. The conjunctions that cause the eccentricity of the moons are displayed [Wikipedia, 2005]

mass radius distance to rotational eccentricity Jupiter period M[kg] R[m] r[m] [days] e[−] Io 8.94 × 1022 1815 4.22 × 108 1.769 .0041 Europa 4.80 × 1022 1569 6.71 × 108 3.551 .0093 Ganymede 14.8 × 1022 2631 1.07 × 109 7.155 .002 Callisto 10.8 × 1022 2400 1.88 × 109 16.69 .007

Table 4.1 Data of the Galilean moons

clockwise is larger than the force that pulls Europa clockwise. The net result drags Europa counter clockwise. By observation of the cracks, the non-synchronous rota- tion time of the surface must be larger than 104 years [Greenberg, 1998]. Whether the mantle co-rotates with the surface is unclear. The non-synchronous rotation will be discussed more in detail in section 4.6.

Tidal forces by the Galilean moons

Next to Jupiter the neighbouring moons also generate a tidal force. Their tidal forces are small compared to Jupiter’s and are therefore often discarded. Since future missions to Europa might be able to detect very small deformations, it is important to assess the influence by these moons.

Io, Europa and Ganymede are in a 1:2:4 orbital resonance (see table 4.2 and figure 4.4). When Europa is in apojovium and Io in perijovium, the moons are closest. Similarly, two moons meet when Europa is in perijovium and Ganymede is in apojovium. The tidal variation is caused by relative movements of the moons. When Io is in superior conjunction its tidal force is smallest, whereas the force is largest when it is in inferior conjunction. For the outer moons, the tidal difference occurs between conjunction and opposition. The period of these mutual motions is the period between two successive conjunctions, or an epicycle. In other words, this is the period when the inner moon completed exactly one orbit more than the outer moon, 1 t = , (4.1) 1 − 1 Tinner Touter

where T is the rotational period of the inner or outer moon, and t the time of the epicycle. Examining table 4.2, one interesting feature becomes clear. The epicycles of Europa with Io and Ganymede are 1% shorter than what is expected 4.2 Tidal potential of Jupiter, Io and Ganymede 37

constellation smallest largest epicycle tidal distance distance period variation 2 1 1 r[m] r[m] [days] GMR ( 3 − 3 )[N] rmin rmin

Europa-Jupiter 6.65 × 108 6.77 × 108 3.525 5.76 × 10−5 Europa-Io 2.49 × 108 10.93 × 109 3.525 9.40 × 10−7 Europa-Ganymede 3.99 × 108 1.74 × 109 7.051 3.78 × 10−7 Europa-Callisto 1.21 × 109 2.55 × 109 4.511 1.23 × 10−8

Table 4.2 Data of the Galilean moons

for orbital resonance. In this manner, the conjunctions rotate clockwise in about one revolution per Earth year. Still, the constellation of the moons remains the same, because the perijoviums rotate by the same speed. By this effect the period between two subsequent perijoviums of Europa is also shorter. This effect is well known in orbital mechanics and is often referred to as perihelion precession. This system is very complex for the constellation of the Jovian moons and is still barely understood. How this precession works in detail and how the stability of the Galilean system is guaranteed is beyond the scope of this thesis.

The tidal variation by the Galilean moons can be estimated using the differ- ence between maximum and minimum tidal force. In table 4.2, an overview of the different tidal forces is given. As can be seen, the tidal effects by the Galilean moons are at least 60 times weaker than the tidal effect of Jupiter. The ampli- tude of the variable potential is shown in figure 4.5 for different influences using equation 2.41. The region where the amplitude is largest is not concentrated at the subjovian point, but spread longitudinally. For the case of Jupiter, this is caused by the eccentricity, see figure 2.3. Hence, the largest displacement of the surface will be at coordinates [90◦, ±35◦] and [90◦, ±145◦] (see appendix B.1 for the definition of the spherical coordinates). For the Galilean moons, this has a different cause. While in orbit, Io overtakes Europa, and Europa overtakes on its turn Ganymede and Callisto. Therefore the neighbouring moons do not approach Europa at its subjovian point or antijovian point, but at an angle. Furthermore, the tidal variation is minimum for a region around the coordinates [35◦, 0◦] and [145◦, 0◦]. This is not surprising: at these coordinates the Legendre polynomial P2(cos ψ) equals zero. This point can be a good reference point for an altimetry mission.

Fourier analysis of orbital motion

As mentioned before, the tidal effect of the other Galilean moons is neglected in most other analyses because their magnitudes are negligible compared to the effect of Jupiter. However, the tidal potential of the moons does not vary close to sinusoidal like the one of Jupiter, but shows an interesting pattern. For example, for 72% of the time, Jupiter is closer to Europa than Io. This means that Io stays behind Jupiter for most of the time, and suddenly overtakes Europa in its race around Jupiter. The variation of tidal potential by Io has the form that is depicted in figure 4.6.

From section 3.3 it has become clear that an analytical description of tidal 38 Europa

Figure 4.5 The maximum difference of geoid height during one epicycle for Jupiter, Io, Ganymede and Callisto

Figure 4.6 The variation of tidal potential caused by Io, together with two different partial sums of the Fourier series. 4.3 Normal mode analysis for fluid layers 39

Figure 4.7 The Fourier series of the tidal potential by Jupiter, Io and Ganymede at the subjovian point.

deformation can be constructed when the tidal force is expanded into a Fourier series. Besides the mathematical advantage the model can easily be restricted to a certain degree of accuracy. Figure 4.7 shows the expansion of the tidal potential of Jupiter, Io and Ganymede. While the orbit of Jupiter relative to Europa is nearly circular, the Fourier series of Jupiter converges rapidly. In contrast, the relative movements of Io and Ganymede are less circular and higher degree terms are more dominant. The effects of Io and Ganymede for the first degree are nearly negligible, but they have already a substantial contribution for the second-degree term. Moreover, the influence of Jupiter is completely negligible for higher degrees. Note that the epicycles of the moons are different, and therefore the period 2L that is used in the Fourier analysis is different.

4.3 Normal mode analysis for fluid layers

The normal mode analysis as explained in chapter 3 propagates the surface forces to the centre of a celestial body, and propagates the displacements back to the sur- face. Unfortunately, this model becomes degenerate at low viscous zones, because these zones are unable to propagate the six quantities from equation 3.20. To overcome these problems, extra boundary conditions are introduced to overcome this problem. Two methods to do this are described in [Van Barneveld, 2005].

The first method uses a 4 × 4 matrix to propagate the conditions through the fluid layer. Here, radial stress and displacement are removed, and inertia of the fluid layer is introduced to prevent the fundamental matrices to become singular. However, for the elastic response, the inverse relaxation time has to be set to s → −∞, which creates infinite entries in the fundamental matrix. This makes this approach invalid.

The second method assumes that the boundaries of the fluid layer will be close to a hydrostatic equilibrium. Only the potential and the potential stress are propagated through the fluid layer. The radial stress is derived from the difference between the radial displacement and change of potential rather than using inertia forces. The way how is being dealt with the boundary conditions is the following: the six unknowns are: 40 Europa

K1,K2,K3 at the core, just as in the case of a body without fluid layers,

K4 for the radial offset between the equipotential plane and the crust-fluid boundary,

K5 for the tangential (free) displacement of the crust with respect to the mantle,

K6 for the radial offset between the equipotential plane and the fluid-mantle boundary.

These freedoms are constrained by the following six properties:

At the free surface,τ ˜rr, τ˜rθ and Q˜l are constrained by the external load. These three quantities are zero if no mass load or potential load is exerted on the body. The fluid-mantle boundary produces three analogous constraints as the free surface. In this case,τ ˜rθ must be zero, since the fluid layer cannot propagate shear stress. The radial stressτ ˜rr and potential stress Q˜l are constrained by the fact that they are related by the offset of the fluid-mantle layer from the equipotential.

First, it is shown how the potential can be propagated through a fluid layer, and how. Next, the new boundary conditions for the fluid-solid transitions are composed, and the new constraints are inserted. Finally, the Love numbers can be calculated.

Gravity potential propagation through fluid layers

The Poisson equation is used to propagate the gravity potential for solid layers. Since the mechanical quantities U˜l and V˜l are undefined in the fluid, this approach is invalid in fluid layers. For these layers the mechanical quantities are decoupled and only the gravity potential and potential stress will be propagated. This is done by using the Laplace equation instead of the Poisson equation [Wieczerkowski, 1999]: ∂2Φ˜ 2 ∂Φ˜ l(l + 1) l + l − Φ˜ = 0 (4.2) ∂r2 r ∂r r2 l

The potential stress Q˜l in this propagation is replaced by: ∂Φ˜ Q˜∗ = − l , (4.3) l ∂r

with l + 1 Q˜∗ = Q˜ + Φ˜ − 4πGρU˜ . (4.4) l l r l l

This conversion has to be used when switching from Poisson’s equation to the Laplace equation and vice versa. This happens when a solid-fluid boundary is crossed. Using these relations, the fundamental matrix with the potential and potential stress yields  ˜   l −(l+1)      −Φl r r C1 ˜ i C1 ˜∗ = l−1 −(l+2) · = Hl(ri,s) · . (4.5) Ql lr −(l + 1)r C2 C2 4.3 Normal mode analysis for fluid layers 41

˜∗ Unfortunately, the adapted potential stress Ql is not continuous over two subse- ˜∗ quent fluid layers. Therefore, Ql should be adapted for each internal layer bound- ary by [Wieczerkowski, 1999]

˜∗(i) ˜∗(i+1)  (i+1) (i) ˜(i+1) Ql = Ql + 4πG ρ − ρ Al , (4.6)

where the superscripts i and i + 1 represent the indices of two subsequent fluid layers. For simplicity the ocean of Europa will be modelled as one layer. The conditions through the single fluid layer are propagated by the matrix

−1 f ˜ (i)  ˜ (i)  Bl = Hl (ri, s) Hl (ri+1, s) . (4.7)

Boundary conditions for a fluid layer model

After the propagation through the fluid layer is constructed, the propagation through a celestial body with water layer can be constructed. Consider the general case of a body with N layers. From r1 to rup the body consists of solid layers. The fluid ocean is located between rup and rlo, under which a solid mantle is located. The fluid core is located under rN . This structure is outlined in figure 4.8. Note that [Van Barneveld, 2005] uses a different indexing: there the index of a boundary and the top layer corresponds. Here, the boundary and the layer under it have the same index in order to be consistent with chapter 3. At the free surface the boundary conditions are the same as for a solid body, namely:

 (up−1)   (1)  U˜ (rup) U˜ (r1) l l  (up−1)  (1) V˜ (rup)  V˜ (r1)   l   l   (up−1)   0   τ˜ (rup)    = Bs ·  rr,l  , (4.8)  0  l  (up−1)     τ˜rθ,l (rup)   ˜(1)   (up−1)  −φ (r1) −φ˜ (r ) l  l up  0 ˜(up−1) Ql (rup)

s where matrix Bl is the propagator matrix for the upper (ice) layer. Under the upper solid layers the conditions should match the upper fluid layer,

   φ˜(up)(r )  U˜ (up−1)(r ) l up + K l up g(rup) 4  ˜ (up−1)     Vl (rup)   K5   (up−1)   (up)   τ˜ (r )   ρ g(rup)K4   rr,l up     (up−1)  =  0  .  τ˜rθ,l (rup)       −φ˜(up)(r )  −φ˜(up−1)(r )  l up   l up    ˜(up)  (up−1)  ∗(up) l+1 (up) (up) φl (rup)  ˜ Q˜ (rup) − φ˜ + 4πGρ + K4 Ql (rup) l rup l g(rup) (4.9)

The adaptations for the conditions from solid to fluid can be interpreted in the following way: the fluid tends to form to an equipotential. The crust can force the fluid to deflect from this equipotential. Therefore a constant K4 has to be 42 Europa

Figure 4.8 The indexing of the layers of Europa. Tt is assumed that the four different layers (ice, ocean, mantle, core) can consist of more separate layers. Since little is known about the inner structure of Europa, it will be sufficient for now to assess only a four-layer model.

introduced for the radial displacement and radial stress. Since the upper layer can move freely over the fluid, the tangential stress is zero, and a constant K5 is in- troduced to account for the unconstrained tangential displacement. The potential is the only parameter that can be propagated normally. The potential stress must be transformed because the Laplace equation is used in the fluid, while the poten- tial stress in the solid layers is propagated by the Poisson equation. In the model by [Van Barneveld, 2005] the last row contains −K4 instead of +K4. During the validation process as described in section 4.4 it became clear that the viscoelastic response was not consistent when comparing with the case of a viscoelastic ocean. After checking thoroughly it was discovered that this correction factor of −K4 was inconsistent with equation 4.4 and the first row of equation 4.9. The validation of this method will be discussed more in detail in section 4.4. After propagating over the solid-fluid boundary, the conditions can be easily propagated through the fluid by ! ! −φ˜(up)(r ) −Φ˜ (lo−1)(r ) l up = Bf · l lo . (4.10) ˜∗(up) l ˜∗(lo−1) Ql (rup) Ql (rlo)

A similar conversion has to be made between the fluid layers and the solid mantle     ˜ (lo−1) U˜ (lo)(r ) + K Ul (rlo) l lo 6  ˜ (lo−1)   −   V (rlo)     l   (lo)   τ˜(lo−1)(r )   ρ g(rlo)K6   rr,l lo     (lo−1)  =  0  (4.11)  τ˜rθ,l (rlo)   ˜ (lo)   (lo−1)   −Φl (rlo)  −Φ˜ (r )   (lo)  l lo (lo) (lo) Φ˜  ∗(lo−1)   ˜ l+1 ˜ (lo) l  ˜ Ql (rlo) + r Φl − 4πGρ g(r ) + K6 Ql (rlo) lo lo

Again, the fluid layers want to form an equipotential, but will be forced to de- flect slightly from this case. Therefore the radial displacement and radial stress are dependent on the equipotential with an extra introduced constant K6. The 4.3 Normal mode analysis for fluid layers 43

boundary is frictionless, hence the fluid layer can move freely over the mantle. This requires that the tangential stress is zero. The tangential displacement from the crust is already introduced as constant K5. Since the tangential movement of the water itself is undetermined, the tangential displacement is not propagated here. Finally, the potential and potential stress are converted back in the same manner as the upper fluid boundary.

After passing the fluid layer, the conditions are propagated in the way as described in section 3. At the fluid core, three constants K1,K2,K3 describe the core behaviour:  ˜ (lo)  Ul (rlo)  ˜ (lo)  Vl (rlo)  (lo)  K  τ˜ (r ) 1  rr,l lo  c  (lo)  = Bl · IN,l(rN ) · K2 (4.12) τ˜ (rlo)  rθ,l  K3 Φ˜ (lo)(r )  l lo  ˜(lo Ql (rlo)

Boundary constraints for a fluid layer model

The propagation matrix with all boundary conditions is composed. However, six unknowns are introduced, while the surface boundary constrains the model only for three parameters. The lower fluid boundary layer produces analogous boundary constraints as the surface, but then related to an equipotential boundary, no free boundary, as explained in the beginning of this section.

In this paragraph multiple substitutions are done to cast the previous boundary conditions in a fundamental matrix. Only the constant K4 is eliminated rather than solved: combining the third row of equations 4.11 and 4.12 facilitate this

  K1 (lo) c c c
ρ g(rlo)K1 = (Bl · IN,l(rN ))31 (Bl · IN,l(rN ))32 (Bl · IN,l(rN ))33 · K2 . K3 (4.13)

The constant K6 can hereby be expressed in three other constants K1, K2 and K3. Next, using the property

Φ˜ (lo)(r ) ˜ (lo) l lo Ul (rlo) = , (4.14) g(rlo) 44 Europa

the rows 1, 3 and 5 of equations 4.11 and 4.12 yield the following condition   K1 c c c
0 = (Bl · IN,l(rN ))11 (Bl · IN,l(rN ))12 (Bl · IN,l(rN ))13 · K2 K3 K  1 1 + (Bc · I (r )) (Bc · I (r )) (Bc · I (r ))
· K (lo) l N,l N 31 l N,l N 32 l N,l N 33  2 ρ g(rlo) K3   K1 1 c c c
− (Bl · IN,l(rN ))51 (Bl · IN,l(rN ))52 (Bl · IN,l(rN ))53 · K2 . g(rlo) K3 (4.15)

To determine the eigenmodes, a matrix that relates the five constraints to the external deformation will be made, similar to equation 3.37. In this case, five unknowns are used instead of three, while only the three deformations at the surface are of interest. A characteristic equation has to be formed from the given equations, boundary conditions and constraints. First, the constrained parameters and the unconstrained parameters are separated. For the constrained parameters, equations 4.8 and 4.9 can be rewritten as

  s (up) s (up) s  s  − Bl,31 + ρ g(rup)Bl,33 − 4πGρ Bl,36 · K4 − Bl,32 · K5      s (up) s (up) s s  − Bl,41 + ρ g(rup)Bl,43 − 4πGρ Bl,46 · K4 − Bl,42 · K5   s (up) s (up) s  s  − B + ρ g(rup)B − 4πGρ B · K4 − B · K5 l,61 l,63 l,66 l,62 (4.16)  ˜ (up−1)  Φl (rup) g(rup) R1  (up−1)  = Bl ·  ˜  −Φl (rup) ˜(up−1) Ql (rup)

with 1 0 0 0 0 0 R1 s   B = P1 · B · 0 0 0 , (4.17) l l   0 1 0 0 0 1

P1 is defined in equation 3.32. The ice-water boundary conditions can be expressed in the conditions at the mantle-ocean boundary by making use of equations 4.9 and 4.10, resulting in:    − 1 Bf Φ˜ (lo−1)(r ) + Bf Q˜∗(lo−1)(r ) g(rup) l,11 l lo l,12 l lo R1   d~ = B ·  Bf Φ˜ (lo−1)(r ) + Bf Q˜∗(lo−1)(r )  (4.18) l  l,11 l lo l,12 l lo  f ˜ (lo−1) f ˜∗(lo−1) Bl,21Φl (rlo) + Bl,22Ql (rlo) + X

where the left-hand side of equation 4.16 is denoted by d~ for simplicity of notation and (lo) ! l + 1 4πGρ  f ˜ (lo−1) f ˜∗(lo−1)  X = − Bl,11Φl (rlo) + Bl,12Ql (rlo) . (4.19) rlo g(rlo) 4.3 Normal mode analysis for fluid layers 45

The vector on the right-hand side can be related to the inner core boundary con- ˜ (lo−1) ditions by making use of equations 4.11 and 4.12. The expressions for Φl and ˜∗(lo−1) Ql in equation 4.18 then become   K1 ˜ (lo−1) c −Φl = (Bl · IN,l(rN ))5 · K2 (4.20) K3

and     K1 (lo−1) ! K1 ˜∗(lo−1) c 4πGρ l + 1 c Ql = (Bl · IN,l(rN ))6 · K2 + − (Bl · IN,l(rN ))5 · K2 g(rlo) rlo K3 K3   K1 4πG c − (Bl · IN,l(rN ))3 · K2 g(rlo) K3 (4.21)

c c where (Bl · IN,l(rN ))n denotes row n of the 3×6 matrix Bl ·IN,l(rN c). Combination of equations 4.18, 4.20 and 4.21 yields an expression in which the free surface
T conditions are related to the three constants of the vector K1 K2 K3 . From this expression a matrix Gl,1 can be constructed in such a way that the following relation holds:     G1,11 G1,12 G1,13 K1 ~ d = G1,21 G1,22 G1,23 · K2 . (4.22) G1,31 G1,32 G1,33 K3

From equations 4.18, 4.20 and 4.21 it can be derived that the elements of this matrix are given by

R1 Bv1  f c f  G1,vw = − Bl,11(Bl · IN,l(rN ))5w + Bl,12Yl, w g(rup) R1  f c f  + Bv2 Bl,11(Bl · IN,l(rN ))5w + Bl,12Yl, w R1  f c f  + Bv3 Bl,21(Bl · IN,l(rN ))5w + Bl,22Yl, w (up) ! R1 l + 1 4πGρ + Bv3 − · rlo g(rup)  f c f c Bl,11(Bl · IN,l(rN ))5w + Bl,12 (Bl · IN,l(rN ))6w

(lo) ! 4πGρ l + 1 c 4πG c + − (Bl · IN,l(rN ))5w − (Bl · IN,l(rN ))3w , g(rlo) rlo g(rlo) (4.23)

in which:

(lo) ! c 4πGρ l + 1 c 4πG c Yl,w = (Bl ·IN,l(rN ))6w+ − (Bl ·IN,l(rN ))5w− (Bl ·IN,l(rN ))3w g(rlo) rlo g(rlo) 46 Europa

(4.24)

where 1 ≤ v ≤ 3 and 1 ≤ w ≤ 3. The secular determinant can be formed by adding the two extra unknowns and two extra constraints in the matrix. Using the following substitutions:

s (up) s (up) s Kl,v = Bl,v1 + ρ g(rup)Bl,v3 − 4πGρ Bl,v6, (4.25)

and: c c (B · I (rN )) (B · I (rN )) L = (Bc · I (r )) + l N,l 3v − l N,l 5v , (4.26) l,v l N,l N 1v (lo) ρ g(rlo) g(rlo)

the matrix that relates the constrained parameters at the surface with the five unknowns takes the form:

 0 0 Ll,1 Ll,2 Ll,3  c c c  0 0 (Bl · IN,l(rN ))41 (Bl · IN,l(rN ))42 (Bl · IN,l(rN ))43  s  W1 = Kl,3 Bl,32 Gl,11 Gl,12 Gl,13  . (4.27)  s  Kl,4 Bl,42 Gl,21 Gl,22 Gl,23  s Kl,6 Bl,62 Gl,31 Gl,32 Gl,33

The inverse relaxation times of the system are the non-zero solutions of the secular equation:

|W1| = 0, (4.28)

which can be solved numerically using the bisection algorithm, similar to section 3.2. To construct the matrix that describes the relation between the five unknowns and the displacements at the surface, W1 must be slightly altered:

 0 0 Ll,1 Ll,2 Ll,3  c c c  0 0 (Bl · IN,l(rN ))41 (Bl · IN,l(rN ))42 (Bl · IN,l(rN ))43  s 0 0 0  W2 = Kl,1 Bl,12 Gl,11 Gl,12 Gl,13  , (4.29)  s 0 0 0  Kl,2 Bl,22 Gl,21 Gl,22 Gl,23  s 0 0 0 Kl,5 Bl,52 Gl,31 Gl,32 Gl,33

0 where the terms Gl,vw are constructed in the same way as in equation 4.23, but R1 Bvy is replaced by 1 0 0 0 0 0 R2 s   B = P2 · B · 0 0 0 . (4.30) l,1 l   0 1 0 0 0 1

Combining W1 and W2 gives the complete relation between the load and the deformation:   U˜l ˜ −1 ~  Vl  (R, s) = P35 · W2(R, s) · W1(R, s) · P53 · b. (4.31) ˜ −φl 4.4 Validation of the method 47

Two extra matrices are introduced to get rid of the rows and columns that represent the boundary layer unknowns and constraints. They are defined as

0 0 0   0 0 0 0 0 1 0 0   P53 = 1 0 0 , P35 = 0 0 0 1 0 . (4.32)   0 1 0 0 0 0 0 1 0 0 1

After the solution is transformed back from Laplace domain to time domain, the solution has the form

 U˜  M l K~ j(R) ˜ ~ e X l  Vl  (R, s) = Kl (R) · + . (4.33) s − sj −Φ˜ l j=1

The matrix describing the elastic response is formed by:

e −1 K~ = P53 · lim W2 · (W1) · P35. (4.34) l s→−∞

In order to prevent numerical problems in the program, the elastic response can also be modelled by replacing the compliance for the rigidity by the normal rigidity. Analogous to chapter 3, the matrix that describes the viscoelastic response is formed by

† ! ~ j W2 · (W1) Kl = P53 · d · P35. (4.35) det(W1) ds s=sj

† −1 By rewriting the viscoelastic response matrix in terms of W1 rather than with W1 (see [Van Barneveld, 2005], the problem of having singular matrices is avoided.

4.4 Validation of the method

Before results of the model are produced and interpreted, it is important to test the validity of the method and the software that is used. When assuming only viscoelastic layers without ocean, the model does nothing new compared to existing software to calculate the Love numbers of the Earth. These software packages can be used as benchmark. Inserting different Earth models in the program showed the same Love numbers and relaxation times as other software, so that program errors in these routines are unlikely.

Several science groups calculated the elastic response of Europa with a liquid ocean [Yoder and Sjogren, 1996], [Moore and Schubert, 2000], [Tobie and Mocquet, 2005] and [Hurford, 2005]. When possible, their results are compared with this model. It was more difficult to find good comparison for the viscoelastic response. The viscoelastic response is normally envisaged as a phase lag on the tidal bulge. But since this tidal bulge consists only of the elastic tidal response, these models are very inaccurate. More complicated models decompose the tidal force into several 48 Europa

different tidal bulges [Greenberg, 1998], rotating around Europa with different magnitudes, rotational speeds and directions. To account for viscous effects every tidal bulge has an extra secondary bulge that rotates a certain angle behind the main tidal bulge. Although this model shows some insight for the response of a viscoelastic body, this model also does not show the quantitative relation between the composition of Europa and the tidal deformation. These models are difficult to compare with the results in this chapter, since the normal mode analysis computes the lag of the tidal bulge in the time domain, not in terms of rotational delay. A recent paper shows the viscoelastic response using normal mode analysis, but how the fluid layer is dealt with is not mentioned [Tobie and Mocquet, 2005]. Therefore, the viscoelastic response is validated by comparing the results of a fluid ocean with a viscoelastic “ocean”.

Elastic response

Different assumptions have been tried to overcome the problem of internal fluid layers. [Yoder and Sjogren, 1996] modelled the ocean as a solid layer for low rigidities. For a real fluid, the rigidity must be equal to zero. The fluid limit was not evaluated in the research of [Yoder and Sjogren, 1996]. Since a lot of parameters in the model (e.g. density and rigidity of the mantle, rigidity of the surface ice) are not specified in this reference, the results are almost impossible to compare.

The fluid limit was described more in detail by [Hurford, 2005]. The solution for h, k and l were solved partially numerically, partially analytically using Math- ematica. The rigidity of the ocean was kept as unknown. The solution of the Love numbers for small values of µocean were evaluated concerning convergence to a limit at µocean = 0. In some cases, the solution diverged into noise, caused by either numerical errors or singular matrices. Instead of using the Love numbers for µocean = 0, the Love numbers were predicted using the extrapolation of the regime of µ where the Love numbers were more realistic.

A model that treats the ocean really as an fluid layer, is described by [Moore and Schubert, 2000]. The propagation technique by [Wolf, 1994] is used. Although the basic set up of the method is similar, this technique is different on four ma- jor points: firstly, the mechanical quantities are related to the isopotential field and the gravitational quantities are related to the local reference frame. The use of isopotential and local fields has decoupled the mechanical quantities from the gravitational quantities, resulting in a 4 × 4 propagation matrix and a separate 2 × 2 matrix. Secondly, the Fourier transform is used to simplify the Maxwell model, rather than the Laplace transform. Thirdly, unknown constants are not retrieved using the inverse of the matrix propagation, but using a shooting tech- nique. Finally, the extra constraints and unknowns are introduced differently. For each fluid-solid boundary, two constraints (zero shear stress and continuity of nor- mal stress) set the two unknowns (radial and tangential displacement). How this is done in detail is not described.

11 The results by [Moore and Schubert, 2000] are only for µmantle = 10 P a. They match the result by [Hurford, 2005] by less than one percent, which makes it unnecessary to consider them further. In figure 4.9, only the comparison with 4.4 Validation of the method 49

Figure 4.9 Elastic Love numbers as function of the ice thickness in kilometres for different rigidities of the mantle and ice. The solid line represents the results by [Hurford, 2005] and the dashed line represents the method of this thesis.

Hurford and this model is depicted. The method in this thesis follows the results of Hurford but a certain offset that is independent on the ice thickness. For 11 10 µmantle = 10 P a the offset is only 0.01, but the offset for µmantle = 10 P a has grown to 0.06. When the rigidity of the mantle is decreased by a factor of 10, the Love numbers increase roughly 0.06 for Hurford’s method, while the method of this thesis shows an increment of 0.12. Apparently, the mantle is more constrained by its upper fluid layer in Hurford’s method. Which of the two models is best is impossible to decide without comparing more results.

Viscoelastic response

As explained before, no real viscoelastic method for a body with an ocean under the surface exist. The only qualitative way to validate the model is by comparing the case of a liquid ocean with the case of a viscoelastic “ocean”. The viscosity of the ocean will be varied in order to see if the liquid ocean matches the limit of infinitely small viscosity. The comparison between liquid ocean and viscoelastic ocean will only be shown for the second degree deformation. The model of Europa as described in table 4.4 is used for the comparison as well as for the next chapter, where the sensitivity of the method is discussed. The varying of the viscosity of the ocean has not only a theoretical purpose: since the ocean can be slush rather than a liquid, the influence of ocean viscosity can be assessed.

The model of Europa contains a liquid core, a viscoelastic mantle and an ocean covered with an ice layer. Since this model has a lot of uncertainties, the sensi- tivity to variations are discussed in section 4.5. Using the rules in section 3.2, the 50 Europa

layer radii density rigidity viscosity state r[m] ρ[kg/m3] µ[P a] η[P a · s] 1 1.569 − 1.549 × 106 950 1010 1013 solid 2 1.549 − 1.399 × 106 1000 109 - 1010 - solid fluid 3 1.399 − 0.588 × 106 3300 1011 1021 solid 4 .588 − 0.0 × 106 8000 - - fluid

Table 4.3 Model of Europa which is used to test the validity and sensitivity of the model.

Figure 4.10 Relaxation times as function of ocean viscosity.

model with liquid ocean contains four buoyancy modes: one is contributed by the surface and three by the density jumps. From surface to core, they are tagged M0, M1−2,M2−3 and C0 respectively. Moreover, the viscoelastic ocean model contains four extra transition modes, two for each jump in Maxwell time. They are tagged T1−2 and T2−3, corresponding to their neighbouring layers. The pair of transition modes have the same index, because no distinction can be made in the output between a pair.

The relaxation times are shown in figure 4.10 as function of the viscosity of the ocean. In order to understand the mechanism and the role of different modes, the modes should be identified. The identification of the modes can be established in the following way: it is possible to neutralise an eigenmode by changing the model parameters. By equalising the density of two subsequent layers, this particular mode disappears from the figure. Similarly, a pair of transition modes drops out when two subsequent layers have the same Maxwell relaxation time. The latter 19 12 effect is visible in figure 4.10 for ηocean = 10 P as and ηocean = 10 P as. In this way, all buoyancy modes and pairs of transition modes could be retrieved. The identification seems logical: the two modes that are triggered by the two lower layers are only influenced by the viscosity of the ocean at high viscosities, while 4.4 Validation of the method 51

Figure 4.11 Relaxation time and modal strength as function of the degree

the two upper buoyancy modes are more influenced at lower viscosities. In general, as the viscosity is decreasing, Europa responds faster to loads, caused by shorter relaxation times.

Higher degree effects are another identification for the different buoyancy modes. Generally, the mode that is triggered by the core decreases rapidly for higher degrees. This effect can be seen in the right figure of figure 4.11. The modes belonging to the ice layer shows the opposite: for higher degrees, the modal strength is increasing. This might have a relation with the fact that the ice can easily form itself to different shapes. Higher degree deformations will only be in- teresting in the future. After a new visit to Europa the surface can be studied more in detail. The surface deformation by local differences in ice thickness and viscosity can be modelled by considering the abundance of ice as surface loading. The dimension of the spatial details determine up to which harmonic degree the deformation should be modelled. It should be noted that the normal mode analysis assumes lateral homogeneity, thus effects of lateral heterogeneity will be neglected by this analysis. Numerical software is better equipped to model a heterogeneous surface.

By decreasing viscosity, the propagation matrices become more and more sin- gular. The modes that are contributed by the inner layers become gradually im- possible to retrieve. An interesting fact is that the modes corresponding to the lower layers are the first to be not retrieved by the model, while the mode that corresponds to the surface is retrievable for all viscosities. The relaxation times of the lower layers are very long compared to the ones of the upper layers, thus the inverse relaxation times sj are smaller. The derivative of the determinant in equation 3.41 is more susceptible for numerical errors when sj is small. The four modes in the fluid case have a good match with the modes of the viscoelastic case, which is a good measurement for the validity of the model.

In figure 4.12, the strength of the modes corresponding to the model above are depicted. The mode corresponding to the surface shows a very good relation between the fluid case and the viscoelastic case. The values are more difficult to compare for the modes by the mantle and core since the viscoelastic model cannot 10 find relaxation times for ηocean < 10 P as. For the case of diurnal tides the relaxation times are several orders of magnitude longer than the revolution time of Europa making their contribution negligible. However if Europa is rotating non-synchronously relaxation times can be of interest. 52 Europa

Figure 4.12 Strength of modes as function of ocean viscosity: a comparison of 8 modes.

Figure 4.13 Strength of modes as function of ocean viscosity: a comparison of the buoyancy modes, tidal Love number, and total deformation. 4.4 Validation of the method 53

9 Figure 4.14 Strength of modes as function of ocean viscosity for ice with τM = 10 s.

19 Interesting features are the jumps and drops in the lines. For ηocean = 10 P as the transition modes corresponding to the water-mantle transition disappear be- cause both layers have the same Maxwell relaxation times. Similarly, the transition 12 modes corresponding to the ice-water boundary disappear for ηocean = 10 P as. 17 The peak for M1−2 at ηocean = 10 P as is difficult to connect to a physical effect. It is probably caused by the fact that the relaxation times of M1−2 and M2−3 are almost equal there.

In figure 4.13, 4 buoyancy modes are depicted together with three other lines of interest. A very obvious line is the one for the tidal Love number. This shows no relation with the variation of viscosity, since its value represents the instant deformation. The jump from viscoelastic model to inviscid model can be explained by the following: in figure 4.10 the transition modes become infinity fast for the low-viscous limit. This means that the residues corresponding to these modes are relaxed in a very short time. Since the fluid case can be interpreted as the case with infinitely small viscosity, the relaxation times become infinitely short, hence they become an instant response. For this reason the term “quasi elastic response” is introduced, which represents the sum of the tidal Love number and the strengths of the transition modes. This line matches the load Love number for the fluid case. Finally, the sum of all strength of modes and the tidal Love number is shown in the figure. It should be noted that the total deformation is only depicted if the program finds all modes, otherwise the value is not representable. The variation of ocean viscosity does not effect this total deformation, because after an infinite time interval, all layers are deformed to a hydrostatic equilibrium. The viscosity and the rigidity only change the instant when this equilibrium occurs, not the amount of deformation. This conclusion implies that the method to find the normalised 54 Europa

axial moment of inertia by observing the deformation [Kaula, 1968] is also valid for bodies with fluid layers.

The influence of a slushy ocean

Concerning the regimes of different ocean viscosities the physical consequences of a slushy ocean can already be discussed. In order to predict the viscosity of the ocean it is interesting to see where the transition occurs from viscoelastic 3 behaviour to pure fluid behaviour. For the case of τM,ice = 10 s the mode M0 is 2 −1 the governing factor for τM,ocean > 10 s. Only below τM,ocean = 10 s the elastic deformation is dominant over the viscoelastic responses of the ice layer. Whether the viscosity of the ocean is a dominating factor is highly dependent on the Maxwell relaxation time of the ice surface. For a high viscous ice layer, the ocean must have a substantially higher viscosity in order to have effect in the viscoelastic response 9 of Europa. According to figure 4.14 the transition for τM,ice = 10 s occurs at 5 7 10 < τM,ocean < 10 s. As a rule of thumb the ocean can be regarded as fluid if the Maxwell relaxation time of the ocean is 3 orders of magnitude lower than that of the ice layer. In reality the temperature gradient in the ice will cause a variation of viscosity over the depth. As a consequence, the surface ice will behave more elastically that the inner ice. The ocean will be regarded as fluid in the next sections.

4.5 Sensitivity to changes in Europa’s rheology

After the model is validated, the relation between the potential load and the tidal deformation can be calculated. However, considering the large range of parameters that can be varied it is impossible to retrieve all properties of Europa by only the deformation.

Therefore, different parameters are firstly assessed separately in order to see how strong they influence the deformation of Europa. This is not always possible. For instance, when the ocean thickness is increased, the mantle density should be increased to correct the global density, and the core size should be decreased in order to keep the axial moment of inertia at the same level. It is preferred to change only some parameters in order to isolate cause and effect of the tidal deformation. The Europan model is then no longer in accordance with the observed data of Europa, but this is not of concern at this phase.

Sensitivity to ice thickness and ocean thickness

The relaxation times and deformation as function of ice thickness and ocean thick- ness are depicted in figure 4.15 respectively. For the case of Europa without an ocean two extra transition modes are triggered by the different Maxwell relaxation times of the ice and the mantle. Since these modes are not comparable with the case of a fluid ocean, only M0 and M1−2 are depicted. The values that showed numerical errors are filtered out. According to the graphs the variation of ice thickness is dominant compared to the variation of ocean thickness if an ocean is 4.5 Sensitivity to changes in Europa’s rheology 55

Figure 4.15 The two shortest relaxation times and the corresponding strength of modes as function of ice thickness and ocean thickness. The surface has the shortest relaxation time; the ice-ocean transition is slower. The elastic deformation is almost independent on the ice thickness, but highly dependent on the presence of an ocean. The contribution of the surface mode is larger than the contribution of the ice-ocean boundary.

present. Apparently, the ocean gives a certain freedom to the ice shell, indepen- dent on the ocean thickness. The only requirement is that the ocean is sufficiently deep to prevent the ice from touching the mantle. Note that the normal mode analysis assumes small deviations from a stratified body and becomes unrealistic for very thin oceans, because it cannot deal with possible ice-mantle interactions.

For the case that no ocean is present, the elastic response is about 30 times less due to the absence of a fluid that deforms almost instantly. The viscoelastic response is surprisingly very similar to the case with an ocean. The contribution of the different layers is independent on the presence of the ocean, except for one contribution: the relaxation time of the surface has become very short. In this case, no relaxation times are in the order of magnitude of the orbital rotation of Europa, which is about 10−2 years. Therefore, Europa without an ocean will have a very small phase lag. This will be discussed more in detail in section 4.6.

As mentioned before in section 4.1, a local ocean is unstable and therefore impossible. Considering the large effect of the presence of an ocean this will be the first distinction that can be made with an altimeter mission to Europa. After confirming or rejecting the possibility of an ocean, the details about the ice thickness and ice rheology will be determined.

Sensitivity to rigidity and viscosity of ice

As shown in previous sections, properties of ice can vary over a large range, de- pendent on the amount of pollution and porosity. Especially the viscosity is very uncertain and is still poorly understood. Figure 4.16 show the effect when the ice properties are varied. The following relations can be observed:

The relaxation time contributed by the surface is proportional to the Maxwell relaxation time of the ice. The modal strength of the surface is inversely proportional to the rigidity. The relaxation time triggered by the ice-ocean boundary is proportional to the ice viscosity. 56 Europa

Figure 4.16 Relaxation time and modal strength as function of ice rigidity and viscosity. The core and mantle have the longest relaxation time; the surface has the shortest. The elastic deformation has the largest contribution, followed by the mantle, the core and finally the ice-ocean boundary. The strength of mode of the surface is highly dependent on the rigidity.

The viscosity has no effect on the strength of the modes.

The Maxwell relaxation time of the ice is very close to the relaxation time of 9 the surface, albeit with a distinguishable offset. For µice = 10 P a the surface relaxation time is only 1.7% longer; this offset is increased to 17.8% for µice = 1010P a. Apparently, the deformation of the ice is more constrained by the rest of the model at high ice rigidities. The offset is an important outcome, since many scientists related the phase lag on the rigidity of the ice.

Sensitivity to changes in lower layers

As mentioned before, the mantle and core, or the lowers layers in general, will have a too large relaxation time to contribute to a phase lag for diurnal tides. Therefore, a fluid core is assumed in previous sections to reduce the complexity. Here their behaviour is studied in order to get more insight in the normal mode analysis.

The normalised axial moment of inertia of Europa was used to put constraints on the material profile [Anderson et al., 1998]. Three different core properties are suggested: solid or Fe-FeS eutectic or a solid Fe core. When the ocean thickness is held constant only one solution exists for the core radius and mantle density in order to match the measured density and axial moment of inertia. The three different configurations are shown in table 4.5.

The relaxation times and modal strength are shown in table 4.5. Two extra transition modes occur for case 2 and 3. Considering their strengths and relaxation times, they are almost negligible for all geological processes. The buoyancy mode that is triggered is very different from the fluid core case. This is to be expected by the difference in densities of the core and mantle, which are the same for case 1 and 2. The effect of M2−3 is larger for a liquid core. In general a solid core restricts the mantle more from moving than a liquid core. The consequences of different cores are discussed at the end of section 4.6. 4.6 Results 57

layer radii density rigidity viscosity state r[m] ρ[kg/m3] µ[P a] η[P as] Case 1: liquid Fe-FeS core Ice 1.569 − 1.549 × 106 950.0 1010 1013 solid Water 1.549 − 1.449 × 106 1000.0 - - fluid Rock 1.449 − 0.704 × 106 3300.0 1011 1021 solid Fe-FeS .704 − 0.0 × 106 5150.0 - - fluid Case 2: solid Fe-FeS core Ice 1.569 − 1.549 × 106 950.0 1010 1013 solid Water 1.549 − 1.449 × 106 1000.0 - - fluid Rock 1.449 − 0.704 × 106 3300.0 1011 1021 solid Fe-FeS .704 − 0.0 × 106 5150.0 109 1030 solid Case 3: solid Fe core Ice 1.569 − 1.549 × 106 950.0 1010 1013 solid Water 1.549 − 1.449 × 106 1000.0 - - fluid Rock 1.449 − 0.496 × 106 3315.0 1011 1021 solid Fe .496 − 0.0 × 106 8000.0 109 1030 solid

Table 4.4 Three configurations of Europa with different cores.

Case 1 Case 2 Case 3 mode rel. time. h2 rel. time. h2 rel. time. h2 [kyrs] [kyrs] [kyrs] elastic 1.0808 1.0807 1.0766 1 3.711 × 10−8 0.1851 3.711 × 10−8 0.1851 3.713 × 10−8 0.1848 2 9.323 × 10−4 0.0027 9.323 × 10−4 0.0027 9.346 × 10−4 0.0027 3 14.147 0.7606 103.33 0.2817 62.050 0.0272 4 209.4 0.0144 1.490 × 1011 0.0210 1.039 × 1011 0.0066 5 5.107 × 1013 0.0508 6.501 × 1013 0.0082 6 1.001 × 1016 0.0854 2.708 × 1014 0.0080

Table 4.5 The relaxation time and strength of modes for the three cases from table 4.5.

Conclusions

The difference between a global ocean and no ocean is substantial, however the thickness of the ocean is barely observable. The relaxation time and strength of modes are largely dependent on the ice rheology.

Since the sensitivity to various different material properties is known, the tidal deformation can be discussed by only varying the parameters that have a large influence. When more is known about the surface it is possible to assess, for instance the effect of a layered ice structure from cold, elastic ice at the surface to warm slushy ice below.

4.6 Results

After the model is validated for realistic rheologies, the response can be combined with the real tidal potential in order to find a relationship. Furthermore, the con- 58 Europa

Figure 4.17 Four graphs that relate the tidal deformation and phase lag at the subjovian point. The upper left graph shows the phase lag as function of ice thickness and ice rheology; the upper right graph shows the deformation as function of the same properties. The graphs below the inverse of the upper graphs: here the ice properties are shown as function of the deformation and phase lag. For the viscosity of ice, only values of 1011 < η < 1016P as are shown, while the results for deformation and phase lag are nearly constant beyond this interval.

nection between the deformation of the mantle and the possible non-synchronous rotation is discussed.

Determining the tidal bulge

The elastic and viscoelastic response can be combined according to section 4.3 to calculate the total tidal bulge and its phase lag. For a first insight the tidal deformation in the subjovian point is taken. According to figure 4.5 the maximum variation of tidal potential is not located at the subjovian point, but at the loca- tions with coordinates [θ = 90◦, ϕ = ±35◦]. This is unimportant for the general picture of phase lag and amount of deformation. Furthermore, only the primary Fourier component, which is governed by Jupiter’s relative cyclic motion is taken. The result is depicted in figure 4.17. Note that the phase lag that is defined here is in the time domain, not in terms of angular displacement (e.g. [Moore and Schubert, 2000]). 4.6 Results 59

µ[N/m2] η[P as] Maxwell maximum relaxation time [years] phase lag [days] 1.0 × 109 6.9 × 1013 2.2 × 10−3 .024 3.0 × 109 1.0 × 1014 1.1 × 10−3 .076 5.0 × 109 1.9 × 1014 1.2 × 10−3 .15 1.0 × 1010 5.1 × 1014 1.6 × 10−3 .19

Table 4.6 Maximum phase lags for different rigidities of ice

Most aspects that are shown in this figure seem logical. First, as the ice thickness is increased, Europa is more rigid and the deformation is decreased. Simultaneously the phase lag is larger because the viscous response dominates 13 the elastic response. Second, for low viscosities (ηice < 10 P as), the ice has a short relaxation time and Europa will show a tidal bulge of almost 30 metres, almost independent on the ice thickness. The surface will deform to a hydrostatic equilibrium rapidly and therefore almost no phase lag is visible. The viscous response is too slow for high ice viscosities and the elastic response is dominant. The thickness of the ice becomes therefore much more important.

Table 4.6 shows the maximum phase lag for different ice rigidities. The max- imum temporal phase lag for a realistic Europan model is 0.194 days, or 5.5% of the orbital revolution time. According to this table the Maxwell relaxation time of the ice for maximum phase lag is not equal to the orbital period of Europa, which was claimed by [Moore and Schubert, 2000]. Instead, the Maxwell relaxation times are about 4 to 10 times shorter.

The upper graphs in figure 4.17 predict the ice thickness and ice rheology implicitly: for a certain configuration of Europa the deformation and phase lag are determined. The inverse is preferred: what kind of ice can be expected when a certain deformation and phase lag are observed? The lower graphs in the same figure show this inverse. Here, the regimes of viscosity, rigidity and thickness are related to the observable properties. Unfortunately, these graphs do not provide an explicit solution. Especially for large deformations and small phase lags many different ice properties are possible. The situation is better when the deformation is less. This fact became already clear from the upper graphs: in many situations the ice deforms about 30 metres. An overview of the solutions is given in figure 4.18

Based on only the tidal deformation of the primary component of Jupiter’s tidal variation, the ice parameters cannot always be found explicitly. It is therefore interesting to evaluate the influence of higher degrees of the Fourier coefficients of the tidal variation.

The effect of higher degree Fourier coefficients

The first degree Fourier coefficients represent the tidal variation with a period sim- ilar to the tropical orbital period. It is uncertain of how much use this analysis is, because only particular ice rheologies show an interesting result for this frequency.

In section 4.2 it is shown that the influence by the other Galilean moons is substantial for higher degree Fourier coefficients, and they will therefore be taken 60 Europa

Figure 4.18 A schematic overview that shows the relation between deformation, phase lag and ice properties.

Figure 4.19 Temporal phase lag and deformation as function of the viscosity of ice for 100 km ice, µ = 1010P a and for different Fourier coefficients.

into account. For comparison with the first Fourier coefficient the situation with 10 maximum deformation and phase lag is taken, namely µice = 10 P a and ice thickness 100 km. The result is depicted in figure 4.19.

The phase lag and deformation appears to have the same shape for the whole Fourier series, except for the magnitude and a shift in ice viscosity. The phase lag for different Fourier series follows a clear pattern

degree of coefficient ∝ (maximum temporal phase lag)(4.36)−1 degree of coefficient ∝ (viscosity of ice at which maximum phase lag occur)−1 (4.37)

When the period of the tidal variation changes, the Maxwell relaxation time of the ice for maximum phase lag changes proportionally. Similarly, the temporal phase lag also changes proportionally. This can be explained by the fact that the phase lag remains the same when expressed in a percentage of the period of tidal variation, see figure 4.20. 4.6 Results 61

Figure 4.20 A comparison between two frequencies in order to illustrate the different between absolute and relative phase lag. Although the phase lag in the right picture is half the phase lag of the left picture, the relative phase lag is the same, since the period twice as short.

The deformation by the second degree Fourier coefficient is about 1 metre, which may be still measurable. But in most cases the deformation by this co- efficient is very similar to the primary deformation, as shown in figure 4.17. In general, this contribution to tidal deformation will only provide new informa- tion about the structure of Europa if the temporal phase lag is maximum, or 13 14 6.9 × 10 < ηice < 5.1 × 10 P as, see table 4.6. When considering all assump- tions made in the normal mode analysis (incompressible, homogeneous layers of constant thickness), the effect of the Galilean moons is negligible compared to the inaccuracies of the model.

On the non-synchronous rotation of Europa

In the previous sections the deformation caused by the upper layers of Europa played a superior role. Next to the high modal strengths of the upper layers, the relaxation times are close to the orbital period of Europa. When the orbital period and relaxation time match, the phase lag is maximum. One mechanism where the mantle and the core could be dominant is the non-synchronous rotation. If Europa would have a slow rotation with respect to Jupiter, the mantle and the core are forced to deform according to this movement. If the mantle and core would show a phase lag, the upper layers will deform likewise since the lower layers disturb the gravity potential. The phase lag might then be observable form an orbiter around Europa. If no phase lag is observed two options are possible. Either the lower layers have not the required properties to cause a phase lag, or the lower layers might be totally separated from the ice surface. In this case only the ice surface could rotate with respect to Jupiter, while the mantle and core remain tidally locked in Jupiter’s gravity. The latter explanation is not unlikely. If an ocean is present, the diurnal tidal bulge is mainly caused by the flexibility of the ice and ocean, while the mantle and core are too sluggish to respond to a low periodical variation, as explained in section 4.5. Taking into account the different relaxation times of the different layers, only the surface experiences a substantial phase lag by the diurnal tide. The torque created by Jupiter is therefore only applied on the ice layer, not on the core and mantle (see section 4.2). The mantle and core will rotate synchronously, unless they are forced by the surface to rotate non-synchronously.

In order to quantify the phase lag by non-synchronous rotation the phase lag 62 Europa

Figure 4.21 The phase lag and amount of deformation as function of the non-synchronous rotation period.

will be described in geometrical degrees, in contrast to the temporal phase lag that was used to describe the viscous response by diurnal tides. This has two reasons. First, the non-synchronous rotation of Europa is assumed to be constant. The tidal bulge would rotate over the surface of Europa with a constant speed, similar to the rotation speed with respect to Jupiter. Assuming a counter clockwise rotation, the tidal bulge will be located to the east of the subjovian point. Second, the non- synchronous rotation time is about three orders longer than the duration of an orbital mission. As a conclusion, the phase lag will be measured in longitudinal degrees.

The phase lag and deformation are depicted in figure 4.21. In case of a liq- uid core, the phase lag is about 9 degrees, thus clearly observable. The assumed minimum rotation time of 104 years is about an order of magnitude lower than the regime of rotation times where maximum phase lag occurs. Despite the fact that the phase lag will not be distinguishable for this rotation period, another interesting aspect would occur. If the rotation time of Europa is substantially faster than the relaxation time of the central layers, the body is not in hydrostatic equilibrium, which is assumed when estimating the axial moment of inertia [An- derson et al., 1998]. To cause a deformation that is observed by the spacecraft Galileo, the normalised axial moment of inertia should be larger than the predicted I/MR2 = 0.346.

If the latter would be the case, this would change the whole adopted model of Europa: a different axial moment of inertia can only be caused by a differ- ent density gradient, putting new constraints to the materials properties and the corresponding viscoelastic response. Chapter 5 Phobos

Phobos is a small irregular moon, situating very close to Mars. The history of Phobos is wrapped in mysteries. The main questions are how it ended up in its present orbit and what has caused the surface features on Phobos. In this chapter some theories on Phobos will be summarised and discussed using normal mode analysis.

This chapter is built up in the same manner as chapter 4. First a general introduction is given about the moon. The outline of this section is based on a literature study [Kleuskens, 2006]. Next, the potential theory and normal mode analysis are applied for this case. Using these theories, new insight about the history of Phobos can be deduced. Material properties can be found using the long-term deformation of the moon. Finally, the different causes of the forming of grooves are assessed.

5.1 Introduction

Although several astronomers and science fiction writers predicted the existence of tiny moons around Mars (e.g. by Kepler in the 17th century and by Jonathan Swift in the novel “Gulliver’s Travels”, published in 1726), it was Asaph Hall who discovered two small moons around Mars in 1877, and named them Phobos and Deimos. Phobos and Deimos are the sons of Ares and Aphrodite, the Greek ver- sions of Mars (the god of war) and Venus (the god of love). Phobos and Deimos are the personifications of fear and dread [Wikipedia, 2005]. Soon after the discovery, Asaph Hall could already distinguish a large crater on the surface of Phobos. He called it Stickney, after his wife.

The only interesting data about Phobos that could be observed before the space age was its orbit. Phobos revolves around Mars in only 7 hours and 40 minutes, even faster than the rotation speed of Mars itself (24 hours, 40 minutes). In practice this means that Phobos does not rise on the east and sets in the west like most other celestial objects, but travels from west to east almost twice per day. Furthermore, measurements between 1877 up to now showed that the revolution speed slowly increases. It followed Kepler’s law by decreasing its orbital distance to Mars. The reason for this orbital decay is the following: even though Phobos is very small, its low gravitational force deforms Mars slightly. Since Phobos revolves

63 64 Phobos

Figure 5.1 The mechanism that causes the orbital decay of Phobos.

around Mars faster than the rotation speed of Mars, the tidal bulge on Mars lags behind the location of Phobos, see figure 5.1. The extra gravitational force by the tidal bulge has a component in the opposite direction compared to the velocity vector of Phobos. As a result, the moon is decelerated in its orbit and the radius of the orbit will decrease. As the orbital speed will become only faster, the orbit becomes a spiral motion that eventually comes to a disastrous end. Based on the past measurements and predictions for tidal deformation, Phobos is expected to crash on Mars in about 32 million years (see section 5.4).

But before this catastrophic event takes place, another mechanism will strike first. With a present distance between Mars and Phobos of 9377 km the tidal forces by Mars are substantial. [Roche, 1850] formulated the equilibrium figures for ellip- soids rotating around a parental body. He proved that tidal forces disrupt a body when comes too close to another body. Phobos is already positioned inside the classical Roche limit, which means that it would already have been disintegrated if the moon would have a low viscosity. Apparently, Phobos is substantially viscous to stay intact for the present situation. Details on the Roche limit are described in section 5.4.

Surface

Detailed research on Phobos could only start during the space age. When the US spacecraft Mariner 9 had to wait for a storm on Mars to pass in 1971, it aimed its instruments to Phobos and sent back the first close up pictures. The first really detailed pictures were taken by the Viking 2 orbiter in 1975. The USSR made two attempts to visit Phobos and even penetrate the surface, but both missions failed. Only the spacecraft Phobos 2 photographed Phobos before it failed. In the last few years, the US Mars Global Surveyor and ESA’s Mars Express studied Phobos more in detail (figure 5.2).

The observed shape of Phobos is highly irregular. A first approximation is a tri-axial ellipsoid, with principal axes of 13, 11.5 and 9.3 km [Dobrovolskis, 1996]. The main axis is pointed towards Mars, the intermediate axis is pointed to the direction of movement and the minor axis is parallel to the rotation vector of the moon’s orbit. Commonly, the coordinate system [x1, x2, x3] is chosen in the same direction. The values for the principal axes differ in various references, for instance [Veverka and Burns, 1980] adopted 13.5, 10.7 and 9.6 km. The shape is close to 5.1 Introduction 65

Figure 5.2 Phobos seen by Mars Express [ESA, 2005].

a Roche ellipsoid [Chandrasekhar, 1969], which is an equipotential shape when taking into account the self gravitation, centrifugal and tidal forces. This matter will be discussed more in detail in section 5.4.

The most accurate mass prediction has been done observing the orbit of the Soviet spacecraft Phobos 2. A 3D model could be made by combining photographs from different angles. In this way the average density of Phobos was found to be only (1.9 ± 0.1) × 103kg/m3 [Duxbury, 1991]. Spectral properties together with the measured low density suggest a porous, carbonaceous chondritic composition of the moon. Comparing with asteroids that have fallen on Earth, the rigidity is in the order of 108 − 109P a [Lambeck, 1979]. Furthermore, Phobos has a regolith, a layer of loose material covering the rock underneath. The layer is thinner than at Deimos, with less boulders [Davis and Greenberg, 1981]. A part of the regolith could be pulled off by tidal dynamics or blasted of by an impact [Horstman and Melosh, 1989]. The question remains whether the regolith retained on the surface after the impact of Stickney or whether in blasted off and reaccreted later. If the rejection speed would be low enough, the regolith debris could remain in an orbit around Mars.

Craters

Phobos is heavily cratered. The main crater Stickney has rim-to-rim dimensions of 9.6 ± .5 km north-south and 9.2 ± .5 km east-west [Thomas, 1998], a little less than the radius of Phobos itself. The centre is located at colatitude 95.6◦ and longitude −53.3◦ [Fujiwara, 1991]. The origin of this reference system is taken as pointing towards Mars. The rim is about 1.3±.3 km higher than the bottom of the crater, while the rim is about 0.3 km above local surface level. The western rim is sharp edged in contract with the round eastern edge. An interesting point is that 66 Phobos

[Hartung, 1975] proposed that a body would be destroyed by an impact that could have produced a crater larger than one third of the body diameter. With other words, a crater larger than one third of the body diameter could not exist. Phobos and the Jovian moon Amalthea violate this rule. The estimated volume loss from the crater is ∼ 5 × 1010 m3, about 1% of the total volume of Phobos [Thomas, 1998]. This represents ∼ 1014 kg if compression of material is neglected. If this material would have been spread over the surface of Phobos, the depth would be around 30 metres.

Other big craters are Roche (6 km), Hall (5 km) and d’Arrest (2 km) [Thomas and Duxbury, 1979]. Craters with diameters below 200 metres occur less than “empirical saturation” expects [Robinson et al., 2003]. This can be explained by the regolith: shocks in Phobos cause the fine grained material to wipe out traces of small craters. At the east side of the Stickney crater the population of small craters is even lower than the moon’s average. The regolith is expected to be thickest there: 300 metres.

Based on the density of craters on the surface, the age of Phobos can be approximated [Thomas and Veverka, 1980]. Since the crater density and impact hazard is comparable to the Moon the age is expected to be comparable, about 3 × 109 years. In case of heavy bombardments after a disaster the time is shorter, but no less than 109 years.

Grooves

The Viking Orbiter discovered long linear depressions on Phobos, initially called “striations”, and later assigned as “grooves” [Thomas and Duxburry, 1978]. This has become the formal name. In figure 5.3 the global distribution of the grooves is shown. In some areas some grooves are left out for clarity. The lines on the figure are based on Viking pictures. In the region in the west of Stickney there is a clear scarcity of grooves. At first it was thought it was the cause of the resolution of the pictures, but recent pictures confirm the differences in groove distribution around Stickney.

As can be seen in figure 5.3, most of the grooves seem to originate from crater Stickney. They are best visible near the crater, and seem to die out near its antipode. The crater on its turn is mostly grooved outside the eastern rim. Those grooves are closest to the surface of Mars. However, in figure 5.2 taken by Mars Express, the grooves are more parallel than radial, where the main direction of the grooves is parallel to the intermediate axis. The grooves stretch up to more than the radius of Phobos. They even seem to be able to continue in the same plane after they cross a relief with elevation changes over 1 km [Drobyshevski, 1988].

The grooves have a smooth topography; most of the grooves have a width between 100 and 200 metres and are about 20 to 30 metres deep, with local maxima of 100 metres deep [Thomas and Duxbury, 1979]. Usually the slope is less than 10◦, but near Stickney the slopes are around 30◦. They do not have significant raised rims. The smoothness can be explained by the regolith. Only some grooves near Stickney have a V-shaped cross section. In the zone east of Stickney, the grooves are often very wide, up to 600 metres in which younger, smaller grooves occur. 5.1 Introduction 67

Figure 5.3 The grooves of Phobos. The contour of Stickney is included as reference [Thomas and Duxbury, 1979].

Figure 5.4 The grooves of Phobos, categorised in 5 groups [Thomas and Duxbury, 1979].

A grouping is done by [Thomas and Duxbury, 1979] on the basis of location, morphology, or parallelism to other grooves. Five groups are designated from A to E, depicted in figure 5.4. Sets A, B and C are grouped on the basis of orientation. These groups form sets of parallel grooves and their planes are parallel to the intermediate axis. Sets D and E follow the surface topography rather than the global geometry. Most of them begin and end at a crater’s rim.

The age of the grooves can be estimated by counting the craters superimposed to them. Comparing the number of craters with those on other celestial bodies like the Moon, the average age of the grooves is no less than 109 years [Thomas and Duxburry, 1978]. An important aspect is that the grooves are superimposed on all big craters, but the three small craters inside Stickney are superimposed on those grooves on their turn. This can imply that the grooves were formed relatively 68 Phobos

short after the crater itself [Thomas and Duxbury, 1979]. An interesting feature is that craters that are crossed by grooves do not show any deformation.

Finally, relative ages can be estimated by examining which grooves are super- imposed on others. By inspection of detailed pictures, the most obvious order of creation of the grooves is A-C-B. Unfortunately, many spots on Phobos were found where apparent younger grooves were put to a stop at a groove from an older category. This can be explained by the fact that the grooves could have been formed nearly at the same time, and that some grooves took a long time before evolving to their actual size.

5.2 Theories about Phobos

In this section, the two most important questions about Phobos are discussed, namely:

What is the origin of Phobos, and how did it end up in the present orbit? What is the origin of the grooves?

These two questions are in some way tangled; many theories claim that both events are the consequence of the same mechanism. In this chapter, both questions are dealt separately for clarity.

What is the origin of Phobos?

Since the orbits of Phobos and Deimos are nearly circular, prograde and almost coincide with the plane of Mars’ equator, the moons were logically formed during the accretion process in our early solar system.

However, the moons are very small and have nearly identical composition of carbonaceous chondrites. This material is only found in the outer parts of the asteroid belt. During the formation of our solar system at the distance of Mars, the solar nebula temperatures were too high for carbonaceous material to be condensed out [Lambeck, 1979]. Therefore, they could not have formed around Mars. Phobos and Deimos could rather be asteroids that were captured by Mars’ gravity.

The question remains how the moons ended up in such circular and planar orbits. Especially the situation for Deimos is difficult, the distance from Mars is too large to create substantial tidal forces. Phobos’ orbit could have been damped, however if Phobos’ orbit was highly eccentric, it would cross the orbit of Deimos. When only damping of eccentricity by Mars is assumed, Phobos must have been crossing the orbit of Deimos for at least 108 years [Yoder, 1982].

The predictions about the dampening of the eccentricity based on the present eccentricity of only 0.015 are very disputable. Resonance is an important factor that makes previous calculations even less accurate. At the time when r = 3.8RM or r = 2.9RM the rate between the orbital period of Phobos and the rotational 1 1 period of Mars was 2 and 3 respectively. Zonal variations in Mars’ gravitational field cause the eccentricity to increase substantially [Yoder, 1982]. 5.2 Theories about Phobos 69

Different other theories on how the moons could end up in the present orbits have been posed. The capturing process could have been done by collisions with other asteroids or by drag in the gaseous surroundings of proto-Mars. If a proto- moon would orbit in a cloud of remnants, the damping of eccentricity could already been established on the order of 10 years [Drobyshevski, 1988].

The moons can also be a remnant of a large bombardment on Mars by material from the asteroid belt. The rotation time of Mars is almost as short as on Earth. However, in the case of Mars the rotation is hard to be explained by the natural collision of particles during accretion. Models of natural accretion show that the measured spin rate in rotations per sidereal day is about 5 times more than sto- chastically expected. The extra spin could be induced by Phobos-like bodies that collapsed on Mars in the early stage of our solar system. The amount of mass required to cause the present spin of Mars is about 1000 times as much as the mass of Phobos and Deimos [Craddock, 1994]. This would mean that the moons are only a small portion of the original mass.

What caused the grooves on Phobos?

What caused the numerous grooves on Phobos’ surface is still not understood. Out of all theories two major ones can be distinguished; the impact theory and the tidal theory. Other theories are combinations or derivations of these two.

A widely accepted theory to cause the grooves is the impact by Stickney [Thomas and Duxburry, 1978]. As mentioned in section 5.1, the grooves were formed shortly after the impact of Stickney. Many grooves seem to originate from the crater and seem to die out near its antipode, although this is not the case for all grooves. Furthermore, the grooves seem to become wider when they are at a certain distance form the impact. This can be caused by the ejection of volatiles [Ill´esand Horv´ath, 1981].

However, the connection between the grooves and the crater is often disputed [Soter and Harris, 1977]. Only some grooves are radial with respect to Stickney. The alignment with the y-axis seems suitable for the majority of the grooves.

Tidal forces have a severe influence in geological process, not only on earth, but on all bodies close to each other. On the Moon and other bodies many twin craters are found, probably caused by asteroids that were tidally disrupted prior to impact. Some of the features are related to readjustment of the satellite’s figure with increasing tidal stress as the orbit evolves inwards under the action of tidal friction. If the width of the grooves is in fact due to tidal readjustments of the figure of Phobos, older craters should be systematically deformed from their original circular shape. But as the rate of orbital decay increases with time, this mechanism would require the grooves to be extremely young features. Camera’s on the Viking Orbiter revealed a high number of craters superimposed on the grooves. Their implied age is > 109 years, ruling out this explanation [Weidenschilling, 1979]. Another argument against the tidal stress theory is the fact that grooves are also found on bodies which are not deformed by tidal forces. The grooves on the asteroids 951 Gaspra, 243 Ida and 443 Eros are more likely to be a result of collisions and impacts than tidal forces. 70 Phobos

The third theory is that the grooves are caused by tidal forces after the impact [Weidenschilling, 1979]. The longest axis of the satellite points towards Mars due to tidal forces. Shortly after the Stickney impact the orientation of Phobos was temporally offset. Until the gravity of Mars synchronised again, Phobos underwent extra tidal stresses.

The problem is that the orbit of Phobos was much larger at the time of impact, around 3 × 109 years ago. According to Drobyshevski [Drobyshevski, 1988], the misalignment of the major axis could be at most 4◦. At that time the semi-major axis was around 4.45RM . This would cause only twice the amount of stress that is to be seen today. If this would have caused grooves, it would also cause grooves nowadays.

Finally, some theories are related to impacts, but not directly by the impact of Stickney. The first is supported by [Veverka and Duxbury, 1977] and [Wilson and Head, 2005]. It states that the grooves are no fractions caused by Stickney, but lines caused by ejecta and rolling and bouncing boulders. It was calculated that boulders could roll the full length of an average groove and the width-to- depth ratio is also consistent. Furthermore, tracks from boulders on the moon are found with consistent appearances. However, the grooves are not all in the radial direction and no boulders are found at the end of the grooves. It is also doubtful that a boulder would travel more than half the diameter of the moon before it comes to a stop. An explanation for the absence of boulders could be found in the fact that the surface gravity on Phobos varies considerably. This could cause the boulders to leave Phobos at a certain point. The boulders could also be eroded after the long journey through the regolith. The point of the lack of radial grooves is defended by the fact that the pre-existing structure of Phobos will dictate the direction of the boulders, not the impact itself. Since boulders can theoretically move in all directions due to accidental collisions it is very easy to explain mysteries in this way. No numerical simulations have been produced yet to verify this theory.

Several other theories have been proposed since the discovery of the grooves, like drag forces generated during the capturing of the moons [Pollack and Burns, 1977] or a layered structure of Phobos [Ill´esand Horv´ath, 1981]. A recent theory connects the grooves with ejecta from impacts on Mars [Murray et al., 2006]. The impact that caused Stickney was probably accompanied by a cloud of rocks that mostly impacted on Mars. Simulations show that almost all families of grooves could be caused by ejecta from these impacts. This cause would also explain the abundance of grooves around the submartian point that die out near the leading point and trailing point of Mars.

If the grooves are formed by tides or by impacts, different processes could establish the grooves. The meteorite impact could have initiated an explosion of ice inside the moon [Drobyshevski, 1988]. This explosion cracked the surface through the grooves of B, E and D. The grooves of A on their turn were caused by the contraction of the moon because of the loss of gasses. Furthermore, the lack of ice can explain the low density of Phobos. This theory supplies no explanation for the C grooves. These grooves could be formed by the layered structure of Phobos. But no hydrated minerals are detected from Near-Infrared Spectrometry, cancelling out this mechanism [Rivkin et al., 2002]. 5.3 Normal mode analysis for a homogeneous sphere 71

Next, the grooves could be the lack of regolith where regolith is excavated by venting gas [Thomas and Duxbury, 1979]. On several images the grooves have an irregular shape, which could be explained by pits. This could also explain that other grooves have upstanding edges, just like at Europa [Horstman and Melosh, 1989]. Furthermore, the grooves at the rim of Stickney must be formed well after the impact, which is in accordance with gas venting. Venting gas would probably produce even more raised rims than observed on Phobos. [Siegal and Gold, 1973] carried out experiments with venting gas to prove if this process caused particular craters on the moon. Here the theory was in accordance with reality. However, there are obvious differences in shape and pattern: the craters on the moon that are presumably caused by gas venting have a large variety of sizes, craters close to each other do not form grooves and the distribution is highly irregular [Horstman and Melosh, 1989]. If the grooves of Phobos were caused by this process, the pits from which the gas was venting should be entirely different. Finally, if Phobos has a thick regolith of 300 metres as predicted by [Horstman and Melosh, 1989], gas would blow up in a random direction and would produce meandering grooves rather than straight lines.

Vice versa, the regolith could be drained inside cracks [Thomas and Duxbury, 1979]. This requires the “drainage pits” to be open widely in order to be filled with regolith. Drainage pits usually have steep, conical cross sections whereas the grooves of Phobos are shallow. However in lava tubes, drainage of surface material produces shallow, elongated craters. Moreover, the steep conical pits could be degraded into more shallow grooves after impact processes elsewhere on the moon. Horstman and Melosh showed that the spacing of drainage pits in loose material is approximately equal to the depth of loose material draining into deep fractures. Based on this, the regolith is about 300 metres thick. This is about twice the amount that could be produced by all craters on Phobos [Goguen and Duxbury, 1978]. A correlation between the spacing of the pits and the dynamic height was found [Horstman and Melosh, 1989]. The correlation is best near rims of craters. Flat areas correlate worse. This could be caused by the difference in gravity field in the past by for instance tumbling. The discontinuities in the regolith that are observed suggest many cratering events as large as Stickney.

5.3 Normal mode analysis for a homogeneous sphere

It is unlikely that Phobos has a layered structure. The average density of 1900 kg/m3 is is almost unrealistically low. If the there would be a density gradient, the density near the surface would be even lower. Furthermore, the self-gravitation of Phobos is probably not sufficient to differentiate the inner material. Only the regolith can be regarded as an extra layer, but the thickness of the regolith is negligible compared to the radius of Phobos. Without evidence for a layered structure, it is sufficient to consider Phobos as a homogeneous body.

For a homogeneous sphere the normal mode analysis can be simplified so that the deformation can be written in an explicit form. The ellipticity is included as deformation on the sphere. The linearity of the model is not sufficient for accurate calculations on Phobos, but as a first approximation, it can give insight in the overall structure of Phobos. 72 Phobos

Elastic response

For a one-layered body, the matrix propagation of equation 3.26 can be reduced to   K1 ~ ˜ y˜l(r, s) = Ic,l · K2 . (5.1) K3

Here, the core is the whole sphere. Since the “core” is solid, the left part of the propagator matrix of equation 3.24 is used rather than the matrix of equation 3.30. This matrix can be split up directly into a part for the free displacements (row one, two and five) and a part of constrained quantities (row three, four and six). Analogous to equation 3.37, the free quantities are related to the constrained quantities by:

 ˜   lRl+1 l−1  Ul 2(2l+3) R 0 ˜  (l+3)Rl+1 Rl−1   Vl  (R, s) =  0  · ˜ 2(2l+3)(l+1) l −Φl 0 0 −Rl

2 l −1  (lρ0g0R+2(l −l−3)˜µ)R l−2 l  2(2l+3) (ρ0g0R + (2l − 1)˜µ)R −ρ0R  l(l+2)˜µRl 2(l−1)˜µRl−2  ·~b.  (2l+3)(l+1) l 0   l+1  2πGρ0lR l−1 l−1 2l+3 4πGρ0R −(2l + 1)R (5.2)

The inverse of the 3 × 3 matrix can be found analytically using Mathematica. By using the gravity at the surface for a homogeneous sphere, GM G 4 4 g(R) = = πρR3 = πρGR (5.3) R2 R2 3 3

setting s → −∞ and making the deformation dimensionless by using equation 3.49, the tidal Love numbers become

2(2l + 1) h = T 2l2+4l+3 µ (l − 1)(4 + 3 l GπR2ρ2 ) 2(2l + 1) l = T 2l2+4l+3 µ l(l − 1)(4 + 3 l GπR2ρ2 ) 6 k = , (5.4) T 2l2+4l+3 µ (l − 1)(4 + 3 l GπR2ρ2 )

which are depicted in the left part of figure 5.5. All quantities in the factor GπR2ρ2 are known quite accurately and the factor is found to be 9.23 × 104kgm−1s−2 (see appendix A.2). If µ  GπR2ρ2 the first term in the denominator of equation 5.4 5.3 Normal mode analysis for a homogeneous sphere 73

Figure 5.5 The Love numbers as function of their degree for Phobos with an overall rigidity of µ = 1011P a. Left, tidal Love numbers, right load Love numbers.

is negligible. The Love numbers are then approximated by

2l + 1 h ≈ T 2(l − 1)µ0(l) 2l + 1 l ≈ T 2l(l − 1)µ0(l) 3 k ≈ , (5.5) T 2(l − 1)µ0(l)

with 2l2 + 4l + 3 3µ µ0(l) = . (5.6) l 4GπR2ρ2

These results are similar to the analytical results by Kelvin (see e.g. [Munk and Macdonald, 1960]). The load Love numbers are found in a similar way (see the right part of figure 5.5):

0 0 hL = (2l + 1)/(1 + 3µ (l)) ≈ (2l + 1)/(3µ (l)) 0 0 lL = (2l + 1)/l(1 + 3µ (l)) ≈ (2l + 1)/(3lµ (l)) . (5.7) 0 0 kL = 1/(1 + µ (l)) ≈ 1/(µ (l))

Viscous response

The viscous response can be calculated in the same manner. Since a homogeneous sphere has no inter-layer boundaries or core-mantle boundary, only the surface contributes a mode. The inverse relaxation time can be calculated analytically using Mathematica

−µ −4GπR2ρ2l s = ≈ . (5.8) j ν(1 + µ0(l)) 3(2l2 + 4l + 3)ν 74 Phobos

The inverse relaxation time is mostly dependent on the viscosity and is not sensitive to a change in rigidity. The modal strength for a tidal load becomes:

 3(2l+1)   2l+1  K~ j(R) 2(l−1)(3+1/µ0) 2(l−1) l =  3(2l+1)  ≈  2l+1  . (5.9)  2l(l−1)(3+1/µ0)   2l(l−1)  −sj 9 3 2(l−1)(3+1/µ0) 2(l−1)

For high rigidities, the modal strength are only a function of the harmonic degree itself. When a model shows a result independent on input of the material properties of the moon, the most logical conclusion is that the program contains an error. But after checking the derivation several times, no error was found. The effort was in vain, because in the end, a physical phenomenon caused the numbers. For a fluid body, it has been shown by Kelvin that hT,2 = 5/2 (e.g. [Munk and Macdonald, 1960]). Every sphere with constant density would eventually form itself to a hydrostatic equilibrium. A rigid body that is subjected to a Heaviside load will initially deform only a elastically. The viscous creep completes the deformation to a hydrostatic equilibrium when time elapses. If the elastic deformation is very small, the modal strength is almost equal to the tidal Love number for a fluid. According to equation 5.8, only the time to fully relax is dependent on viscosity of the material. The modal strength for a load is not discussed, since it has no application for Phobos.

Deformation inside a body

Maximum stress by an external load does not necessary has to occur at the sur- face of the body. Here the deformation inside a body is discussed. The matrix propagation in equation 5.2 is adapted. The external force is propagated to the centre, but after that, it is only propagated to the required radius,

 ˜   lrl+1 l−1  Ul 2(2l+3) r 0 ˜  (l+3)rl+1 rl−1   Vl  (r, s) =  0  · ˜ 2(2l+3)(l+1) l −Φl 0 0 −rl

2 l −1  (lρ0g0R+2(l −l−3)˜µ)R l−2 l  2(2l+3) (ρ0g0R + (2l − 1)˜µ)R −ρ0R  l(l+2)˜µRl 2(l−1)˜µRl−2  ·~b,  (2l+3)(l+1) l 0   l+1  2πGρ0lR l−1 l−1 2l+3 4πGρ0R −(2l + 1)R (5.10)

where R is the radius of the homogeneous sphere, and r the radius where the deformation needs to be known. The elastic Love numbers become:  h   h  r(l−1)R(−l−1) (1 − l2)r2 + l(2 + l)R2
l (r) = · l (R). (5.11)   2l + 1   k k

By substituting r = R, it can easily be seen that this equation matches the surface boundary condition and that the deformation is zero at the centre. Figure 5.6 5.4 Determining material properties by the deformation 75

Figure 5.6 The normalised deformation as function of the normalised radius.

Figure 5.7 The variation of radial, colateral and longitudinal stress as function of the radius of a homogeneous sphere for different harmonic degrees.

visualises this. Although the deformations die out in the centre, the stresses are not linearly proportional to the deformation according to equation 3.57. The radial stress is proportional to the radial derivative of the deformation, while the colateral and longitudinal stress is proportional to 1/r times the deformation. The relative change of stress over the radius is depicted in figure 5.7.

5.4 Determining material properties by the deforma- tion

As mentioned in section 5.1, the global shape of Phobos represents a Roche el- lipsoid, although not the ellipsoid that is expected for this orbital radius. In this section it is calculated which rock viscosity belongs to the present shape if Phobos is only deformed by tidal forces and the possible consequences for the history and rheology of Phobos are discussed. First, the mathematical background about the Roche ellipsoid and the spiral motion is given. Next the viscoelastic deformation of Phobos is modelled using normal mode analysis. Finally the validity of this method is discussed. 76 Phobos

Roche ellipsoid

Ellipsoidal figures of hydrostatic equilibrium have attracted the interest of many mathematicians [Chandrasekhar, 1969]. Their main question was: what kind of stable ellipsoidal bodies can exist in real physics next to a spherical body. The study began with the investigation of the figure of the Earth [Newton, 1687]. Newton showed a linearised relation between the ellipticity and the centrifugal acceleration with respect to the rotational axis. The theory was expanded for the case that the rotation cannot be considered small [Maclaurin, 1742]. While is was already theorised that ellipsoids with three unequal axes could satisfy the requirements of equilibrium, it took almost a century before the corresponding requirements were constructed [Jacobi, 1834]. By using the gravity potential field Φ inside a tri-axial ellipsoid

3 Φ = GM[−Φ + x2Φ + x2Φ + x2Φ ] g 4 0 1 x1 2 x2 3 x3 Z ∞ dα Φ0 = s=0 ∆ Z ∞ dα Φxi = 2 s=0 (xi + α)∆ q 2 2 2 ∆ = (x1 + α)(x2 + α)(x3 + α), (5.12)

it was proven that solutions only exist for 1 1 1 2 > 2 + 2 . (5.13) x3 x1 x2

Finally, Roche applied the theory on a body rotating in a Keplerian orbit around a parental body [Roche, 1850]. Consider Phobos, approximated by a tri-axial ellipsoid. Using the data of appendix A.2, the gravity potential inside Phobos is

Φ = Φg + Φc + ΦT −6 2 −6 2 −6 2 Φg = −98.84 + .215 × 10 x1 + .269 × 10 x2 + .418 × 10 x3 1 Φ = − ω˙ 2(x2 + x2) c 2 1 2 2 2 2 2 ΦT = −ω˙ (x1 − x2/2 − x3/2) (5.14)

and GM ω˙ 2 = mars , (5.15) R3

this becomes  3GM  Φ = − 98.84 + .215 × 10−6 − x2 2R3 1 (5.16)  GM  + .269 × 10−6x2 + .418 × 10−6 + x2. 2 2R3 3 5.4 Determining material properties by the deformation 77

If Phobos has no cohesion it will start to disintegrate when the gravity becomes zero. In all cases the regolith will be sucked away by Mars. This will happen firstly at the x1 axis, where the tidal forces are maximum

dΦ = 0. (5.17) dr x2=x3=0

Substituting this into equation 5.16 yields 3GM 2(0.215 × 10−6 − )x = 0. (5.18) 2R3 1

23 For MMars = 6.421 × 10 kg, the orbital radius becomes 6685 km. As the actual mean distance is about 9380 km, Phobos is still outside the Roche limit for the rigid case.

The traditional Roche problem deals with a liquid body that is subjected to tidal forces. These equations are described in detail in [Chandrasekhar, 1969]. Here it is shown that a tri-axial ellipsoid is a solution for a body that is in hydrostatic equilibrium. Although this is obviously not the case for Phobos, some interesting things can be concluded based on the equations by Roche. First, the present shape of Phobos closely resembles the hydrostatic equilibrium shape at an orbital radius of 12360 km. It could be possible that the viscosity of Phobos causes a delay in the adjustment to a hydrostatic equilibrium. If this would be the only cause, the present shape of Phobos is a good measurement for the viscosity. However, extreme ellipsoidal asteroids show that tidal forces are not necessary to cause oblate shapes. Especially for small bodies like Phobos it is doubtful that the self-gravitation is sufficient to deform the overall shape. Second, it can be derived that the equations do not have a solution if the distance to Mars is smaller than a certain value. This value is dependent on the parameters ρP hobos, ρMars and radius RMars [Roche, 1850]:

r ρ 3 Mars r = 2.423RMars . (5.19) ρP hobos

The Roche limit is therefore 10470 km. If Phobos would be liquid, it would already have been torn apart.

Spiral motion

The spiral motion of Phobos was already discussed qualitatively in section 5.1. Here, the process is dealt with in a quantitative way. The torque acting on the tidal bulge of Mars is approximated by [Goldreich and Soter, 1966]

2 ¯ 3 k2Mars 5 MP hobos T = GRMars 6 , (5.20) 2 QMars r

where the uppercase R stands for the mean radius of a body, the lowercase r stands for the orbital distance and Q is the quality factor of Mars. The higher this 78 Phobos

dimensionless number, the slower an oscillation is damped. The torque must be equal to the loss of orbital energy [Goldreich and Peale, 1966] r 1 GM dr T¯ = M Mars , 2 P hobos r dt

hence dr k r G M − = 3 2Mars R5 P hobos . Mars 11/2 dt QMars MMars r

Using Mars Orbiter Laser Altimeter data from the Mars Global Surveyor, the estimated approach velocity is −(1.28 ± .01) × 10−9 m/s [Bills et al., 2005]. With this value an expression for the tidal dissipation can be found k 2Mars = 2.1 × 10−19[−]. QMars

When only the orbital distance changes over time, the time between two different radii becomes

−31 2/13  13/2 13/2 t = 5.40 × 10 m s · r2 − r1 .

The orbital radius as function of time is depicted in figure 5.9. The time to reach the “rigid Roche limit” is 32×106 years. Furthermore, Phobos had an equilibrium shape about 179 × 106 years ago, which means that Phobos has the shape that it would have 179 × 106 years ago if it would be fluid. Based on these values, it can be calculated for which rheology Phobos would deform with this speed.

The total deformation of Phobos using Normal mode analysis

According to section 3.3 the convolution of the Love numbers with an external force generates the deformation. The function that relates the orbital distance to time is too complicated to solve analytically. The tidal potential will therefore be modelled as Heaviside functions representing small increments of potential, see the left part of figure 5.8. The sum of all responses is depicted in the right part of the same figure.

Since the present form of Phobos represents a Roche ellipsoid, it is only neces- sary to consider the deformation at one axis. Assuming a present primary radius of 13500 metres and an average radius of 11040 metres, the tidal force should uplift the submartian point by 2460 metres. The deformation of the submartian point consists of two terms: the tidal force and the centrifugal force by the rotation.

Since the approach is partially numerical, a trade off must be made between accuracy and computational time. To evaluate the step size it is sufficient to look at the last time step, because then the orbital radius is decreasing fastest (see figure 5.9). Taking a step size of 10 million years, the orbital radius decreases with 4% in the last period. This is well within the order of accuracy of this analysis, since many factors (anisotropy, ellipticity) cause a substantially higher error. The 5.4 Determining material properties by the deformation 79

Figure 5.8 A schematic overview of modelling the tidal deformation. Left, the potential as function of time is approximated by a stepwise function. This can be split up into Heaviside functions. Right, the convolution with the Heaviside load produces instant elastic responses and asymptotic viscoelastic responses. The sum of all these responses is the total deformation.

second problem is the start time of the calculation. At the initial position of Phobos the tidal forces already contribute to deformation. If Phobos solidified in a tidal field it is possible that it deformed already to a hydrostatic equilibrium. According to section 5.1 the age of Phobos is expected to be around 3 × 109 years. If Phobos would always have been orbiting Mars in the described spiral motion, the primordial orbital radius was about twice as large as nowadays. Using equation 2.22, the tidal force was about eight times lower. Although the effect of the shape of the primordial Phobos is relatively small it is taken into the calculations, of which the results are depicted in figure 5.10.

For ν < 1018P as the moon behaves as a fluid on the geological timescale. When higher order effects are taken into account, the moon would have been disintegrated by tidal forces [Roche, 1850]. For ν > 1022P as the moon is sufficiently viscous that the moon deforms only elastically. The elastic deformation of Phobos is on the order of one metre for realistic rigidities. Therefore, this influence can be completely neglected. The effect for the case that Phobos was formed to a hydrostatic equilibrium three billion years ago is only visible for high viscosities. In that case the total deformation is completely defined by the primordial shape. The line representing the present uplift and the line representing the uplift as function of viscosity intersect at ν = 4 × 1019P as. With this viscosity, the moon will deform to the present shape. The fact whether Phobos was already an ellipsoid or not does not matter for this rheology.

Discussion

The normal mode analysis assumes a spherical layered structure loaded by self- gravity and external forces. It is questionable whether this assumption is valid for Phobos. The moon can be composed of several rocks, loosely connected. Next, the ellipticity of Phobos does not necessarily need to be a result of tidal forces, but can simply be the primordial shape of the rock. Furthermore, the deformation of Phobos is substantially large that second order effects can not be neglected, as is done in the normal mode analysis. Finally, the spiral motion is based on an undisturbed history, hence it could have happened entirely different. Nevertheless, 80 Phobos

Figure 5.9 The orbital radius, elastic deformation and viscous deformation as function of time for µ = 109P a and ν = 1020P as.

Figure 5.10 The total uplift of the submartian point as function of the viscosity, compared with the real uplift of 2460 metres.

the assumption is most likely valid for the last period in Phobos’ history, which was also the most important part with respect to tides.

Also plenty of other events during the history of Phobos could have triggered the moon to deform. Firstly, the impact that resulted in the Stickney crater could caused seismic waves that harm the structural integrity of the moon. Such sit- uation could have deformed the moon to a hydrostatic equilibrium much faster. Secondly, the libration by the orbital eccentricity could have caused diurnal vari- ation in tidal forces (see chapter 5.5). Variable stresses will yield the material by fatigue failure.

Nevertheless, while all the previously mentioned events and mechanisms are unsure, the calculations that are presented in this section are valuable for a first approximation of the rheology. The viscosity for the mantle of the Earth are in the order of magnitude of ν = 1021P as [Vermeersen, 2002]. The value of ν = 4 × 1019P as is more than an order of magnitude lower and can be explained by the difference between a fractioned or anisotropic body and a monolithic rock. 5.5 The causes of the grooves 81

Figure 5.11 A 3D model based on a model by Tony Dobrovolskis. The crater Stickney is clearly visible in the topographic map.

Therefore, it can be realistic for the average rheology. Not the derived viscosity itself is remarkable, but the fact that the deformation can be the cause of tidal forces. Although the gravity on Phobos is minimal and the material is highly viscous, it could well be shaped by gravity potential over a long time. To show this, it can be noted that the length of the principal axis increased by 120 metres in the last 10 million years according to the calculations. The uplift of 12 µm per year indicates that the deformation is not distinguishable on a human timescale.

5.5 The causes of the grooves

In this section the cause of the grooves is discussed more in detail. Several theo- ries about the origin of Phobos and its grooves are discussed in section 5.2. Most research is done in a qualitative manner by assessing the photographs that have been taken by spacecrafts and by comparison with other celestial objects. Be- sides intuitive reasoning, experiments were conducted on scaled models of Phobos [Fujiwara and Asada, 1983]. In this case cracks were found to disrupt the whole object, which makes it difficult to compare with the present cracks. Numerical simulations were conducted for a tri-axial ellipsoid [Holsapple, 2001], [Dobrovol- skis, 1982]. They took only into account the constant tidal stress, which is only one of the possible explanations.

Here different theories are tested on their plausibility using normal mode analy- sis. The main criteria will be whether an external force can cause the main group of grooves, which run parallel to the y-axis, with the main concentrations near the submartian point and near the geometric north pole.

This approach neglects other aspects compared to previous research. For in- stance, a spherical body is assumed rather than a tri-axial ellipsoid. But there are also advantages: more different influences can be combined, like an impact and a tidal load. Although the calculations are based on a sphere, the results are plotted on a 3D model which is depicted in figure 5.11. It was made by Tony Dobrovolskis. 82 Phobos

Only the theories about tidal forces and an impact are discussed. The other theories which were discussed in section 5.2 have no relation with the inner struc- ture of Phobos and will therefore not be treated here.

Material failure

The way in which material fails is dependent on the material properties. For brittle materials the yield criterion is only dependent on the maximum tensile stress, whereas ductile materials fail when maximum shear stress occurs. The failure criterion is often expressed as

p 2 p 2 D = 1 + f (σ1 − σ3)/2 + f((σ1 + σ3)/2 = ( 1 + f − f)τ + fσ1, (5.21)

where σ1 is stress in the direction where the maximum stress occurs, σ3 the min- imum stress, and τ = (σ1 − σ3)/2 is the maximum shear stress. The dimension- less parameter f is called the coefficient of internal friction. D is also knows as “duress”. Fracture will occur when this parameter reaches a certain value. This linear function is often called the Mohr-Coulomb friction law. For perfectly ductile materials f = 0 and for brittle materials only σ1 is of importance, hence f → ∞. In dry rock f ∼ 1.0. This value will be adopted for the model of Phobos. The value for D varies more, from virtually zero for granulated material up to several hundred bar in solid rock. This value for Phobos is up to now difficult to predict, although the low bulk viscosity suggests a low yield stress.

Description of the program

In order to show the stress in and on Phobos, the following procedure is adopted. Firstly, the stress tensors at the surface and inside the body are calculated using equations 3.58 and 3.59. Here the angle θ in equation 3.58 is not the angle with respect to the geometrical pole, but with respect to the location where the load is applied. The impact load is applied at colatitude 95.6◦ and longitude −53.3◦. The time dependent tidal load is composed of three terms according to 2.40. Since the Maxwell rheology is linear, all different influences can be added linearly. Before adding different contributions the stress tensors σ are transformed to the same coordinate system to be able to add them properly. The principal stresses σ1, σ2, σ3 are defined by the eigenvalues of the stress tensor, where the stresses are ordered from maximum tension to maximum compression. The plane of failure is assumed to be perpendicular to the maximum principal stress.

Impact cratering

The load Love numbers are often used to study the deformation caused by a superimposed load, e.g. ice. The rheology of the Earth can be predicted by observing post-glacial rebound. An impact on Phobos can best be described by a sudden load that is instantly removed again. The load vector for a point mass load is represented by equation 3.34. Since it is not a load but rather can impulse, the load can best be represented by a delta force. Since a Maxwell rheology is assumed, 5.5 The causes of the grooves 83

Figure 5.12 The stress concentrations and crack direction in the case of an impact. The contribution up to harmonic degree l = 10 is taken into account. Red indicates a high stress, whereas blue is low stress.

Figure 5.13 The stress concentrations, top view. Left, a cross section though the xy-plane is depicted; right depicts the stress and crack direction at the surface.

the moon will only deform elastically. The load vector assumes a vertical load and therefore neglects the fact that the impact could have been oblique.

The results are depicted in figure 5.12 and 5.13. As expected, the lines that represent the cracks move radially away from the crater, with maximum stress concentrations near the crater. The coloured “waves” on the globe are caused by the expansion in Legendre polynomials up to the tenth degree. The waves will smooth out if an infinite amount of harmonic degrees would be used. Since the grooves do not occur radially from the impact and most grooves are further away from the impact, this cause is unlikely.

Some remarks can be made based on this figure. Firstly, the load Love numbers take into account that the surface is not only deformed directly by the impact, but also indirectly since the body is attracted to the gravity of the superimposed load. The stress caused by the impact is much higher than this change in gravity potential and should be neglected. Furthermore, an impact would cause stress waves that propagate through Phobos [Melosh, 1989]. The waves can reflect at the surface and can therefore trigger failure in various directions. This cannot be modelled by assuming only an impulsive load. 84 Phobos

Tidal load

Next, the stress by tidal load is discussed. Based on crater counts and the number of craters that is superimposed on the grooves, the grooves are expected to be about three billion years old (see section 5.1). At this time, the distance to Mars was still 1.5 × 107 metres, corresponding to a revolution time of 15 hours, 30 minutes. The eccentricity could have had a maximum value of e = 0.2 [Yoder, 1982]. Since the Maxwell relaxation time of Phobos is many orders of magnitude longer than the orbital period, the rock behaves elastically according to section 3.4.

The stress and crack direction are depicted in the left column of figures 5.14 and 5.15. The maximum stress does not occur at the tidal bulge itself, but at the ring at 90 degrees from the tidal bulge. This is the case in the pericentre, where the tidal force is largest, but also at the moment when Phobos is in apocentre, when the tidal bulge is negative compared to the average tidal force. Since the stress changes from tension to compression over an epicycle, the direction where the grooves would occur changes.

Summarising, through one orbital cycle the maximum tensile stress changes, which can cause material fatigue. From this point of view the poles are most vul- nerable to stress, since the stress is always higher than average, while the direction of the maximum stress changes continuously. The local radius could be the only reason that the submartian point and the north pole are more subjected to stress than other regions.

A combination of tidal load and impact load

Besides from taking the tidal load or impact load separately, it is possible that both forces had a contribution to the grooves. The first cause could be that the asteroid impacted when the material of Phobos was already subjected to tides. The second cause is that the periodical load cause fatigue of the material in a certain direction. When the impact occurred, it would behave anisotropically.

The stresses by the two different contributions are scaled in order to see the effect of the combination most clearly. Nevertheless, the two extremes (only impact or only tidal force) were already discussed previously. The results are depicted in the right column of figures 5.14 and 5.15. The impact load only amplifies the stresses near the impact, but does not change much to the overall pattern of grooves.

Conclusion

Based on the presented model, the connection with the grooves and tides or impact cannot be proven. Apart from the fact that the crack direction is not in accordance with the observed grooves, no concentration of grooves near the geometric north pole is observed. Still, two causes that are validated are not ruled out. The irregular shape or anisotropy of the material can give a preference for certain areas to initiate cracks. Only numerical methods can provide more insight in the cause of the grooves. 5.5 The causes of the grooves 85

Figure 5.14 The stress concentrations and crack direction. Left represents only tidal load, right is the tidal load combined with an impact. The upper figure represents the situation in pericentre. Every step is a shift of 1/8 of a complete orbital cycle. The viewpoint is chosen such that the most grooves should be visible. 86 Phobos

Figure 5.15 Same as figure 5.14, but now starting from apocentre to almost pericentre again. Chapter 6 Conclusions and recommendations

Tides are one of the most important forces in our solar system. This report deals with modelling the tidal deformation of Europa and Phobos.

Since Europa most likely contains a fluid or slush layer under the surface, the normal mode analysis is adapted for internal fluid or low-viscous layers. The elastic deformation that is found is in accordance with other literature. The viscous de- formation is validated by gradually decreasing the viscosity and checking whether the fluid solution matches the low viscous limit.

13 If µice < 10 P as the surface deforms instantly to the tidal potential. The deformation of the surface is then 30 metres in the case of an ocean, while the case without ocean deforms only in the order of one metre. This is in accordance with the results by [Van Barneveld, 2005]. For higher ice viscosities, the ice does not adjust completely with the tidal variation. The deformation is then highly dependent on the ice thickness, ranging from 15 metres for 100 kilometres of ice till 30 metres for virtually no ice. Although the presence of an ocean can be easily distinguished, the deformation is independent on the ocean thickness. When the ice has the highest realistic rigidity, the tidal bulge has a maximum phase lag of about five hours on an orbital period of 3.55 days. The maximum phase lag does not occur when the relaxation time of the ice is similar to the orbital period, but it should be about one order shorter.

The tidal effect of the other Galilean moons must be taken into account for maximum accuracy, but the influence of the Galilean moons separately will not provide extra constraints on the properties of the surface.

It is theorised that Europa is rotating slightly faster than its revolution time. Relatively to Jupiter the moon would rotate about its axis in 104 years. It is possible that the core cannot catch up with this rotation. For a liquid core it can have a phase lag of about 8 degrees, while a solid core has a maximum of 4 degrees. Besides the properties of the ice an altimetry mission to Europa could also put extra constraints on the core.

The shape of Phobos is close to a tri-axial ellipsoid. When the bulk viscosity is about 4 × 1019P a, which is lower than for average rock, the moon would have deformed in the last three billion years to its present shape by tidal deformation.

87 88 Conclusions and recommendations

Modelling a destructive impact like on Phobos shows cracks that radiate from the crater and die out at its antipode. Since the impact is modelled as a load, the analysis gives a very inaccurate result. The tidal load on Phobos generates cracks mostly at the poles, but there is no reason for the observed abundance of grooves at the north pole and the submartian point. Therefore, this method can give no exclusive cause for the grooves.

Recommendations

This thesis raises several questions for further investigation. For the case of Europa:

Up to now, the thickness of the ice is predicted independently in three ways: assessing the cracks of the ice, conservation of heat and normal mode analy- sis. When these calculations are combined, the properties of the ice can be constrained better. The accuracy of the normal mode analysis can be improved if compressibility is taken into account. The temperature of the ice varies from 100 Kelvins to the melting point of ice. Therefore, the material properties of the ice can vary from cold rigid inviscid ice at the surface to sluggish viscoelastic ice. This gradient of material properties has to be taken into account for accurate prediction of the deformation. The orbit of a Europa mission should be investigated in detail in order to assess up to which accuracy the altitude can be observed.

For Phobos:

If the impactor that caused the Stickney crater impacted under an angle, the extra toroidal deformation and stress component could be modelled using the toroidal solution of the normal mode analysis. The solution can be expanded for an ellipsoidal case. Compare the results with a finite element method. Appendix A Data of moons

A.1 Europa

Discovered by Galileo Galilei Date of discovery 1610 Mass m 4.80 × 1022 [kg] Size R 1565 × 103 [m] Volume V 1.593 × 1019 [m3] Surface Area C 3.1 × 1013 [m2] Normalised Momentum C/MR2 .346 [-] Mean density ρ 3014 [kg/m3] Mean distance from Jupiter r 671034 × 103 [m] Rotational period T 3.551 [days] Orbital eccentricity e 0.0093 [-] [Moore and Schubert, 2000] Orbital inclination i .47◦ Gravitational acceleration g 1.314 [m/s2] Escape velocity 2.025 × 103 [m/s] Visual geometric albedo 0.67 [-] Magnitude 5.3 [-]

89 90 Data of moons

A.2 Phobos

Discovered by Asaph Hall Date of discovery 1877 Mass m (1082 ± 0.1) × 1013 [kg] Size a 13.05 × 103 [m][Thomas, 1993] b 11.48 × 103 [m][Dobrovolskis, 1996] c 9.31 × 103 [m][Dobrovolskis, 1996] 23 2 Momentum I11 4.77 × 10 [kgm ][Simonelli et al., 1993] 23 2 Momentum I22 5.61 × 10 [kgm ][Simonelli et al., 1993] 23 2 Momentum I33 6.60 × 10 [kgm ][Simonelli et al., 1993] Volume V 5.748 ± 0.19 × 1012 [m3][Thomas, 1993] Surface Area C 1.633 ± 0.0035 × 109 [m2][Thomas, 1993] Mean density ρ (1.9 ± 0.1) × 103 [kg/m3][Duxbury, 1991] Density of regolith ρ 1.3 × 103 [kg/m3][Kuzmin et al., 2003] Thermal conductivity k 1.65 × 10−5 [kg/m3][Kuzmin et al., 2003] Mean distance from Mars r (9379.4 ± 0.21) × 103 [m][Szeto, 1983] dr −9 Decrement of distance dt −(1.28 ± .01) × 10 [m/s][Bills et al., 2005] Rotational period T 27570 [s][Szeto, 1983] Rotational speed ω 2.279 × 10−4 [rad/s][Szeto, 1983] dω −21 2 Rotational acceleration dt (47.9 ± 0.2) × 10 [rad/s ][Bills et al., 2005] Orbital eccentricity e .014979 ± 0.000029 [-] [Jones et al., 1989] Orbital inclination i 1◦02 ± 0◦01 [deg][Szeto, 1983] Gravitational acceleration at sub-Mars point g 3.3 × 10−3 [m/s2] at poles g 6.3 × 10−3 [m/s2] Escape velocity 10.3 [m/s][Szeto, 1983] Visual geometric albedo 0.05 [-] [Szeto, 1983] Magnitude 11.3 [-] [Szeto, 1983] Inclination i 1.02◦ [deg][Szeto, 1983] Appendix B Mathematical tools

B.1 Spherical coordinates

Spherical coordinates are displayed in a lot of different way, which leads to con- fusion. In this thesis report, the coordinate system [θ, ϕ, r] is adopted. Here, the colatitude θ is the angle with respect to the north pole, the longitude ϕ is the angle with respect to the meridian and radius r is measured with respect to the centre. The transformation between cartesian and spherical is

r = px2 + y2 + x2 y
θ = arctan x z
ϕ = arccos r , (B.1) or

x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ (B.2)

The Laplacian in spherical coordinates is:

1 ∂  ∂  1 ∂  ∂  1 ∂2 ∇2 = r2 + sin θ + r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂ϕ2

B.2 Spherical harmonics

Here, the solution for the Laplace equation in spherical coordinates is expanded in spherical harmonics. First consider the Laplace equation in spherical coordinates:

1 ∂  ∂Φ 1 ∂  ∂Φ 1 ∂2Φ ∇2Φ = r2 + sin θ + = 0 r2 ∂r ∂r r2 sin θ ∂θ ∂θ r2 sin2 θ ∂ϕ2

91 92 Mathematical tools

By using the separation of variables, the potential can be written as:

Φ(r, θ, ϕ) = R(r) · Θ(θ) · Λ(ϕ)

Substituting this into the Laplacian differential and dividing by RΘΛ/r2, this yields:

1 d  dR 1 d  dΘ 1 d2 r2 + sin θ + = 0 (B.3) R dr dr Θ sin θ dθ dθ Λ sin2 θ dϕ2

The first term of equation B.3 is independent on the second and third term, so it must be a constant. Writing out the left term yields the ordinary differential equation

r2 d2R 2r + dRdr − C = 0 (B.4) R dr2 R

Since the derivatives of R are accompanied by an equivalent power of r, a polynomial is a solution. Substituting C = l(l + 1), the solution becomes:

l −l−1 R = C1r + C2r (B.5)

Since the gravity potential of a body in free space must vanish at infinity, the first constant must be equal to zero for outside a body. Next, the term Λ can be separated. By multiplying equation B.3 with cos2 θ, one obtains

cos θ d  dΘ 1 d2Λ l(l + 1) cos2 θ + cos θ + = 0 (B.6) Θ dθ dθ Λ dϕ2

The function Λ is now completely isolated, and must therefore be equal to a constant. Substituting −m2 and multiplying with Λ leads to the ordinary differ- ential equation

d2Λ + m2Λ = 0 (B.7) dϕ2

Since standard differential equation has the solution:

Λ = C3 cos(mϕ) + C4 sin(mϕ) (B.8)

What is left of the equation is the following:

cos θ d  dΘ l(l + 1) cos2 θ + cos θ − m2 = 0 (B.9) Θ dθ dθ B.2 Spherical harmonics 93

It is quite laborious to solve the function for Θ. For m = 0, the function is called the Legendre polynomial, after Adrien-Marie Legendre, who laid the base for spherical analysis. The Legendre polynomial is defined explicitly by:

1 dl(− sin θ)2l Pl(cos θ) = 2ll! d cosl θ −1 d  2 dPl(cos θ)  = l(l+1) d cos θ sin θ d cos θ (B.10) (B.11)

For 0 < m ≤ l, the associated Legendre function is valid: dm P (cos θ) = sinm θ P (cos θ) (B.12) l,m d cos θm l

or in recursive relations:

P00(cos θ) = 1

P10(cos θ) = cos θ (2l − 1) cos θP (cos θ) − (l − 1 + m)P (cos θ) P (cos θ) = l−1,m l−2,m , l 6= m lm l − m l l Y 0 Pll(cos θ) = sin θ (1 − 2l ), l > 0 l0=1

The derivative of the Legendre polynomial with respect to the colatitude is, recur- sively: dP (cos θ) l cos θP (cos θ) − (l + m)P (cos θ) lm = lm l−1,m , 0 < θ < π dθ sinθ

The first Legendre functions are listed in table B.2. The solution for a potential outside a body becomes:

∞ l l GM X X R Φ = (C cos mϕ + S sin mϕ)P (cos θ) (B.13) r r lm lm lm l=1 m=0

The Legendre functions are often normalised by [Schrama, 2005]: s (l − m)!  sin mθ  m < 0 Y (θ, ϕ) = (2l + 1)(2 − δ ) P (cos θ) , (B.14) lm 0m (l + m)! lm cos mθ m ≥ 0

When a field is known, the coefficients can be found using the orthogonality of the Legendre polynomials:

 2 (l+m)! (2l + 1) (θ, ϕ)Y dS, m < 0  (l−m)! sS lm   C = (2l + 1) H(θ, ϕ)Y dS, m = 0 lm sS lm    2 (l−m)! (2l + 1) H(θ, ϕ)Y dS, m > 0 (l+m)! sS lm 94 Mathematical tools

l \ m 0 1 2 3 0 1 1 cos θ − sin θ 1 2 2 2 2 (3 cos θ − 1) −3 sin θ cos θ 3 sin θ 1 2 3 2 2 3 3 2 cos θ(5 cos θ − 3) − 2 (5 cos θ − 1) sin θ 15 cos θ sin θ −15 sin θ

Table B.1 Legendre functions Plm(cos θ)

A distribution over a globe, e.g. the topographic height, can be represented by an expansion in spherical harmonics (e.g. [Torge, 1991]):

∞ m=l X X H(θ, ϕ) = ClmYlm(θ, ϕ) l=1 m=−l

here, the surface spherical harmonic Ylm(θ, ϕ) is defined as:   Plm(cos θ) cos(mϕ), m ≥ 0 Ylm(θ, ϕ) =  Pl,−m(cos θ) sin(−mϕ), m < 0

B.3 Fourier series

Consider a periodic function g(x) with with a period of 2L. Then, this function can be represented in the form [Boyce and DiPrima, 1997]: ∞   a0 X fπx fπx g(x) = + a cos + b sin 2 m L m L f=1

since the sine and cosine functions are orthogonal:

Z L mπx nπx  0, m 6= n cos cos dx = −L L L L, m = n Z L mπx nπx cos sin dx = 0 −L L L Z L mπx nπx  0, m 6= n sin sin dx = −L L L L, m = n

it can be proven that:

1 Z L fπx af = g(x) cos dx, f = 0, 1, 2, 3 ... L −L L 1 Z L fπx bf = g(x) sin dx, f = 1, 2, 3 ... L −L L

Fourier series make use of the orthogonality relationships of the sine and cosine functions. It can be proven (e.g. [Boyce and DiPrima, 1997]) that: B.3 Fourier series 95

∞   G0 X kπx kπx f(x) = + G cos + H sin (B.15) 2 k L k L k=1

f(x) has a period of 2L, and the terms of the series are always a convergent and defined by:

1 Z L kπx Gk = f(x)cos dx, k = 0, 1, 2, ... L −L L 1 Z L kπx Hk = f(x)sin dx, k = 1, 2, 3, ... L −L L

List of symbols

e Eccentricity of an orbit [-] f Fourier order [-] G Universal gravitational constant [kg−1m3s−2] g Gravitational acceleration [ms−2] hj Love number [-] i Inclination of an orbit [deg] 2 Iij Moment of inertia [kgm ] kj Love number [-] l Harmonic degree [-] lj Love number [-] M Mass of a body [kg] m Harmonic order[-] R Radius of a body [m] r Distance to the centre of a body [m] S Surface [m2] s Laplace transformed time [t−1] T Rotational period [s] t Time [s] δij Kronecker delta [-] ε Strain [-] µ Rigidity [P a] ν Viscosity[P as] ρ Density [kg/m3] τ Shear stress [P a] τM Maxwell relaxation time [s] σ Normal stress [P a] θ Colatitude, 0(north pole) < θ < π(south pole) [rad] Φ potential [m2s−2] ϕ longitude, −π < φ < π (positive West) [rad] ψ Angle with respect to the mean subplanetal point [rad] Ω˙ Revolution speed [rad/s] ω˙ Rotational speed [rad/s]

97 98 List of symbols

List of subscripts

1 With respect to the axis pointing to the parental planet 2 With respect to the axis pointing in the opposite direction as the average velocity vector planet 3 With respect to the axis pointing perpendicular to the other axes, in the northern direction c Centrifugal f Order (for Fourier series) L Load l Degree (for Legendre polynomials) m Order (for Legendre polynomials) T Tidal up Upper boundary of the fluid layer lo Lower boundary of the fluid layer c Core Bibliography

Anderson, J. D., G. Schubert, R. A. Jacobson, E. L. Lau, W. B. Moore, and W. L. Sjogren (1998), Europa’s Differentiated Internal Structure: Inferences from Four Galileo Encounters, Science, 281, 2019–2022. Barneveld, P. W. L. van (2005), Europan Tidal Deformation: Providing a theoret- ical framework for altimetry data to determine ocean presence, Master’s thesis, Delft University of Technology. Bills, B. G., G. A. Neumann, D. E. Smith, and M. T. Zuber (2005), Improved esti- mate of tidal dissipation within Mars from MOLA observations of the shadow of Phobos, Journal of Geophysical Research (Planets), 110, 7004. Boyce, W.E., and R.C. DiPrima (1997), Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons, Inc., New York. Campbell, J. K., and S. P. Synnott (1985), Astronomical Journal, 90, 364. Chandrasekhar, S. (1969), Ellipsoidal Figures of Equilibrium, Yale University Press, New Haven and London. CODATA (2006), www.codata.org. Craddock, R. A. (1994), The Origin of Phobos and Deimos, in Lunar and Planetary Institute Conference Abstracts, p. 293. Davis D.R., Housen K.R., and R. Greenberg (1981), The Unusual Dynamical En- vironment of Phobos and Deimos, Icarus, 47, 220–233. Ditmar, P. (2006), AE3-E01: Earth and Planetary Observations, Lecture notes on the Earth’s gravity field. Dobrovolskis, A. R. (1982), Internal stresses in Phobos and other triaxial bodies, Icarus, 52, 136–148. Dobrovolskis, A. R. (1996), Inertia of any Polyhedron, Icarus, 124, 698–704. Drobyshevski, E.M. (1988), The origin and specific features of the martian satel- lites in the context of the eruption concept, Earth, Moon, and Planets, 40, 1–19. Duxbury, T. C. (1991), An analytic model for the Phobos surface, Planet. Space Science, 39, 355–376. ESA (2005), www.esa.int. Fujiwara, A. (1991), Stickney-Forming Impact on Phobos: Crater Shape and In- duced Stress Distribution, Icarus, 89, 384–391. Fujiwara, A., and N. Asada (1983), Impact fracture Patterns on Phobos Ellipsoids, Icarus, 56, 590–602.

99 100 Bibliography

Goguen J., Veverka P. Thomas P.C., and T. Duxbury (1978), Phobos: Photometry and origin of dark markings on crater floors, Geophysical Research Letters, 5, 981–984. Goldreich, P., and S. Peale (1966), Spin-orbit coupling in the solar system, Astro- nomical Journal, 71, 425. Goldreich, P., and S. Soter (1966), Q in the Solar System, Icarus, 5, 375–389. Greenberg R., Geissler P. Hoppa G. Tufts B.R. Durda D.D. (1998), Tectonic processes on Europa: tidal stresses, mechanical response, and visible features., Icarus, 135, 64–78. Greenberg, R (2005), Europa - The Ocean Moon, Praxis Publishing Ltd, Chich- ester, UK. Han, D., and L. Wahr (1995), The visco-elastic relaxation of a realistically strati- fied Earth and a further analysis of Post-glacial rebound, Geophysical Journal International, 120, 287–311. Hartung, J.B. (1975), Evidence for early solar system history from lunar craters and implied planetary collision, Lunar Science, 6, 337–339. Holsapple, K.A. (2001), Equilibrium configurations of solid cohesionless bodies, Icarus, 154, 432–448. Horstman, K.C., and H.J. Melosh (1989), Drainage pits in cohesionless materials: Implications for the surface of phobos, J. Geophys. Res., 94, 12433–12441. Hurford, T. A. (2005), Tides and tidal stress: Applications to Europa, Ph.D. dis- sertation, The University of Arizona, United States. Hussmann, H., T. Spohn, and K. Wieczerkowski (2002), Thermal Equilibrium States of Europa’s Ice Shell: Implications for Internal Ocean Thickness and Heat Flow, Icarus, 156, 143–151. Ill´es, E., and A. Horv´ath(1981), On the origin of the grooves on Phobos, Advances in Space Research, 1, 49–52. Jacobi, C. G. J. (1834), Uber¨ die Figur des Gleichgewichts, Poggendorff Annalen der Physik und Chemie, 33, 229–238. Jones, D. H. P., A. T. Sinclair, and I. P. Williams (1989), Secular acceleration of Phobos confirmed from positions obtained on La Palma, Royal Astronomical Society, Monthly Notices, 237, 15–19. Kaula, W. M. (1964), Tidal dissipation by solid friction and the resulting orbital evolution, Reviews of Geophysics, 2, 661–685. Kaula, W.M. (1968), An introduction to Planetary Physics: The Terrestrial Plan- ets, Wiley, New York. Kleuskens, M.H.P. (2006), Origin of the grooves on Phobos, Unpublished. Kuzmin, R. O., T. V. Shingareva, and E. V. Zabalueva (2003), An Engineering Model for the Phobos Surface, Solar System Research, 37, 266–281. Lambeck, K. (1979), On the orbital evolution of the Martian satellites, Journal of Geophysical Research, 84, 5651–5658. Love, A.E.H. (1927), A Treatise on the Mathematical Theory of Elasticity, Dover Publications, New York. Maclaurin, C. (1742), A Treatise on Fluxions. Bibliography 101

Melosh, H.J. (1989), Impact Cratering, A Geological Process, Oxford University Press. Moore, W. B., and G. Schubert (2000), The Tidal Response of Europa, Icarus, 147, 317–319. Munk, W.H., and G.J.F. Macdonald (1960), The rotation of the earth, Cambridge University Press. Murray, J. B., J. C. Iliffe, J. A. L. Muller, G. Neukum, S. Werner, and M. Balme (2006), The parallel grooves of Phobos: new evidence on their origin from HRSC Mars Express, in European Geosciences Union Conference, p. 2195. NASA (2006), www.nasa.gov. Newton, I. (1687), Philosophiae Naturalis Principia Mathematica, vol. III. Peltier, W.R. (1974), The impulse response of a Maxwell Earth, Reviews of Geo- physics and Space Physics, 12, 649–669. Pollack, J. B., and J. A. Burns (1977), An Origin by Capture for the Martian Satellites?, Bulletin of the American Astronomical Sciety, 9, 518. Ranalli, G. (1995), Rheology of the Earth, W.H.Freeman and Company, San Fran- cisco. Rivkin, A. S., R. H. Brown, D. E. Trilling, J. F. Bell, and J. H. Plassmann (2002), Near-Infrared Spectrophotometry of Phobos and Deimos, Icarus, 156, 64–75. Robinson, M. S., P. C. Thomas, and J. Veverka (2003), Missing Craters on Eros, Phobos, and the Moon – Crater Erasure in a Thick Regolith?, in Lunar and Planetary Institute Conference Abstracts, p. 1696. Roche, M.E. (1850), La figure d’une masse fluide, Acad. Sci. Lettr. Montpellier, 1, 243–262, 333–348. Sabadini, R., and L. L. A. Vermeersen (2004), Global Dynamics of the Earth, Ap- plication of Normal Mode Relaxation Theory to Solid-Earth Geophysics, Kluwer Academic Publisher, Dordrecht. Schrama, E.J.O. (2005), Tides, Delft University of Technology. Scienceworld (2006), scienceworld.wolfram.com. Siegal, B. S., and D. P. Gold (1973), Parameters Critical to the Morphology of Fluidization Craters, Moon, 6, 304. Simonelli, D. P., P. C. Thomas, B. T. Carcich, and J. Veverka (1993), The gen- eration and use of numerical shape models for irregular Solar System objects, Icarus, 103, 49–61. Solarviews (2006), www.solarviews.com. Soter, S., and A. Harris (1977), Origin of the grooves on phobos, Nature, 268, 421–422. Szeto, A.M.K. (1983), Orbital Evolution and Origin of the Martian Satellites, Icarus, 55, 133–168. Takeuchi, H., M. Saito, and N. Kobayashi (1962), Statical Deformations and Free Oscillations of a Model Earth, Journal of Geophysical Research, 67, 1141–1154. Thomas, P., and J. Veverka (1980), Crater densities on the satellites of Mars, Icarus, 41, 365–380. 102 Bibliography

Thomas, P. C. (1998), Ejecta Emplacement on the Martian Satellites, Icarus, 131, 78–106. Thomas, P.C. (1993), Gravity, Tides, and Topography on Small Satellites and Steroids: Application to Surface Features of the Martian Satellites, Icarus, 105, 326–344. Thomas P., Veverka J., and T. Duxburry (1978), Are striations on phobos evidence for tidal stress?, Nature, 273, 282–284, May 1978. Thomas P.C., Veverka J. Bloom A., and T. Duxbury (1979), Grooves on phobos: Their distribution, morphology, and possible origin, Journal of Geophysical Research, 84, 8457–8477. Tobie, G., and Sotin C. Mocquet A. (2005), Tidal dissipation within large icy satellites: Applications to Europa and Titan, Icarus, 177, 534–549. Torge, W. (1991), Geodesy, second edition, Walter de Gruyter, Berlin. Tufts, R., R. Greenberg, P. Geissler, G. Hoppa, R. Pappalardo, R. Sullivan, and Galileo Imaging Team (1997), Crustal displacement features on Europa, p. 983. Vermeersen, L. L. A. (2002), Geophysical Applications of Satellite Measurements, Delft University of Technology. Veverka, J., and J. A. Burns (1980), The moons of Mars, Annual Review of Earth and Planetary Sciences, 8, 527–558. Veverka, J., and T.C. Duxbury (1977), Journal of Geophysical Research, 82, 4213. Wakker, K.F (2002), Astrodynamica I, Delft University of Technology. Weidenschilling, S.J. (1979), A Possible origin for the grooves of Phobos, Nature, 282, 697–698. Weimar, P. (2005), Development of a satellite mission determining the chemical makeup of the surface of Europa, Master’s thesis, Delft University of Technol- ogy. Wieczerkowski, K. (1999), Gravito-viskoelastodynamik f¨ur verallgemeinerte Rhe- ologien mit Anwendungen auf den Jupitermond Io und die Erde, Ph.D. disser- tation, University of M¨uster, Germany. Wikipedia (2005), www.wikipedia.org. Wilson, L., and J. W. Head (2005), Dynamics of Groove Formation on Phobos by Ejecta from Stickney Crater: Predictions and Tests, in 36th Annual Lunar and Planetary Science Conference, p. 1186. Wolf, D. (1994), Lam´e’s problem of gravitational viscoelasticity: The isochemical, incompressible planet, Geophysical Journal International, 116, 321–348. Yoder, C. F., and W. L. Sjogren (1996), Tides on Europa, Europa Ocean Confer- ence, California, pp. 89–90. Yoder, C.F. (1982), Tidal Rigidity of Phobos, Icarus, 49, 327–346.