A Model for Moon Formation Around Jupiter Like Planets

Institute for Computational Science at the University of Zurich

Bachelor thesis

A model for moon formation around giant planets formed by gravitational instability

Supervisors Author Dr. Judit Szul´agyi Cassandra Inderbitzi Prof. Dr. Lucio Mayer

August 28, 2018 Contents

1 Introduction 4 1.1 Planets and their satellites in our Solar System...... 4 1.1.1 Jupiter...... 5 1.1.2 Saturn...... 6 1.1.3 Uranus...... 6 1.1.4 Neptune...... 7 1.1.5 Moons of the terrestrial planets...... 7 1.2 Star formation...... 7 1.2.1 Jeans criterion...... 8 1.3 Terrestrial planet formation...... 8 1.3.1 Goldreich-Ward mechanism...... 9 1.3.2 Streaming instabilities...... 10 1.4 Giant planet formation...... 11 1.4.1 Core accretion...... 11 1.4.2 Disk instability...... 12 1.4.3 Comparison of CPDs formed by core accretion and disk instability.... 13 1.5 Satellite formation...... 13

2 Semianalytic model 14 2.1 Disk...... 14 2.2 Disk evolution...... 15 2.2.1 Gas and dust densities...... 15 2.2.2 Temperature...... 16 2.3 Satellite formation...... 16 2.3.1 Formation model...... 17 2.4 Migration...... 17 2.4.1 Type I migration...... 17 2.4.2 Type II migration...... 18 2.4.3 Migration model...... 19 2.5 Accretion...... 20 2.5.1 Accretion model...... 21 2.5.2 Depletion...... 21 2.5.3 Dust reﬁlling...... 21 2.6 Resonance trapping...... 21 2.7 Collisions...... 22 2.8 Population synthesis...... 22

3 Results 23 3.1 Number of satellites...... 23 3.2 Mass distribution...... 24 3.3 Positions...... 28 3.4 Survival timescales...... 32 3.5 Formation temperature...... 35 3.6 Formation timescales...... 36

4 Discussion 36 4.1 Comparison to a Jupiter-analog model...... 36 4.1.1 Number of satellites...... 37 4.1.2 Masses...... 38 4.1.3 Survival timescale...... 39 4.1.4 Formation temperature...... 40 4.1.5 Formation timescale...... 41

1 4.1.6 Summary of the comparison between the Jupiter-analog model and my model...... 41

5 Conclusion 42 5.1 Outlook...... 42

Bibliography 43

2 To my best friend, without whom I wouldn’t be who I am today.

3 1 Introduction

In the second millenium BC, the ancient Babylonians identified the first 6 planets of our Solar System [1]. This shows the longstanding interest of humanity in the mysteries of outer space. With the advance in technology, the telescope allowed for a continued study of the bodies in our solar system. In the 17th century, for the first time moons around other planets were observed, first the 4 big Jupiter satellites by Galileo Galilei [2] and later the largest of Saturn’s moons, Titan, by Christiaan Huygens [3]. It took another century for Frederick William Herschel to discover Uranus [4] and another 30 years for Giuseppe Piazzi to discover Neptune [5] and finally, in the beginning of the 20th century, Clyde William Tombaugh discovered the then 9th planet, which was later reclassified as a dwarf planet, Pluto [6]. And while we are still discovering new objects within our solar system, our focus has mostly shifted to objects that aren’t within our Sun’s sphere of influence. Since the first discovery of an exoplanet in 1992, orbiting around the pulsar PSR B1257+12 [8], and the discovery of the first planet orbiting a main sequence star, Pegasi 51 [9] , we have since steadily expanded that number and as of June 5th, 2018, there are 3786 planets [10] orbiting various types of stars. With technology becoming ever better, it is only a matter of time until we start detecting moons around such planets. In fact, there is already work being done on this, and preliminary results suggest that we might have discovered one already [11].

1.1 Planets and their satellites in our Solar System

The International Astronomical Union divides the objects in a solar system broadly into 4 categories [12]: 1. “A ’planet’ is a celestial body that (a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, and (c) has cleared the neighbourhood around its orbit. 2. “A ’dwarf planet’ is a celestial body that (a) is in orbit around the Sun, (b) has sufficient mass for its self-gravity to overcome rigid body forces so that it assumes a hydrostatic equilibrium (nearly round) shape, (c) has not cleared the neighbourhood around its orbit, and (d) is not a satellite. 3. “All other objects, except satellites, orbiting the Sun shall be referred to collectively as ’Small Solar-System Bodies’.” (The International Astronomical Union,[12]) 4. Satellites, which are objects whose orbit is around planets or asteroids. They are more colloquially called moons. This means that there are 8 planets, from Mercury to Neptune, a number of dwarf planets, such as Pluto and Ceres, and a host of small solar-system bodies, like in the asteroid belt between Mars and Jupiter, in the Kuiper-belt past Neptune, which Pluto is a part of, and in the spherical Oort-cloud, which is the source of comets. While this categorization is a good baseline, it still does not quite work in every case. Pluto in particular, as Charon is designated a satellite, even though the mass ratio is 8:1 and thus the two objects orbit a point that lies outside of either surface, which would make it more accurate to call it a twin system of two dwarf planets. The different types of planets in our Solar System are further subdivided into categories: 1. Terrestrial planets are planets whose mass is mostly compromised of solid material like silicates and metals. These are planets like Earth or Mars and at least in the solar system they all seem to be made up of three basic layers: a metallic core, a silicate inner mantel and a solid outer mantle. Many satellites also share this characteristic composition and in some cases even the sizes are comparable (for example, Mercury is slightly smaller than Jupiter moon Ganymede (see table1)) 2. Gas giants are planets whose mass is mostly made up of gaseous Hydrogen and Helium. Although we do not have a clear picture of their internal structure, the theory is that gas

4 giants have an outer layer made up of gas, which is kept in that state due to internal and external heating, a middle layer of metallic hydrogen, kept in that state by pressure and maybe some solid inner core. In our solar system Jupiter and Saturn are the gas giants. 3. Ice giants are planets whose mass is mostly made up by heavier elements (such as oxygen, hydrogen or carbon, as well as water ice) which is the main diﬀerence between them and gas giants. These planets are Neptune and Uranus.

1.1.1 Jupiter

The largest planet in our solar system is Jupiter. It has so far 69 conﬁrmed moons, although its four main moons (called the Galilean satellites, after their discoverer) make up the vast majority −4 of the combined satellite mass, which is around 2 · 10 MJupiter (as a comparison, the Moon is −2 about 10 MEarth.

Name Diameter [km] Mass [Mjup] Orbital radius [Rjup] Io 3660 4.47 · 10−5 6 Europa 3121.6 2.4 · 10−5 9.6 Ganymede 5262.4 7.4 · 10−5 15.3 Callisto 4820.6 5.4 · 10−5 26.9

Table 1: Jupiter’s moons [26]

Io is the innermost of Jupiter’s satellites and the second to lightest. Its density suggests that Io is 100% rocky, being mainly composed of silicates [27]. Models based on measurements taken by the Voyage and Galileo missions suggest that it is has a similar inner structure as Earth with a silicate/rocky crust, a mantle and an iron core [28]. Gravitational interactions with Jupiter and the outer Galilean satellites, called tidal heating, causes Io to be extremely geologically active, resulting in over 400 active volcanoes. Europa is the lightest Galilean satellite. It is in a 2:1 mean-motion resonance with Io, which means Europa completes one orbit in the same time that Io completes two. Europa’s density suggests that it is about 90% rocky and 10% ice or water. The smooth surface, which is in fact the smoothest of any body in the Solar System, suggest not only that it has a crust of frozen water, but also a layer of liquid water beneath that. This is further supported by magnetic-field data from the Galileo mission, which shows that Jupiter induces a magnetic field in Europa. This could be explained by a conductive layer of salty water, which is kept liquid due to tidal heating, under the frozen surface [29]. This ocean is of great interest, as it could be a potentially host life. Ganymede is the heaviest of the Galilean satellites. It is in a 4:2:1 resonance with Io and Europa, so for every orbit Ganymede makes, Europa makes 2 and Io makes 1. It has a radius bigger than Mercury’s, but only about half the mass. This suggests that Ganymede is composed of about 50% rock and 50% water ice [30]. Its inner composition is thought to be an iron core, a silicate mantle and an outer layer of ice and potentially liquid water [31]. Callisto is the outermost of Jupiter’s satellites. It is not in a resonance with the other Galilean satellites and is not tidally heated, as there are no signs of geological activity. It’s density suggests that Callisto is also about fifty-fifty rock-ice [30]. However data from the Galileo mission also suggest that its interior composition is an even mix of ice and rock, although the rock ratio slightly increases towards the center [32]. Magnetic measurements also suggest a subsurface liquid water layer [33].

5 1.1.2 Saturn

Saturn is the second largest planet in our Solar system, with about 30% of Jupiter’s mass. It has so far around 60 conﬁrmed satellites and is unique with its very pronounced ring system. It has 6 major satellites, where the heaviest of them, Titan, is in the same mass range as Jupiter’s moons. This means Titan is about two orders of magnitude heavier than the rest of Saturn’s major satellites, which is a very diﬀerent system than Jupiter’s, where all the major satellites are −4 in the same mass range. The combined mass of all its satellites is also around 2 · 10 Msaturn, which is in the same relative range as in the case of Jupiter.

Name Diameter km [40] Mass Msaturn [41] Orbital radius Rsaturn [42] Enceladus 504 1.94 · 10−7 4.09 Thethys 1062 1.09 · 10−6 5.06 Dione 1123 1.94 · 10−6 6.48 Rhea 1527 4.05 · 10−6 9.05 Titan 5150 2.38 · 10−4 20.98 Iapetus 1470 3.17 · 10−6 61.14

Table 2: Saturns moons

Enceladus is the smallest of Saturn’s main satellites. Data from the Cassini mission suggest that it is mostly made up of water-ice, with a center slightly enriched with silicates [34]. Cassini also discovered water-rich plumes venting from Enceladus [35] and as of today, around 100 such geysers have been discovered [37]. It is thought that tidal heating creates sub-surface oceans and eruptions are facilitated by cracks in the surface due to tidal stress. Recently, it has been discovered that there are complex organic compounds in those water plumes. This, together with the evidence of liquid water makes Enceladus a very good candidate to support life. Titan is the largest Saturnian satellite and the second-largest satellite in the Solar System. It is similarly composed as Callisto and Ganymede, about fifty-fifty rock-ice [39]. Titan is the only moon with a dense atmosphere and the only object other than Earth with confirmed surface liquids, in Titans case liquid hydrocarbons [38]. The combination of dense atmosphere, surface liquids and the presence of complex hydrocarbons make Titan the most likely place in the Solar System, other than Earth, to find extraterrestrial life.

1.1.3 Uranus

Uranus is the the lighter one of our Solar System’s two planets classiﬁed as ice giants. It weighs about 15 Earth masses, with about a third of that contributed by rocky material ([49],[50]). The rest is made up of ice, about 90% of which is made of water, ammonia and methane. The rest is hydrogen and helium [48]. It has about 27 moons and a lighter ring system than Saturn, with the integrated mass of the satellites following the trend of our Solar System with about −4 10 Muranus . Uranus is also unique in that its rotation axis is tilted by 98 degrees compared to the rotational axis of the Solar System. The reason for this tilt is thought to be a collision with another planet early in its development [51].

6 Name Diameter km [43] Mass Muranus [44] Orbital radius Ruranus [45] Miranda 471.6 7.60 · 10−7 5.1 Ariel 1157.8 1.56 · 10−5 7.53 Umbriel 1169.4 1.35 · 10−5 10.50 Titania 1576.8 4.06 · 10−5 17.19 Oberon 1522.8 3.47 · 10−5 23.0

Table 3: Uranus’ moons

1.1.4 Neptune

Neptune is the heavier of the two ice giants, with about 17 Earth-masses. Its composition resembles Uranus’ ([48],[49]). It has about 14 moons, with Triton being by far the most massive. Triton is also very special in that its orbital rotation is counter to that of Neptune itself [52]. That is why it is believed to be a captured object from the Kuiper-belt [53].

Name Diameter km [46] Mass Mneptune [58] Orbital radius Rneptune [59] Triton 471.6 2.09 · 10−4 5.1

Table 4: Neptunes moons

1.1.5 Moons of the terrestrial planets

The moons of the terrestrial planets are very different from giant planets’ moons, because their formation process is different. Mercury and Venus do not have any satellite to call their own. Mars has 2 very small satellites that were captured from the asteroid belt. Earth’s moon is by far the heaviest moon relative to its host planet with about 2% of Earth’s mass. There were a lot of very different theories on how Luna formed, such as the capture of an already formed object [56] or the formation through material ejected from Earth’s surface [57], but most of them have significant flaws. The most accepted theory nowadays is that the young Earth collided with an planet roughly the size of Mars, called Theia. This impact ejected enough material from Earth to form a debris field in which Luna eventually formed. This is called the giant-impact hypothesis [55].

Planet Name Diameter km Mass Mplanet Orbital radius Rplanet Mars Phobos 22.2 1.69 · 10−8 2.7 Deimos 12.6 3.13 · 10−9 6.92 Earth Luna 1737 1.23 · 10−2 60.34

Table 5: Terrestrial moons

1.2 Star formation

As planets form around stars, they are a side-product of star formation. Stars can have a mass between around 0.1 Msun and 100 Msun and their formation and evolution depend on that mass. The currently accepted theory of how stars develop was ﬁrst proposed by the German philosopher Immanuel Kant [13]. It says that stars form as a result of the collapse of a gas cloud. The current model suggests that our sun for example started as gas cloud of about 3 light-years in diameter. Its composition was most likely about the same as our sun’s today, that is mostly hydrogen and

7 helium, and its mass was slightly higher than today’s sun, where part of the excess mass was used in the creation of the objects in our solar system. If such a cloud is concentrated in a small enough volume, it will start to collapse because of gravity. Conservation of angular momentum will then cause the gas particles to start to rotate faster and the compression will cause the gas to heat up. As the centripetal force of rotation only balances out the radial force of gravity collapse, over the course of time the gas cloud will eventually settle into a disk, called a circumstellar disk, which will be supported in the vertical direction via gas pressure.

1.2.1 Jeans criterion

The conditions for a gas cloud to collapse in on itself can be derived, as done ﬁrst by physicist Sir James Jeans, with simple force balance estimations. Consider an isothermal spherical gas cloud of radius R, mass M, and temperature T . Such a cloud is will not collapse as long as the pressure force due to gas expansion is bigger than the gravitational force, as the direction in which these two act are opposed. The gas force Fp and the gravitational force FG per unit volume can be written like

F p p ∼ (1) V R F GM 2 G ∼ ∼ Gρ2R (2) V R2V Where G is the gravitational constant, ρ is the density of the cloud and p is the pressure, which 2 can be written as p = csρ, with cs the speed of sound. Together, this gives a criterion for the radius below which a spherical gas cloud collapses, which is

c R < √ s (3) Gρ

This is an order of magnitude estimation, as it considers a spherical and isothermal gas cloud, but more complete derivations show that this expression is within a factor of the order of unity.

1.3 Terrestrial planet formation

For terrestrial planets to form in a circumstellar disk, there have to be solid particles. These are either inherited from the interstellar medium,that is the matter between star systems in a galaxy, or condensate out of the disk as it radiates heat and cools down. They start out micron sized and will eventually start to coagulate. At the beginning, dust particles will stick to each other because of surface forces, which leads to porous constructs. As collision energies are low in the beginning, these constructs, at this stage called aggregates, will continue to collide and grow. As they increase in mass, the collisions become too energetic for them to stick together, but not so much that the fragment. During this phase, collisions will cause the aggregates to compact and thus increase their density. This process works fairly well until the dust particles are roughly cm sized, which has been shown by experiments in low pressure, micro gravity environments [17]. However, growth beyond this scale is diﬃcult: 1. As dust particles do not feel the radial gas pressure gradient, they tend to move slightly faster than the gas they are embedded in. This leads to a drag force and will make them drift inwards [15]. This drift depends mostly on the size and density of the particles and is not linear. It is usually fastest for meter sized objects, although this depends a bit on the disk. This means that objects that grow past cm size are the most prone to migrate into the planet.

8 2. As the aggregates increase in mass, collisions will become energetic enough that instead of sticking together, they tend to fragment on impact. This will result in mass loss for usually both partners of a collision, which will lead to the disk being repopulated with smaller grains. It is intuitively clear that dm to m sized objects colliding will not lead to one sticking to the other, although this depends somewhat on the composition, as for example ice is soft and has high surface tension, which would make sticking possible. So for objects to grow past centimeter size there needs to be a process other than just hit-and-stick.

1.3.1 Goldreich-Ward mechanism

The Goldreich and Ward [18] proposed a process that that a dense layer of dust particles in the disk mid-plane can fragment into planetesimals, which are km sized rocks, because the layer becomes gravitationally unstable and collapses. The simplest such model, called the Goldreich- Ward mechanism, can be derived by looking at the stability against gravity of an inﬁnitely thin gas disk.

If we say that this disk has a uniform surface density Σ and rotates with velocity vecΩ = Ω~0zˆ, then we can use Euler’s equations to describe the evolution of density and gas velocity

∂Σ + ∇(Σ~v) = 0 (4) ∂t ∂~v ∇p + (~v∇)~v = − − ∇Φ − 2Ω~ × ~v + Ω2(~x + ~y) (5) ∂t Σ ∇2Φ = 4πGΣδ(z) (6)

Where Φ is the gravitational potential. If we want to see how stable this is, so that we can say how likely regions of high density are for example, we assume that there is a slight perturbation around some equilibrium value

Σ = Σ0 + Σ1(x, y, t) (7)

~v = ~v1(x, y, t) (8)

p = p0 + p1(x, y, t) (9)

Φ = Φ0 + Φ1(x, y, z, t) (10)

2 dp If we then put this into the above equations, assume that (cs = dΣ ) and only look at the linear terms, we can then see how the perturbations change over time

i(kx−ωt) Σ1(x, y, t) = Σαe (11) i(kx−ωt) ~v1 = (vαzxˆ + vαyyˆ)e (12) i(kx−ωt) Φ1(x, y, z, t) = Φαe (13)

Where we can call ω the growth rate and k the length scale of instability. Depending on what the value in the bracket of the exponential is, it can either simply ﬂuctuate around the equilibrium value or it can grow exponentially. We can then devise a dispersion relation for ω and k

2 2 2 2 ω = k cs − 2πGΣ0|k| + 4Ω (14)

According to this, if ω2 > 0 the perturbation is oscillatory and consequentially stable. If not however, then the perturbations grow exponentially, leading to places with extremely high or

9 extremely low density, which we call fragmentation of the disk. With this relation, we can also devise a parameter to estimate how stable a given disk system is. For that, we want the most unstable scale length, because they will grow the fastest, which we ﬁnd with dω2/dk = 0

πGΣ0 kcrit = 2 (15) cs

2 The disk then becomes unstable for ω (kcrit) = 0 and we can deﬁne the Toomre Q parameter, which states that a system is stable if Q > 1, although the exact number varies a bit depending on the exact system (see also section 1.4.2).

c Ω Q = s k (16) πGΣ0

The Goldreich-Ward allows there to be regions with very high density. The length scale k allows us to estimate the size of those regions and ultimately the mass within them. With a disk like our sun is thought to have had, this means that we would have unstable regions of space the size of

2 2cs 8 Rcrit = ' 3 · 10 cm (17) GΣ0

g If we take the mass in this region and equate it to a spherical rock with typical density 3 cm3 , this rock will have a radius of about 6 km. We can also estimate the time the collapse process q 3 λcrit would take via gravitation, tcollaps ' 2GM , which is about a year. That will be slowed down by angular momentum, but even then the time could be as low as 103 years [18]. This shows that on a very short timescale this model allows for planetesimals in the km range to form, thus circumventing the problems coagulation has. There is however a big problem: for this process to occur as described, the disk has to be extremely thin. But the process of thinning creates enough turbulence in the disk, that it could never reach the required thinness.

1.3.2 Streaming instabilities

In general, vertical settling and coagulation can lead to a somewhat thin layer of dust and gas in the mid-plane, but not thin enough that it would become gravitationally unstable. However, since the disk is made up of dust and gas, this mixture would allow for streaming instabilities. That is a catch-all term for instabilities that are caused in coupled mixtures of particles and gas. Analytical models [85] have shown that there are feedback interactions between the embedded dust and the gas, leading to locations in the disk where the dust clumps heavily together, thus allowing for gravitational collapse into planetesimals. One possible way for this over-dense regions to form in a overall stable disk is that radial dust drift leads to dust from the outer region to migrate inwards and increasing the dust density there (see1). It is also possible for such instabilities to be induced from outside of the disk, for example interactions with other, nearby forming or formed stars. Once there are planetesimals, they will become less susceptible to drift because of gas drag. Through collisions with the dust and other planetesimals, they can increase their mass to eventually form the terrestrial planets we can observe.

10 Figure 1: Schematic of how radial drift, caused by aerodynamic forces on dust particles in a gas, might create over dense regions in a globally stable disk, which lead to clumping of dust that allows for gravitational collapse. This would not need a disk as thin as the Goldreich-Ward mechanism. (Graphic is taken from Armitage [64])

1.4 Giant planet formation

1.4.1 Core accretion

One of the theories of how giant planets form is an extension of the terrestrial planet formation and is called core accretion [60]. It says that if a solid core of around 10-20 Earth-masses forms beyond the snowline, which is the distance from the star after which the temperature is low enough for water to freeze, the water ice will help the growth of the core. That is because water ice makes sticking after collisions with dust particles and planetesimals more eﬀective and allow the core to grow heavy enough to start to accumulate a signiﬁcant gas envelope. This envelope will be light at the beginning and support itself by hydrostatic equilibrium. If the planet continues to accrete planetesimals and gas, the envelope will eventually reach a critical mass, usually about as heavy as the solid core [61]. After this it can not support itself any more, contracts heavily and a phase of rapid accretion of gas begins. This phase will last for as long as there is gas in the stellar disk [62]. If there is enough gas, a disk can form around the accreting gas giant, very similar to a disk around a star. It is usually called a Circumplanetary Disk or

11 CPD. This model has one diﬃculty though in that the process of gas accretion has to happen on a timescale short enough that the core planet does not migrate into the star (see section 2.4) or close enough that it could not accrete all the gas it needs.

Pebble accretion The model by Pollack et al. [60], which predicts that core accretion creates planets on a timescale similar to the lifetime of the gas disk in which it is located, assumes that the core grows through accumulation of planetesimals. It has however been shown ([19],[20],[21]) that accumulation of dust in the mm-cm size can speed up that process by a factor of 30-1000 at 5 AU, and 100-10000 at 50 AU. However, this only works if the density of these pebbles is high enough and if there is a seed planetesimal of around 100 km radius.

1.4.2 Disk instability

There is a second way that giant planets can form which works with gravitational instability and is thus called the disk instability model ([63], [65]). The theory is that the disk grows unstable enough that it forms spiral arms, in which eventually clumps form which can collapse into planets (see ﬁg.2)

Figure 2: Results of disk instability simulations show clearly that this type of disks forms spiral arms and clumps which will eventually collapse into planets (Simulations were done by L. Mayer and T. Quinn [70])

The measure for gravitational instability is a parameter known as the Toomre Q eq. (16) and in general, Q < 1 is used as the threshold after which instabilities occur. Although this number was derived for small, axisymmetric disturbances, it can still be applied to most disks at least as a general rule. However, it has been shown that spiral arms form already at Q ≤ 1.7 [68]. In these spiral arms, mass infall and/or rapid cooling can lead to the emergence of clumps, as can be seen in ﬁg.2. Fragmentation can happen for Q ≤ 1.4 [67]. These are not set numbers, as they depend on the structure of the disk, but are rough guidelines one can follow. Any clumps that are formed will collapse into a planet if the can contract fast enough to not be torn apart by disk shear. If they survive, they will form gas giants of mass > 5Mjup within ∼ 104 − 105 years that will be composed of mainly hydrogen and helium and thus have a low metallicity as well as no solid core. The problem with this model is to actually get the instablilies to form planets. Usually, even in a massive disk, this requires very fast cooling. This results in most clumps being formed far away from the star, past ∼ 30AU. In principal, instablilites can also be induced by other means,

12 for example perturbations as a result of a binary companion or interaction with another, close star/disk formation [66].

1.4.3 Comparison of CPDs formed by core accretion and disk instability

In the core accretion scenario, the last phase of rapid gas accretion can form a CPD, while in the disk instability case the disk is formed as a result of the gas collapse. These two types of disk can be vastly different in some properties or very similar in others, as shown in the work of Szulágyi,Mayer and Quinn [70] 1. Temperature: In the case of core accretion, the rapid collapse of the gas envelope will lead to a massive temperature increase and the dense gas near the core makes it hard to cool down, so that the temperature will be ∼ 103 K. The gravitaional instability disks however will be much colder. This is partly because the usual way instabilites are created is by rapid enough cooling, but it will end with core temperatures of ∼ 102 K, so about an order of magnitude less. 2. Mass: There is some difference in disk mass between the two scenarios. Core accretion disks are usually around 12% of the mass of the central planet, where gravitaional instability disk can be up to 100% the mass of the planet. The simulations however also show that, in both formation scenarios, the disk mass is much more sensitive to the mass of the stellar disk. 3. The dimensions are about the same, as the seem to only depend on planet mass and position in the stellar disk.

Figure 3: Comparison of densities and temperatures of disks created by core accretion and gravitational instability. The CA-1 and GI cases are done with the same planet mass, 10 Jupiter masses, and at the same position, 50 AU from the host star. It shows that densities are similar while temperatures are very diﬀerent. (Simulations were done by L. Mayer, J. Szul´agyiand T. Quinn [70])

1.5 Satellite formation

There are three ways how satellites around planets can form: 1. They form in a circumplanetary disk. It is in essence a smaller version of the stellar disk it is embedded in and as such, the same processes that form terrestrial planets can also form satellites in a CPD. This can therefore only happen around giant planets, as only they are massive enough to form a disk around themselves (regardless of what formation scenario created the gas giant).

13 2. They can be captured objects that are created elswhere in the stellar disk. This can happen with both giant and terrestrial planets. 3. They can be a result of a planet-planet collision which creates a debris ﬁeld around a planet in which eventually satellites can form.

2 Semianalytic model

This model is a 1D approximation. I will assume that gas and dust densities as well as the temperature depend only on the radius. The disk is divided into a grid, stretching between 3. − 12555.Rjup, with a cell size of 1Rjup, where density and temperature are constant throughout a cell. The simulation works by calculating the rates of accretion and migration of the satellites, as well as the rate of depletion and reﬁlling of the cells and the evolution of densities and tdisp temperature, assuming they stay constant over a time step dtmax = 1000 , with tdisp a typical timescale over which the disk disperses. Satellite orbits are assumed to be coplanar and circular, with migration manifesting as a change in the radius of said orbits. The simulation ends after 14 tdisp, as that is when gas densities are low enough in the whole disk that migration essentially stops.

2.1 Disk

Figure 4: Plot of the disk surface density as a function of distance from the central planet

The disk that I use in this model is taken from fully hydrodynamical simulation done by Szul´agyi,Mayer and Quinn [70]. In these simulations, four planets were created via gravitational instabilities. One at around 10 Mjup, another at around 20 Mjup and two even more massive than that. For this model, the 10 Mjup planet was chosen, which orbits at around 50 AU. Both the gas density and the temperature exhibit behaviour akin to exponential laws and as such, I ﬁtted curves to it, which look like this:

−6 −1.40· r −4 1.01· r −6 −1.49· r Σgas,0(r) = −6.01 · 10 · e 1 AU · (1 − 2.66 · 10 ) · e 1 AU ) + 9.30 · 10 · e 1 AU (18)

For the dust density, it is usually assumed that a set percentage of the disk is made up of solid particles, called dust (see section 1.3). Often, this is assumed to be 1%, as that is the observed

14 fraction for the interstellar medium. However, observations of protoplanetary disks, that are disks where planets have not formed yet, have shown a wide variety of so called dust-to-gas ratios [71]. As such I will assume that the initial dust density proﬁle corresponds to the initial gas proﬁle, multiplied by a certain fraction.

Figure 5: Plot of the temperature proﬁle as a function of distance from the central planet

The temperature takes the following form

−0.87 r −0.87· r T = 381.14 · e 1 AU + (1 − e 1 AU ) · 11 (19)

This clearly shows a key characteristic of disks formed via gravitational instabilities, which is its very low bulk temperature.

2.2 Disk evolution

2.2.1 Gas and dust densities

During the course of the simulation, the densities and the temperature will change. For the gas density, in principal one would have to solve both Euler’s equation and the continuity equation to determine the change over time. However, it has been shown that, in a very good approximation, the density will decline like

t t Σg(r, t) = Σg,0(r)e disp (20) where tdisp the timescale on which the disk disperses [72]. For the evolution of the dust density, one has to keep in mind that, even though it is treated like a gas, they are still distinct, massive particles. As such, one can look at how coupled these particles are to the gas, which allows to say how much of the dust behaviour is determined by the gas. One possible measure for this coupling is the Stokes number [69]

π ρ r St = d d (21) 2 Σg

15 g where ρd is the typical density of the dust (typically around 2.5 cm3 ) and rd the size of the particle. This shows (reference the to be implemented plot of the stokes number) that particles up to 1 m size are well coupled with the gas and thus I assume that the dust density will decrease like the gas density

t X Ms,i Σ (r, t) = Σ (r)e tdisp − (22) d d,0 4π · r2 i F,i while also taking into account the mass that is accreted by the satellites.

2.2.2 Temperature

In this model I also take into account that the gas will cool over time. This is an important process, as the temperature of the disk will inﬂuence the composition of the forming satellites. A way to model this which is used fairly often is called radiative cooling. Everything that has a non-zero temperature will emit radiation, which will cause the object to cool down. In a disk, where we will have lots of gas particles, the cooling of the disk as a whole will depend on the composition and thickness of the disk. One analytical way to look at this was determined by Wilkins & Clarke [73]. It gives an expression for the change in inner energy of the gas

T 4 − T 4 U˙ = −σ min (23) (τ − τ −1) where σ is the Boltzmann constant and τ the optical depth of the medium. τ is deﬁned by the density and the opacity of the medium and can be calculated with

Z h τ = κρdz (24) 0 where κ is the opacity of the medium and is usually given in tables as a function of P and T or ρ and T in the midplane (for example in [74]). While this is a very good model, it is computationally expensive and also relying on a tabled parameter. Because of this, I will use a simpler model, which has been shown to be a decent approximation [72]

t t T (r, t) = Tmin + (T0(r) − Tmin)e temp (25) where ttemp is taken to be the dipsersion timescale tdisp

2.3 Satellite formation

There are two main theories about how satellitesimals are formed. They are the same as for terrestrial planets. One is that some ultra-eﬃcient coagulation creates planetesimals in the kilometer range. The other says that the planetesimal are a consequence of fragmentation because of gravitational instability in the disk.

16 2.3.1 Formation model

I do not actually model either of those two theories directly. Implementing a proper coagulation routine, which would mean that I have to treat the dust as actual particles, would be beyond the scope of this work. The disk as it is also does not allow a straightforward way to implement a model where coagulation is helped by processes like streaming instabilities. In other works (Cilibrasi et al for example [75]), the disks radial velocity profile has a region in which the sign changes, so that one can construct a dust density profile which has a dust trap at a certain position. This allows for the local dust-to-gas ration to be unity or higher, which would allow to rapidly build a satellite embryo. The disk used here does not exhibit such behaviour and thus I can not use a similar approach. Instead, I use an approach like Miguel and Ida [76], where I seed the disk with a certain amount −7 of satellite embryos with a mass of 10 Mjup. This seems like a reasonable maximum, as making the embryos lighter does not change the results, but making them heavier leads to a noticeable change. I spawn them between [0, tdisp/2], on positions between 1% − 80% of disk radius rdisk = 6 AU. The positions are chosen at the beginning of the simulation such that they are uniformly distributed between [log10(0.01∗rdisk), log10(0.8∗rdisk)]. I choose log-uniform distribution to deal with the fact that the range in which I spawn the embryos spans multiple orders of magnitude. Additionally, I also allowed for embryos to be spawned at a later date, as the disk is in principal massive enough to form possibly several generations of satellites. Such a model is called sequential formation [77]. As this disk does not allow for formation to be helped by gravitational instabilities, I assume that satellite formation will occur on a timescale comparable to what coagulation gives. As such, I assign a cooldown to every initial seed position after which a new embryo will be created. Using the results of Shibaike et al. [78], this cooldown is uniformly and randomly distributed in the range [104, 105].I also scale this exponentially with et/tdisp like I scale the dust density as I can reasonable assume that the less dense the disk is, the harder it is to form satellites via coagulation. This is by no means a definitive model, but rather a way to have sequential formation happening on a reasonable timescale.

2.4 Migration

In general, there are two ways to model how objects in a CPD migrate, the so called type I migration, which is due to angular momentum exchange between gas and objects, and type II migration, which happens when an object is massive enough that the angular momentum exchange between planet and disk is strong enough to push all the gas around the planet away and create a gap in the gas disk.

2.4.1 Type I migration

An object embedded into a viscous disk will experience a certain amount of torque due to gravitational interaction with the particles that make up the disk. Say we have a migrating satellite at a certain position a, moving with the velocity Ω~p and a gas particle moving past it with velocity Ω~ at certain distance b, called the impact parameter. The interaction between the two will lead to a change in the perpendicular velocity component. In ﬁrst order perturbation approximation it can be written as

2 · GM ∆v = p (26) ⊥ b∆v

This being a radial force, it doesn’t change the angular momentum, but since the kinetic energy

17 is conserved, there is a change to the parallel component of the velocity. This is (for small deﬂection angles)

1 2GM 2 δv ' p (27) k 2∆v b∆v and we can write the total change in angular momentum as

2G2M 2a ∆j ' p (28) b2∆v3

Now we can also look at the sign of this momentum exchange. Say we look at the interaction between a satellite initially on a circular orbit and the gas exterior to it. The gas will orbit slower than the satellite. Therefore, as the parallel component of the relative velocity decreases, the angular momentum of the gas increases, forcing the satellite to migrate inwards. Equally, interaction with the gas interior means the gas interior decreases, forcing the satellite to migrate outwards. The net sum of these two processes will eventually decide whether the satellite migrates out- or inward. The total torque on the satellite can be estimated by integrating over all the particles in the gas. We say that the disk is made of annuli of thickness db with dm ' 2πaΣdb. We assume an annulus rotates with a velocity Ω~ , so the planet will encounter all particles in an annulus during a time ∆t = 2π . If we assume b a, we can then integrate from a minimum Ω−Ωp b to ∞ and write the total angular momentum change rate like this:

2 2 dJ 8 · G Mp aΣ ' − 3 2 (29) dt 27 · b Ωp

This being a ﬁrst order perturbation approximation, it is somewhat crude. A better, but ultimately more diﬃcult way to compute the net torque on a satellite in a disk would be with the so called Lindblad resonances, where the torque is decomposed into partial torques at resonant locations ([79],[80]). If we look at a circular orbit of a satellite that orbits with Ωp, we can then say that there exist rotation resonances [81] if m|Ω − Ωp| = ±Ω, for m an integer. The sum over all this torques gives the net torque which ultimately leads to the satellite migration. While this is a very good and accurate way, it is not easy to accomplish analyticallly. You can however do it in 3D hydrodynamics simulations, which can give an approximative expression and the the most commonly used was derived by Tanaka,Takeuchi & Ward [80]

2 aΩp 2 2 T = −b q Σpa Ωp (30) cs where q is the mass ratio between satellite and center planet, cs the local sound speed of the gas and the subscript p means ’at the location of the satellite’. The parameter b is a correction factor which is used to adapt the general formula to diﬀerently structured disks.

2.4.2 Type II migration

Type I migration is valid as long as the satellite does not perturb the disk so much as to open a gap. However, it is possible that, for a satellite that is massive enough, the angular momentum exchange between disk and satellites is big enough that it overcomes gas drift caused by gas internal angular momentum exchange, called viscous drift, eﬀectively pushing the gas out of a certain region which we call a gap. There are two conditions for which a gap can open.

18 The ﬁrst being that the hill radius of a given satellite should be roughly the same as the height of the disk (because the satellite has to realistically aﬀect all the gas of the gap)

q 1 r = a 3 (31) h 3 and the second one being that the tidal torque has to be strong enough to push the gas out of the gap faster than the viscosity can reﬁll it, which we can write as

∆r 1 ∆r 2 tviscous ∼ ≈ 2 2 ∼ topen (32) ν m q Ωp rp with ∆r the gap size and the m coming from the m-th Lindblad resonance. The viscosity in this case is not one caused by gas particles colliding, as this would lead to viscous timescales much larger than the life time of the disk itself. It is instead a way to model angular momentum transport by turbulences in the gas disk as an eﬀective viscosity. The alpha-viscosity model [83] gives the expression ν = αcsh, where cs is the local speed of sound, h is a scale height of the disk and α is a dimensionless parameter which determines how turbulent the disk is. This α is chosen to match the simulations from which I take my disk and has the value α = 4 · 10−4. Crida & Morbidelli [82] combined the two criteria into a single parameter that is very commonly used:

3 c q −1/3 c 2 P = s + 50αq s (33) 4 Ωka 3 Ωka

If P < 1, then it is generally considered that the satellite moves due to type II migration. However, recent work ([84],[86]) has shown that in massive disks, this gap opening criteria makes it too easy for a gap to form. Mueller et al [86] suggest two possible additional criteria, both of which have about the same eﬀect in the I use, making it about 20 % harder to open a gap. In this model, I will use the criteria that is based on a comparison between the viscous timescale 2 (τvisc = a /ν) and crossing timescale, that is the timescale it would take the satellite to migrate da −1 the same distance as the width of the gap, (τcross = RHS dt )[87], with RHS = 2.5RH [88] is the width of the gap. Together, we then say that a gap is opened if

P < 1 and τvisc < τcross (34)

Once a gap is opened by a satellite, the migration of the planet is a result of the barrier at the edge of the gap presented by the torque. The outer gas wants to drift inwards as a result of the viscosity. To prevent the gas from entering the gap, the planet takes angular momentum from the inner gas disk and distributes it outwards. This will cause the gas at the inner edge of the gap to migrate inwards, which will cause the planet to move inwards as well as to maintain the location of the barrier at the inner edge. In practice, this means that the planet will move on the same viscous timescale τvisc as the gas.

2.4.3 Migration model

In this model, I say that the satellites orbit on concentric circular orbits and that migration takes the form of changes of the radius of said orbits. In the case of type I migration, I use the prescription given by Tanaka,Takeuchi & Ward [80], which is

3 MsΣga a 2 v = bI 2 Ωk (35) Mp h

19 with a parameter bI given by Paardekooper [89]. It should be noted that there are many diﬀerent expression one could use for this parameter and that they can be extremly speciﬁc to a certain disk model. I use the one by Paardekooper because it can be reasonably applied here, but it was not derived with a disk as massive as mine. If the satellite opens a gap, it will migrate on the viscous timescale, which looks like this

αc h v = −b s (36) r II a

The factor b = 1 comes from the fact that the type II migration will slow down the II 1+ Ms 4π∗a∗Σg more massive the satellite is [90].

2.5 Accretion

For accretion of satellitesimals of sufficient size (that is usually at least km sized), it is believed that the main mechanism of accretion is direct collision with dust particles and pebbles. This however means that a comprehensive model would involve a full on N-body simulation of the dust and gas particles and satellites. While possible, it would be extremly computationally expensive as well as very prone to integration errors on long scales. Instead, one can use a simple statistical approach. For that, we can look at the collision cross section σ of a satellite. If the mass is low enough, 2 that is simlpy its physical cross section σ = πrs . However, if the mass is significant enough, gravitational effects will lead to the capture and eventually collision of dust that is a certain distance away. This correction to the physical cross section can be attained by using conservation of kinetic energy and angular momentum. Say we have two masses m moving with a relative velocity vrel at infinity on straight trajectories with a separation b. At closest approach, they have a separation of R and a maximum velocity vmax. The conservation laws then state

1 Gm02 1 b mv2 = mv2 − and v = v (37) 4 rel max R max 2 R rel which we can combine to get the maximum separation b within the two masses will collide

2 2 2 4GmRs 2 vesc bmax = Rs + 2 = Rs 1 + 2 (38) vrel vrel where the part in parentheses is called the gravitational focusing term. With this we can then construct a formula that gives the mass ﬂux onto a satellite embedded in a disk with a dust component

dM v2 = ρ vπR2 1 + esc (39) dt s s v2 where v is a typical velocity for the dust, which is usually taken to be the speed of sound cs. Σs We can also assume the density as ρs = 2h , where h is a scale height, which is usually taken as h = cs/Ωk, and we get

2 dM 1 2 vesc = ΣsΩkπRs 1 + 2 (40) dt 2 cs

This is a straightforward, but somewhat simplistic expression for accretion onto a planet. It does not account for the fact that a satellite with signiﬁcant mass can inﬂuence the density of gas

20 and dust around itself. It also assumes that the gravitational focusing term can be obtained by looking at a simple 2D problem between two masses, which may not be the case.

2.5.1 Accretion model

In this model I used the following analytical expression ([91],[92])

r s Rs 2 Ms M˙ s = 2 Σsa Ωk (41) a Mp which is an analytical model based on direct impacts on a satellitesimals. It says that a seed will accrete mass from within its feeding zone, which I take as RF ' 2.3RH [93], until all the mass has been consumed. For the density in the formula above I will average the mass over the whole feeding zone.

2.5.2 Depletion

As the satellites accrete mass, the dust density within the feeding radius will decrease. I do this by subtracting the accreted mass proportionally from each cell, depending on how heavy each cell is, so the mass in the i-th cell Mi will be reduced by

Mi · Macc ∆Mi = (42) Mfeedingzone

2.5.3 Dust reﬁlling

The simulation from which I got my disk parameters shows that there is an influx of gas and dust into the CPD, which is consistent with current literature ([94],[95]). It has a value of ˙ −5 Mjup about Min,0 = 7.44 ∗ 10 year , which should decrease exponentially with time [72] , so that −t t M˙ in(t) = Min,0e disp . This influx, or at least a part of it, will be used by the disk to fill the dents in the dust density caused by the accretion of satellites. It is not really know how this influx reacts to changes in the disk densities. Therefore, I will use a model like Cilibrasi et al. [75]. It assumes that the initial disk is in equilibrium, so that I assign a timescale trefilling on which the disk tries to fill the dents back up to the value they would have if there wasn’t any accretion by satellites. In practice, this is given by

Σs,0 − Σs ∆Σs = dt (43) trefilling

This refilling can only ever be as much as the influx. Should this ∆Mrefilling be bigger, then I scale it based on percentage with the influx as a maximum. Is the ∆Mrefilling smaller than the influx, then the difference between the two will be proportionally added to the whole disk, with the same method as the depletion. I can also say that this influx has the same dust-to-gas ratio as the disk, because the material already in the CPD came from the same vertical influx..

2.6 Resonance trapping

Because of the way I am spawning embryos, I have to keep in mind that they could potentially be migrating close enough to each other to enter mean-motion resonance. If they enter resonance, then they would tend to migrate at the same speed, with a more or less set distance between

21 them. Eﬀectively, you can model this by assuming that 2 converging satellites repel each other a little bit and so I use an analytical approximation [96] which gives an expression to calculate the change in distance between them

−4 2 dbi | ai − aj | rH ' 7 vK,i (44) dt rH ai

1/3 where r = mi+mj is the Hill sphere of their combined system. I will take this as a migration h 3Mcore rate and say that the inner satellite’s velocity will increase by half of it and the outer’s decrease by half of it.

2.7 Collisions

It is also possible that 2 satellites will cross path at some point (this seems to happen to about 10% of spawned embryos). As such, I say that if two satellites are within their Hill sphere, they collide and I simply say that the more massive one survives. This is a really simple model, as the result of a collision depends very much on relative masses, compositions and velocities. However, this will have to suﬃce in the scope of this work, although it is certainly something that could be worth looking at in the future.

2.8 Population synthesis

Figure 6: Visualization of some of the processes and their dependances in a model about satellite formation

The population synthesis model was originally described as a ’link between the the processes that lead to planetary formation and the observed outcomes’ [97]. At that time, more and more exoplanets were discovered, so there was a signiﬁcant statistic about exoplanet properties to discuss. The goal of population synthesis is to take, often simpliﬁed, prescriptions for the processes involved in planetary formation and bring them together in a consistent way. This way, you get a model for how planets (or in the case of this thesis satellites) form without having to run computationally expensive, particle based simulations or the even more expensive hydrodynamical simulations with test particles.

22 Ideally, as many parameters, such as the initial state of the disk, as possible would be taken from either observations or experiments or at the very least have a distribution around which you can vary them.On the other hand this approach can be used to examine possible constraints on those parameters by checking if there are very outlandish outcomes with certain values. In this thesis, I will vary the following parameters: 1. The dust-to-gas ratio of the disk will be in the range [10−3, 10−1] as per observations of protoplanetary disks [71] 2. The dispersion timescale of the disk will be in the range [104, 105], as this are typical timescales for a disk around a planet created by gravitational instability 3. The reﬁlling timescale of the disk will be in the range [102, 105]. This is the widest range possible, as this process is the least known. 4. The positions in which the initial embryos spawn. 5. The amount of initial satelltites.

3 Results

−5 From here on, ”satellite” will refer to objects with M >= 10 Mjup. I would also like to stress that the central planet of this disk is a 10 Mjup planet.

3.1 Number of satellites

Figure 7: This plot shows in how simulations a certain amount of satellites survived. The left panel looks at satellites of around Europa mass, while the right part looks at satellites 10 times more massive

ﬁg.7 shows that the distribution of the number of satellites is symmetric with a peak between 2 and 3, with a maximum of 9. In the tests where the number of seed points was random between 5-20, the only noticeable change was that the maximum number of survived satellites is 10 instead of 9. This strongly suggests that as long as the number of seeds is bigger the peak of my distribution, the results are independent and I can say that my choice of 10 seed points (instead of randomising them)is justiﬁed. The fact that there is no big increase in survived satellites is at least partly explained by the fact that the more objects are in the disk at any given time, the higher the chance is for them to collide. Thus, any embryos that could have potentially grown big enough would collide with one of the bigger satellites which survive. In numbers, the amount of collisions is almost 10% higher when randomising the seed number. The distribution of satellites with a mass bigger than 10−4 is very interesting. In our solar −5 system, in general the moons around gas giants have a mass of roughly 10 Mp. If this were a

23 Satellites > 10−5 > 10−4 0 498 1958 1 1461 2466 2 2084 1866 3 1993 1297 4 1535 833 5 861 417 6 397 129 7 132 30 8 33 4 9 6 -

Table 6: Number satellites

general rule, then in this case, where Mp is an order of magnitude higher, one would expect that distribution to be symmetric around a peak as well. This does not seem to be the case, although one could argue that it is symmetric around one where the left side is obscured because zero is a hard lower limit. This result creates systems that are more like Saturn than Jupiter, with Saturn having 5 satellites −6 that are more massive than 10 Mp and one that is an order of magnitude bigger than that. It also in general creates more Europa sized satellites than in our Solar System. This is because as the central planet is very far out, at 50 AU, and 10 times heavier than any planet in the Solar System. This means that the circumplanetary disk is very big, which allows for more satellites to form. The expectation would be that the farther out a planet is located, the more satellites it should have.

3.2 Mass distribution

Figure 8: The distribution of the masses of all satellites that are active et the end of the simulation −6 over 20000 runs. The masses below 10 Mjup are cropped because of the way I spawn embryos, there are a lot of them that get created late enough that they simply don’t accrete mass anymore, so the numbers are really skewed towards them.

The mass distribution of the survived satellites looks rather ﬂat. There is a very slight peak −4 −2 around 10 Mjup, but it is almost uniform between up until 10 Mjup, after which there is a −1 steep drop and the mass almost never quite reaching 10 Mjup, with maybe 1 or 2 out of 20000 simulations creating a satellite that heavy. The uniformity of the distribution is not inherently a

24 problem or unexpected. This way, it allows to have systems that are like Jupiter, with a few massive moons, as well as systems like Uranus with more, but less massive moons. −2 I want to emphasize the fact that it is possible to have satellites that are 10 Mjup. There are several parameters in my model that should influence how heavy satellites may become: 1. dust-to-gas ratio 2. refilling timescale 3. dispersion timescale 4. mass influx 5. position at which the embryo was spawned 6. creation time I tried to visualize the dependence on the parameters and come up with a way to describe it, but the relationship between the parameters is too complicated to do so, as the figures below show. There are some general statements you can make (although most of them break down the closer −2 to 10 Mjup you get): 1. The heavier a satellite is, the farther out was it created. 2. The heavier a satellite is, the higher the dust-to-gas ratio has to be fig.9. 3. The heavier a satellite is, the longer the dispersion timescale has to be fig. 10. 4. The heavier a satellite is, the more mass influx one needs to have fig. 11. Most of the times the heavy satellites are a combination of several of these parameters , which allows you for example to have a very low dust-to-gas ratio and a very close spawning position and still end up with one of the heaviest satellites. Another explanation for this could simply be that the accretion model overestimates how much mass the heavier satellites can accrete. Certain simulations have shown (citation needed) that heavy satellites will disturb the gas/dust around them much more than in the model used here. This could lead to a slowing down of accretion the heavier a satellite gets. It should also be mentioned that satellites this heavy are rare, as less than 4% of the masses actually do become that heavy.

−2 Figure 9: This plot shows at what positions satellites that grow to mass 10 Mjup and higher are initially spawned. The color of the dots indicates the dust-to-gas ratio of the disk. It shows that in general, the dust-to-gas ratio is higher than 2% to 3%. It also shows that the higher the mass, the farther away form the planet the embryo was created.

25 −2 Figure 10: This plot shows at what positions satellites that grow to mass 10 Mjup and higher are initially spawned. The color of the dots indicates the dispersion timescale of the disk. It shows that in general, the dispersion timescale is longer than around 10−3 years. It also shows that the higher the mass, the farther away form the planet the embryo was created.

−2 Figure 11: This plot shows at what positions satellites that grow to mass 10 Mjup and higher are initially spawned. The color of the dots indicates the amount of mass infall into the disk. It shows only a very small dependence on the mass infall. It also shows that the higher the mass, the farther away form the planet the embryo was created.

26 Figure 12: The integrated mass of the satellite systems produced with 20000 runs. There is a peak around 10−3, which is higher than what is observed in our Solar System, which is about −4 −4 −2 2 ∗ 10 Mp. It also shows a wide spread, with the majority 10 and 10 Mjup

Integrated mass The integrated mass, that is the total mass, of all surviving satellites is −3 −4 spread around a peak of about 10 Mjup, which is 10 Mp in this case. As a comparison, in −4 our Solar System, the integrated mass for the giant planet/moon systems is around 2 ∗ 10 Mp. ﬁg. 12 shows a wide spread of values, with the majority of systems massing between 10−4 and −2 10 Mjup. The reason why most systems are very massive is because the satellite mass should scale with the disk mass, which itself scales with both planet mass and planet distance from the star. The planet in this case is very massive and also far away from the star, which produces a massive disk.

Lost mass Some of the satellites also migrate into the planet over the course of the simulation. Recent works [99] have shown that the atmosphere of Jupiter is about two times more metallic (in this context, metals means elements heavier than helium) than one would expect, that is Jupiter is over-abundant in metals compared to the Sun. An explanation could be that these lost satellites are the cause of this pollution and as such I will also look at this. −2 The median lost mass value is around 2.7 ∗ 10 Mjup which is about 10 Earth masses. At most, −1 around 3 ∗ 10 Mjup can be lost. There seems to be a correlation between how much mass survives and how much is lost, almost linearly, although this correlation is weaker for lower survived masses. The amount of lost mass is mostly independent of the survived mass.

27 Figure 13: This plot shows the amount of mass lost into the central planet against the combined mass of surviving satellites. It shows that survived mass depends almost linearly on lost mass and that usually the survived mass is lower or about the same as the lost mass

3.3 Positions

Figure 14: Positions of the satellites at the end of the simulation. There are two peaks, one close to the planet and one at around 8000 Rjup.

It is also interesting to look at the positions where the satellites end up in the end. 14 shows a peak near the edge of the planet, at around 10 Rjup. This is very close to the edge and also very similar to the gas giants in our solar system. The expectation would be that satellites should end up orbiting farther out, as the central planet is about 10 times heavier.

There is however also a second, more curiously positioned peak at around 8000 Rjup.. To explain this, we can look at the isolation mass, that is the mass that is within the feeding radius of an object at a certain position.

28 −7 Figure 15: Amount of mass within the feeding zone of an embryo with mass 10 Mjup in a disk with 1% dust-to-gas ratio. It shows that an embryo has more mass it can potentially accrete in the region past 1 AU

This shows that in the range where the peak is located, there is a lot of mass inside the feeding zone. But at the same time, gas densities are low, which means long migration times. fig. 16 shows the migration rate of an embryo as a function of the position it is located. The migration rate peaks at around 1 AU, which should make it harder for satellites to end up in this region, as they would migrate away very fast. This would then lead to position distribution like in fig. 14. It should also be kept in mind that the planet around which satellites form can migrate towards the star it orbits. In my case, the core planet does not, but it is in principal possible. Migrating closer to the star would mean that the planet’s Hill-Sphere, within its gravity dominates, becomes smaller. Eventually, it could have a smaller radius than the outermost satellites’ orbital radius, at which point these satellites would be freed from the planet. In the case of my core planet, this would start to happen would it migrate closer than around 20 AU. Even though this model includes a resonance capturing mechanism, which fig. 14 does not show any peaks at resonant locations, which means that it is unusual for satellites to end up in resonance.

29 Figure 16: On the left the migration rate of diﬀerent sized objects at the beginning of a simulation −7 are plotted for every radius. On the right, the migration rate of a 10 Mjup embryo is plotted at diﬀerent times during the simulation at every radius. This shows a maximum in the region where the positions show a minimum and thus can explain the two peaks in the positions.

We can also look at the distribution of mass against positions to also put that second peak into context.

30 Figure 17: This plot shows the mass of satellites against their end position. The two peaks are represented by regions of high point density. Most of the satellites around the outer peak are less massive than the satellites around the ﬁrst peak. However, the satellites with the highest mass also tend to end up far out, presumably because if they were created closer to the planet, they would not be able to accrete that much mass.

ﬁg. 17 shows the two peaks as big concentrations at the inner and outer edge, but it also shows that the outer satellites are lighter than the inner ones. It also shows a hole in the bottom left, which means that light satellites will not be close to the planet. However, the top right is also interesting, as it shows that the heaviest satellites need to be farther out, at least past 1 AU.

31 3.4 Survival timescales

Figure 18: This graph shows the distribution of times at which the oldest of the surviving satellites with mass > 10−5 was created. It shows a slight peak around 3 dispersion times, which matches with the timescale on which I create secondary/tertiary generations. It also shows that in a lot of the cases, a satellite from the ﬁrst survives.

From the way I spawn embryos, it is reasonable to expect that there would be a peak around 2-3 dispersion timescales, which there is. However, there are a lot of cases where one of the ﬁrst generation satellites survives throughout the simulation (see ﬁg. 19), which is interesting enough to look at it a bit more.

Figure 19: This graph shows the number of surviving satellites with mass > 10−5 that were part of the ﬁrst generation created

fig. 20 shows that in general, the disk needs a low dust-to-gas ratio for first generation embryos to survive. This is because low dust means a low change in embryo mass which means a low change in migration rate. Slow migration means higher chance of survival. The figure also shows that the farther out an embryo is spawned, the less important dust-to-gas ratio becomes.

32 Figure 20: This plot shows the initial spawning positions of an embryo that will survive the whole simulation against its mass at the end. The colors represent the dust-to-gas ratio of the disk. It shows that most of the ﬁrst generation survivors formed in a disk with a low dust-to-gas ratio, less than around 2%. It also shows that the farther away an embryo is spawned, the higher the dust-to-gas ratio can be for it to survive.

ﬁg. 21 shows that for embryos spawned close to the planet to survive, the dispersion timescale needs to be low. That is because the dispersion time scale is a measure for the disk’s lifetime and it is easier to survive if the lifetime is short.

Figure 21: This plot shows the initial spawning positions of an embryo that will survive the whole simulation against its mass at the end. The colors represent the dispersion timescale of the disk. It shows that embryos can survive on all dispersion timescales, but the longer the dispersion timescale is the farther away they have to be created.

fig. 23 shows that the refilling timescale does not have a strong influence on whether first generation embryos survive. fig. 22 shows the same dependence on the total mass influx as on the dust-to-gas ratio. This can be expected because the dust-to-gas ratio is one of the parameters that influence the mass influx. Together, fig. 23 and fig. 22 show that the first spawned generation of embryos is mostly influenced by the initial disk and not by the mass that passes through at later times. In general, this means that first generation survivors occur in short lived disks with low dust-to-gas ratios, with the possibility that it occurs in heavier, longer lived disks if they are spawned far out.

33 Figure 22: This plot shows the initial spawning positions of an embryo that will survive the whole simulation against its mass at the end. The colors represent the mass infall into the disk during the whole simulation. It is dependent on dust-to-gas ratio and dispersion timescale. As the dependence mirrors ﬁg. 20, it can be said that the dust-to-gas ratio is more important than the disperion timescale in the survival of embryos throughout the simulation.

Figure 23: This plot shows the initial spawning positions of an embryo that will survive the whole simulation against its mass at the end. The colors represent the refilling timescale of the disk. It also shows no strong dependance of first generation survival on the refilling timescale, which shows that the evolution of first generation satellites is mostly dependent of initial disk structure.

34 3.5 Formation temperature

Figure 24: Formation temperature of the surviving satellites

Thinking of the moons of Jupiter ,two of which are mostly made of rocky material and the other two fifty-fifty rock-ice[ 100], or Saturn, which except for Titan and Enceladus seem to be icy, [101], I look at the temperature the satellites in my model had when they were created. For a satellite to turn out rocky, it can be assumed that its formation temperature, which is the temperature of the disk at the time of creation, should not only be above 180 K (which is the freezing temperature of water in gaseous disk like the one used here), but also high enough that during its migration, it will be able to accrete enough rocky material. On the other hand, if a satellite forms at a temperature already below the freezing point, the satellite will mostly accrete ice during its migration and end up an icy satellite. For example, the Jupiter moons Io and Europa appear to be made mostly out of rock, which would imply their formation temperature should have been comfortably higher than 180 K, while Ganymede and Callisto are about fifty-fifty ice-rock, which implies that their formation temperature would have been close to 180 K to still allow the accretion of rocky material. The formation temperature my model gives yields no big surprises. As the bulk temperature of the disk is very low to begin with, the vast majority of satellites are formed below the temperature where water freezes, which means that they will likely turn out to be icy satellites. This is very interesting, as most of the icy satellites in the Solar System are suspected to have sub-surface liquid oceans, which could potentially contain life.

35 3.6 Formation timescales

Figure 25: Time it takes for an embryo to reach Europas mass

Another property that determines composition, or rather structure, of the satellites is the timescale on which they are create. It is believed that, apart from Callisto , the Jupiter moons are internally diﬀerentiated. That means that they have a layered interior, much like Earth has. For diﬀerentiation to happen, the crust has to melt, which allows for heavier elements to sink towards the core and lighter elements to rise up. In the case of satellites and planets, the energy for this comes mostly from impacts of dust and planetesimals. This means that if a satellite accrets mass very fast, the impacts heat up the crust faster than it can radiate heat away. My results would allow for this to happen, as the most are formed between 104 − 105, with a maximum of around 106, which is an upper limit as this is about the longest a simulation can be. It also allows for very short formation times, going as low as a couple hundred years.

4 Discussion

4.1 Comparison to a Jupiter-analog model

Figure 26: Comparison of densities and temperatures for GI/CA cases

36 To further put the results from this model into perspective, we can compare them to results gained in a model by Cilibrasi et al. [75], which looks at satellite formation around a planet with mass 1 Mjup and at position 5.2 AU, in which they tried to represent Jupiter and the formation of its satellites. The planet used in their model was created via the core accretion scenario. 1. The gas/dust densities and temperatures of the disk are different, as they are around a planet with a mass of 1 Mjup (see fig. 26). 2. As the central planet is a Jupiter mass planet that orbits at around 5.2 AU. This means that the disk is much smaller, only extending around 500 Rjup compared to my 12555 −3 Rjup, and much lighter, around 10 Jupiter masses to mine which is about as heavy as the planet. 3. The disk they used from appropriate hydrodynamic simulations had a dust trap, a region where dust accumulates.[ 62]. This allows for the local dust-to-gas ratio to become unity or higher, at which point streaming instabilities are strong enough to create embryos [85]. Sequential formation happens once an embryo migrates far enough that the dust trap is outside of its feeding radius and the refilling mechanism brings the dust-to-gas ratio to unity again. 4. The dust-to-gas ratios are slightly higher, because the planet is closer to the sun and core accretion needs more solids to create a planet. This means that the lower boundary is approximately 10 times higher than in my case. 5. Typical lifetimes of disks around a planet created by core accretion are about 10 times higher than in the case of disks around planets created by gravitational instability. Although these differences do not allow a direct comparison, one can still try to find differences and similarities between the two models.

4.1.1 Number of satellites

Figure 27: Number of satellites in the CA case (taken from Cilibrasi et al. [75])

Coming from the results presented in section 3, one would expect that,seeing as the disk is closer to the star and thus lighter and smaller, it should be harder to form a lot of satellites. Accordingly, the distribution should look like fig.7 but with the maximum shifted towards a lower number, like 1 or 2. This is not the case, as shown in fig. 27. My model can spawn satellites in almost the whole disk and at different locations, which makes

37 it very easy to end up with more than 5 satellites, as the disk is very big. But because the time between the generations of satellites is long , on similar time scales as the lifetime, it also means it is possible for only one or two satellites to survive. In the model of Cilibrasi et al., it seems to be nearly impossible to end up with only one or two satellites. They explain this by looking at the time between the spawning of two embryos. In their model, this time, which is strongly dependent on the refilling timescale, is either small enough that you will not end up with less than 3 satellites or it is long enough that you end up with no satellites at all, where the switch from one case to the other seems to be very clear cut. In the same vein, the condition that a satellite has to migrate far enough that the dust trap is outside of its feeding radius is a bottleneck that makes it essentially impossible to end up with more than 5 satellites, as it is hard to fit more than this into the space between dust trap and planet. The differences in the two model are thus either because the disk sizes are very different, which means it is easier for my model to create more than 5 satellites, or because the creation mechanism for embryos is different, which means my model can create systems with 1 - 2 satellites more easily than Cilibrasi et al.

4.1.2 Masses

Figure 28: Mass distribution in the Jupiter case. The red lines represent the masses of the Galilean satellites. (taken from Cilibrasi et al. [75])

If we compare the masses, it makes sense to do so relative to the core planet mass. If we do that, we see that in the case of Cilibrasi et al., the distribution is very narrow, with most cases between 10−3 and 10−4, compared to the very flat distribution I got fig.8. This can be explained by the difference in embryo creation. Having a dust trap in which embryos form means that the majority of mass accreted will originate in said trap. This will also mean that even in later stages of the disk evolution, where migration times become slow due to declining gas density, an embryo will still be able to accrete a sizeable amount of mass. In my case, a satellite’s mass is strongly dependent on how far a satellite migrates and it is a lot harder to accrete significant amounts of dust the later they are created.

38 4.1.3 Survival timescale

Figure 29: Number of satellites in the Jupiter case. (taken from Cilibrasi et al. [75])

Because of the disk size difference between the two models, it is also interesting to look at the survival timescale. As the gas densities in the region where satellites move are comparable (fig. 26), the migration rates should also be in the same order of magnitude. This would then mean that it is easier for a satellite to migrate into the planet in a smaller disk. In my model, there are two peaks, one at the beginning of the simulation and one 3-4 dispersion times later fig. 18. That means that it is very likely for a first generation embryo to survive and less likely for second generation and later ones to be the sole last survivors. Cilibrasi et al. find that most of the time, the last generation will form around 3-4 dispersion timescales, and find it very unlikely that embryos survive from start to beginning.

39 4.1.4 Formation temperature

Figure 30: Formation temperatures in the Jupiter case,, where the red line is at 180 K, the temperature at which water freezes in the disk (taken from Cilibrasi et al. [75])

The biggest diﬀerence between the Jupiter-analog and my model are the temperatures. In my case, the bulk of the disk is around 40 K, which means it is virtually impossible to form anything but icy satellites. As the disk closer to the sun is hotter, it should be easier for rocky planets to form in such an environment. And as ﬁg. 30 shows, it is indeed more likely than in my case, but still the vast majority form at a temperature below the freezing point of water in such a disk, which is around 180K. That is because the temperature at the position of the dust trap will, in their model, drop below this point after only about 2 dispersion timescales, whereas most surviving satellites are formed way past this time. This means that in both these models, forming anything but icy satellites is very hard and suggests that there are other mechanisms involved in their creation that were not taken into account.

40 4.1.5 Formation timescale

Figure 31: Formation timescales in the Jupiter case (taken from Cilibrasi et al. [75])

The formation timescales are very similar. Both models predict that most Europa sized satellites are created in a time frame between 104 and 105 years, though it should be noted that in the case of my model this means that most satellites’ formation timescales are on the same order as the disk dispersion timescale. In my model, there are also more satellites that grow very fast, between 102 and 104 years. This is most likely due to the fact that, as most of my embryos are created farther out relative to disk size, the isolation mass is higher (ﬁg. 15) and it is thus easier for embryos to quickly accrete mass.

4.1.6 Summary of the comparison between the Jupiter-analog model and my model

Drawing direct comparisons is made hard because the two main differences, mass and position of the central planet and the way embryos are spawned, both heavily influence the results. It seems that the difference in distribution of number of satellites and mass seem to be a result of the dust trap and its corresponding embryo creation mechanism while the effective masses seem to correspond to planet mass more than disk mass, as for example the maximum mass possible scales roughly with the planet mass, which is about 10 times higher in my model. The survival timescales of the two models disagree only on how many first generation satellites survive, as both have a maximum of around 3 to 4 dispersion timescales. The high number of first generation satellites that survive in my model is to a good part also because I spawn embryos farther out. The differences between the models are thus both because of the disk size difference, the difference in embryo spawning positions and the difference in planet mass. Both the formation timescales and the temperatures seem to end up in the same order of magnitude, with slightly different distributions. This is because both properties should be independent of the disk size and the position of the planet, as the temperature profile of the disk is mostly determined by planet temperature for the maximum, which is dependent on how the planet was formed, and background temperature for the minimum, which depends on the position of the planet in the stellar disk. With these conclusions, the differences between the models are explained by the size of the

41 disk and subsequently embryo spawning positions. This would mean that we can expect more satellites in a planet that orbits far from the star and more massive satellites around a more massive star, while the type of satellites should be only minimally inﬂuenced by size and position of the host planet.

5 Conclusion

My model is a 1D approximation that uses analytic expressions to model the evolution of satellite embryos, which includes migration, mass accretion, collisions, and a mechanism for resonance trapping, as well as the evolution of the planetary disk, which includes gas/dust dispersion and cooling. It uses a population synthesis approach by running simulations with randomized initial conditions such as dispersion and reﬁlling timescales, dust-to-gas ratios, and number and positions of the initial embryos. The goal is to better understand satellite formation around gas giants and also get a rough estimate on exomoon population which could be observed in the future. It allows for a wide range of systems to be created. Satellites can be in the mass range of Europa up until almost Earth sized, which would be easier to detect in an exoplanetary system. They can be formed in a very short time, e.g. hundreds of years, but are most likely to form on a timescale similar to the dispersion of the disk, between 104 and 105 years. This allows for diﬀerent compositions and structures. They are most likely to either end up in a close orbit, between 10 - 20 Rjup, or in a very far orbit, between 3 - 4 AU. They are also almost exclusively icy in composition, as they are formed in a location where the temperature was below the water freezing point.

5.1 Outlook

There are quite a few ways to go from this model, some of which I will talk about here. 1. The biggest caveat is the 1D approximation. It simplifies a lot of things, such as satellite- satellite interactions and satellite movement. Doing it in 3D is too computationally expensive today, but it might be possible to extend this model, or one like it, to 2D. 2. The migration of satellites can also be improved. As already said in the model section, the prescription used here was not derived with a disk as massive as mine in mind, especially the correction parameter bI (see section 2.4.3). There is a big possibility that the satellites will disturb the CPD enough that the usual treatment of type 1 migration does not work anymore, which is something that is worth investigating. 3. The same goes for the accretion. There are already some models that show a partial gap opening that would make it harder for a satellite to accrete mass through collisions. This would definitly be an important thing to consider. 4. Another big question mark is the location I spawn embryos at. I simply spawn a set amount of embryos at the beginning and use a somewhat realistic timescale on which to spawn secondary and/or tertiary generations. This is however a crude way to do it and it also means that varying the parameters in the population synthesis doe not influence how the embryos are spawned. This a disadvantage, as it is reasonable to assume that disk properties influence embryo formation. There are ways to fix this, one of which would be with the use of spiral arms. Simulations of GI disks already show that it is possible that 2 spiral arms are formed early on in the lifetime. Some works [104] show that spiral arms can act as dust traps. It might be possible to create an approximation of spiral arms in a 1D scenario and use the spiral arms as places where streaming instabilities create embryos.

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