A semi-analytical model for the formation of Jupiter's satellites in circumplanetary disks

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Dipartimento di Fisica ”Enrico Fermi”

Tesi di Laurea Magistrale in Fisica

A semi-analytical model for the

formation of Jupiter’s satellites in

circumplanetary disks

Relatore

Prof. Dr. Lucio Mayer

Candidato Co-Relatore

Marco Cilibrasi

Dr. Judit Szul´

agyi

Relatore interno

Prof. Paolo Paolicchi

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Introduction

In the last few years theories about our Solar System formation took a step forward thanks to a more precise comprehension of Jupiter and giant planets evolution within protoplanetary disks1. Today the two most common theories in this field are the Gravitational Instability model (GI) [1, 2], in which a massive disk rapidly cools and fragments into self-gravitating gaseous clumps, and the Core Accretion model (CA) [3], that occurs when collision and coagulation of solid particles form a planetary embryo massive enough to accrete a gaseous envelope2_{. Both of these theories predict the presence of gas and dust disks rotating around}

the forming planets, although their features strongly depend on the formation scenario, whether they are a consequence of gas clumping (GI) or gas accretion onto a core (CA). These disks can be thought as analogous of protoplanetary disks (PPDs) around a star and they are commonly called circumplanetary disks (or CPDs).

Nowadays we do not have any observational evidence of such circumplanetary disks, be-cause of their limited dimension and luminosity, but their existence is suggested by complete sets of 3D hydro and MHD3 _{simulations [4, 5] that predict different structures and features}

(density, temperature, etc.), but the simplest proof of the formation of CPDs can be found in the conservation law for angular momentum. Since the gas around a forming giant planet must have some angular momentum with respect to the planet itself, it can not accrete or clump directly, but it has to form a rotating accretion disk in which momentum is transported outward [6].

An interesting aspect of these disks is that they are believed to be the birthplace of moons around giant planets, similarly to what happens in protoplanetary disks for terrestrial planet formation [7]. Even if we are not able to observe moons in exoplanetary systems (yet), we still have at least 4 examples of satellites systems around giant or icy planets in our own Solar System. First of them, Jupiter has a great satellites system, dominated by four

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A protoplanetary disk is a rotating circumstellar disk of gas and dust, surrounding a young star, in which planets form.

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In case this embryo is not massive enough a rocky planet or a super Earth form.

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large satellites called Galilean (Io, Europa, Ganymede, Callisto). Developing strong satellite formation models and comparing their results to the characteristics of such systems will bring us closer to understand the last stage of giant planet formation, at least in our Solar System. As in the case of planet formation, many mechanisms participate in the evolution of a satellites system, such as, for instance, the circumplanetary disk dissipation and cooling, or protosatellite migration and accretion. Performing 3D gas-dust hydrodynamical or N-body simulations of the entire satellite formation process is not possible today. The only chance we have is to study the details of these mechanisms separately, formulating simpler analytical descriptions for them, and then to produce a numerical simulation including all these analytical prescriptions. This procedure is usually known as a semi-analytical model [8].

This approach has its foundation not only in these analytical descriptions, but also in the constraints used to set initial conditions up, whether they come from observations or simulations. Most of the time we do not have a clear or well-defined knowledge of these constraints, hence the best way to approach the problem is statistically. Observations or simulations may, for instance, provide certain ranges in which initial conditions (e.g. dust-to-gas ratio) or disk features (e.g. disk lifetime) can vary, and sometimes they are also able to define how these parameters distribute in these ranges. Treating them as random variables in a Monte Carlo framework it is possible to produce many different simulations set-ups (even thousands) and to globally study the outcomes. This statistical approach is called population synthesis [8].

In the last few years population synthesis and semi-analytical modeling have been ap-plied to planet formation in PPDs, allowing to have a better understanding of protoplanetary disks structure and satellite formation processes. The essence of these models is to compare the produced planet population to the increasing number of exoplanets we are able to char-acterize today (almost 4000) and consequently to get some very general constraints about, for example, initial conditions in PPDs. In case of satellites, although we do not have an exomoon population at all, comparing population synthesis results to the features of satel-lites systems around giant and icy planets in the Solar System could give a more detailed knowledge at least about initial conditions and processes’ details of giant planet formation in the Solar System itself.

The goal of this thesis is to produce a complete semi-analytical model for Jupiter’s moons formation, taking into account several processes such as CPD dissipation and cooling, pro-tosatellite formation, migration, accretion within the disk, and to perform a population

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synthesis. The main innovations in my work are basically the prescriptions chosen for disk structure and evolution. First, disk density and temperature profiles are modeled fitting the results of 3D radiative hydrodynamics simulations of a CPD formed specifically in a Core Accretion scenario [4] and not, as previously done in literature [9], starting from approxi-mate models such as the Minimum Mass Solar Nebula model4 or viscous 1D radial models [11, 12]. Second, the dissipation of the CPD is computed considering a feeding influx from the evolving PPD [13], while in the same time the circumplanetary disk cools down because of radiative processes [14] and this is something that is not taken into account in previous models.

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The Minimum Mass Solar Nebula (MMSN) is a protoplanetary disk that contains the minimum amount of solids necessary to build the planets of the Solar System. Gas mass is computed assuming a fixed dust-to-gas ratio [10]. The same mechanism applies to the characterization of a Minimum Mass Subnebula for circumplanetary disks.

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Chapter 1

Solar System formation

Figure 1.1: Not-scaled representation of planets in the Solar System. wikipedia.org

The first theories about planet formation in the Solar System came out long time ago. For instance, it is possible to find references to one of the most popular ideas, the so called Nebula hypothesis, in the works of Kant, Laplace and other authors back in the 18thcentury. This idea basically consists in considering that the 8 planets, terrestrial and gaseous, of the Solar System formed from a rotating disk of gas and dust during the formation process of the Sun itself. A description of terrestrial planets formation appeared for the first time when Viktor Safronov published Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets (1972), while the first elements of gas giant planet formation were developed

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in the 1980s.

Recently, new observations and computation techniques have led to a new interest in the problem. In this sense the detection of extrasolar planets can be considered as one of the most important developments in the last years, proceeding from the first planet around a pulsar (PSR B1257+12) [15] and going to the large number of planets orbiting main-sequence stars that are known today, beginning with 51Peg b, the first planet ever found orbiting a star [16]. These observations showed an extraordinary diversity in extrasolar planetary systems and as a consequence, together with the advent of high resolution imaging of protoplanetary disks and the study of minor bodies in the Solar System, they have focused theoretical attention on the initial conditions for planet formation and the role of dynamics in the early evolution of planetary systems.

1.1 Planets in the Solar System and beyond

1.1.1 General definitions

Planets can be defined as large bodies, in orbit around a star, that are not massive enough to derive their luminosity (i.e. energy), or at least not a big fraction of it, from nuclear fusion reactions in their cores. According to this definition there is a maximum mass that a planet can have, i.e. the deuterium burning threshold, which is approximately 13 Jupiter masses1. More massive objects, that have nuclear reactions in their core, are usually called brown dwarfs.

If we want to have constraints also on the mass lower limit and to have a more for-mal characterization, the current definition by the International Astronomical Union (2006) requires a planet to be massive enough to be spherical (because of self-gravity) and to be able to clear its orbit of other large bodies. Usually, objects that are massive enough to be spherical but not enough to have a dominant dynamical influence on close bodies are called dwarf planets. Examples of these objects in our Solar System could be Pluto, which has only the 7% of the total mass of objects in its orbit, Eris and Sedna.

There is also the possibility to have objects with planetary masses that are not bound to a central star, either because they have formed in isolation or they have been ejected from a planetary system. Such objects are usually called free-floating planets.

According to what we observe in our Solar System, planets are usually classified in three main categories, based on composition and dimension:

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• Terrestrial planets: a terrestrial planet, or rocky planet is a planet mainly composed of silicates and metals. Given their composition, they should have all the same basic structure, as it happens in the Solar System, with a central metallic core, a silicate mantle around it and a solid crust as an outer layer. Satellites, such as the Moon or Io and Europa, can also have a very similar internal structures (and size).

• Gas giants: a gas giant is a giant planet mainly composed of hydrogen and helium in gas state. Gas giants also have a three layer structure, consisting of a metallic hydrogen layer surrounded by a large layer of molecular hydrogen without a well-defined surface, with probably a denser core in the inner part. The properties of this dense core are not completely understood yet, because clear observational data are not available today and temperatures and pressure could be so high that it would be difficult even to produce a realistic equation of state for these conditions.

• Ice giants: an ice giant is a giant planet composed mainly of elements heavier than hydrogen and helium (oxygen, carbon, nitrogen), that should be frozen if these planets orbit far enough from the central star, and only a little part of hydrogen and helium in mass. This last feature allows to distinguish them from the gas giants, Jupiter and Saturn in our System, which both have more than 90% of hydrogen and helium in mass.

1.1.2 Planets in the Solar System and their features

The Solar System has eight planets. Two are giants, Jupiter and Saturn, then composed of hydrogen and helium, two are ice giants, Uranus and Neptune, and then we have four terrestrial planets, two of which, Earth and Venus, are substantially more massive than the other two, Mercury and Mars. In addition there are a lot of dwarf planets orbiting the Sun, including for example Pluto, Eris, Ceres and many minor bodies, such as asteroids and comets.

The orbital elements and physical properties of the Solar System’s planets are summarized in Figure 1.2. With exception of Mercury, planets in the Solar System have almost circular and coplanar orbits and another interesting fact about the structure of the Solar System is that giant and terrestrial planets are clearly divided in orbital radius, with terrestrial planets orbiting the Sun much closer than giant planets.

These 8 planets make a negligible contribution (» 0.13%) to the total mass of the Solar System, which is concentrated in the Sun. In contrast to the mass, most of the angular

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Figure 1.2: Orbital and physical parameters of planets in the Solar System. Armitage (2007) [17]

momentum in the Solar System is concentrated in the orbital angular momentum of planets. Assuming rigid rotation the Solar angular momentum can be written as

J@“ k2M@R@Ω (1.1)

where R@ “ 6.96 ˆ 1010cm and Ω “ 2.0 ˆ 10´6s´1. In computing moment of inertia for a

star with a radiative core k2» 0.1 is usually adopted, then we have J@“ 3 ˆ 1048g cm2s´1.

For comparison the orbital angular momentum of Jupiter is

JJ “ MJ

a

GM@a » 2 ˆ 1050g cm2s´1 (1.2)

This issue, known as the angular momentum problem or paradox, leads to the conclusion, which is crucial in studying planet formation theories, that there should be a strong transport mechanism for both mass and angular momentum during Solar System’s formation, that brings mass toward the center and angular momentum toward larger orbital radii.

The planets orbital radii do not show any evident relationships that can be directly linked to their physical properties2_{, but an important phenomenon in the Solar System structure}

and in orbital radii determination is resonance, that occurs when there is a near-exact relation between characteristic frequencies of two bodies and that acts in stabilizing planets orbits while they are migrating, in the first phases of their evolution3. This mechanism is very important because mutual gravitational interactions between planets are generally much smaller than the dominant attracting force from the central body (the Sun in this case) and it is not able, in general, to influence the position of a planet alone. The simplest type of resonance, known as mean-motion resonance, occurs when the period P1 and P2 of two

planets satisfy P1 P2 » i j (1.3) 2

The well-known empirical relation called Titius-Bode law is not thought to have any fundamental basis.

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where i and j are small integers. Jupiter and Saturn are the nearest to a resonance in the Solar System, since their motion is affected by their proximity to a 5:2 mean-motion resonance. Among low mass objects Pluto is in 3:2 resonance with Neptune, but we have many examples of important resonances among satellites (with their planet or between themselves) and even within the asteroids belt.

1.1.3 Extrasolar planets

An exoplanet is a planet orbiting a star other than the Sun. The first confirmed detection of an exoplanet was in 1992, in the pulsar system PSR B1257+12, but we had to wait until 1995 to have the first confirmed detection of an exoplanet orbiting a main-sequence star (51 Peg b) [16]. Today there are 3,671 planets in 2,751 systems, with 616 systems with more than one planet.

About 97% of all the confirmed exoplanets have been discovered by indirect detection techniques, mainly by radial velocity measurements and transit monitoring techniques, since it is very complicated to resolve and directly detect the light of a planet from the light of its parent star. In the first method a star with a planet moves in its own orbit because of planet’s gravity, even if this motion is small, given the negligible mass of a planet compared to the mass of the star. This motion leads to variations in the speed with which the star moves toward or away from Earth, i.e. its radial velocity, that can be deduced from the displacement in the star’s spectral lines due to the Doppler effect. This is the first successful method that led to the first discoveries.

In the second case, if a planet transits in front of its parent star, then the stellar observed brightness drops by a small amount. The depth of this so called dip depends on the relative sizes of the star and the planet and the duration of the transit depends on the inclination of the system with respect to the line of sight. This is the current most successful method, used for example by Kepler (NASA). In both cases the effects are really tiny but nowadays our instruments have the sensitivity needed in order to detect them.

The most important fact about these two methods (and in practice about all detection methods) is that they are strongly affected by the planet’s position, dimension/mass and inclination. For example massive planets close to the star produce a bigger effect on radial velocity, especially when they also have a low inclination in respect to the line of sight (the visible perturbation is 9sinpiq). Then bigger planets produce a bigger effect on transit detection and transit events are more frequent if they are closer to the central star. Again they need to have a low enough inclination to be detected, i.e. to transit between their host

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star and the Earth.

Figure 1.3: Exoplanets distribution in size, that depends on mass and density, vs orbital period, that is linked to semi-major axes by the third Kepler law. Colors indicate the detection method used between radial velocity, transits (that are the most used ones) and also direct imaging, microlensing, pulsar timing. Among transit detection Kepler’s one are highlighted. Lines in the figure represent Earth size, Neptune size and Jupiter size.

www.nasa.gov

This is why the first planets that have been detected are the so called Hot Jupiters, i.e. gas giant very close to their star and, consequently, very hot and inflated. In Figure 1.3 they are in the upper left corner of the plot and they are the the only type of planet we don’t see in the Solar System as describe in section 1.1.1, together with lava worlds, i.e. terrestrial planets that are so close to the central star and then so hot that they are thought to be substantially melted. As showed in Figure 1.3 in the last few years instruments’ detection limit has become lower and lower and now it allows to detect Earth-like (terrestrial) and Jupiter-like (cold giant) planets, at least in their dimensions and orbital parameters.

Not present in the plot we can also have other 2 notable types of extrasolar planets: super Earths and mini Neptunes, that are planets with a mass higher than Earth’s, but lower than Solar System’s ice giants, Uranus and Neptune, which contain 15 and 17 Earth masses respectively. The difference between these two types of planets is basically their

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composition and structure, since super-Earths should have solid surfaces or oceans with a sharp boundary between liquid and atmosphere, while mini-Neptunes should have thick hydrogen and helium atmospheres, probably with deep layers of ice, rock or liquid oceans.

1.2 Solar nebula

Figure 1.4: Fundamental highlights of the nebula contraction that gave birth to the proto-planetary disk in the solar nebula model.

Pearson Prentice Hall

Constraints on theories of planet formation come from both observations of the Solar System and of extrasolar planetary systems. For example space missions (Rosetta, Juno, Cassini just to cite some of the most recent) have collected detailed information on surfaces and interior structures of planets, satellites and other bodies in the System.

Compared to the Solar System, our knowledge of extrasolar systems is much less detailed (in many cases we only know masses and few orbital parameters), but this is partially compensated by the large and rapidly growing number of systems we know and it is only

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by studying extrasolar planetary systems that we can make statistical studies of the range of outcomes of the planet formation process and avoid any bias, caused for example by the fact that the Solar System structure must necessarily be suitable for hosting life.

Today the formation of these exoplanetary systems is thought to have been dominated by the same processes as the formation of the Solar System. As a consequence the nebular model (or hypothesis) can be refined thanks to the huge amount of data coming from exoplanet detection and characterization.

1.2.1 Nebular model

As said in the firs part of this chapter, the nebular model [18] was first developed in the 18th century and it has been slightly changed and refined to take into account new observations during the last decades.

The nebular model says that the Solar System formed from the gravitational collapse of one of the fragments (of about 1pc each4) of a giant molecular cloud (probably 20pc large) and the collapse of such fragments led to the formation of dense cores (0.01 ´0.1pc) [19]. The composition of this region, showing a mass just over that of the Sun (M@), was about the

same as that of the Sun today, with hydrogen, helium and traces of lithium and beryllium produced by the primordial Big Bang nucleosynthesis. Probably an even smaller fraction of the mass consisted of heavier elements that were created by nucleosynthesis in previous generations of stars.

Because of the conservation of angular momentum, the nebula fragments started rotat-ing faster and heatrotat-ing up durrotat-ing collapse. In particular the center, where most of the mass concentrated, became pretty hotter than the surrounding region. After 100,000 years, the competition between gravity and rotation (with some minor contribution by pressure and, possibly, magnetic fields) caused the contracting nebula to flatten out into a spinning proto-planetary disk with a diameter of about 200 AU and surrounding a dense and hot protostar at the center [19].

1.2.2 Jeans criterion

In previous paragraphs we introduced the idea of a gravitational collapse of the solar nebula, but actually we can also estimate quantitatively which conditions could cause a collapse. For example it is possible to estimate which density would trigger it, starting from simple

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A parsec was defined as the distance at which one astronomical unit subtends an angle of one arcsecond, i.e. 1pc » 3 ˆ 1016m » 2 ˆ 105AU » 3.26ly.

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considerations. For example the virial theorem says that d2I

d2_t “ 2ET ` U (1.4)

where I is the moment of inertia of the nebula, ET is its thermal/kinetic energy and U is its

potential energy. The nebula becomes unstable when the second derivative of the moment of inertia is negative, i.e. when 2ET ` U ă 0. Estimating

ET “ 3 2N kT “ 3 2 M 2mp kT (1.5)

for a molecular hydrogen nebula with a temperature of about 10K and U “ ´1 2G żM 0 M prqdM prq r » 1 2G żR 0 4 3πρr 31 r4πr 2_{ρdr “} 1 2 3 5 GM R2 (1.6)

assuming uniform density, we get

ρ ą 10´18ˆ M M@

˙´2

g{cm3 (1.7)

that gives a density threshold for which a collapse should begin.

This is known as Jeans criterion or Jeans instability [20]. Actually the derivation of this criterion could be done in a more precise way using Euler and Poisson equations coupled together in a PDE system such as

$ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ % ρB~v Bt ` ρp~v ¨ ~∇q~v “ ´ρ~∇Φ ´ ~∇P Bρ Bt ` ~∇ ¨ pρ~vq “ 0 ∇2_{Φ “ 4πGρ} (1.8)

First order solution of this equation, assuming uniform density and pressure at the 0th order and an isothermal perturbation with sound speed c “

b

kBT

2mp, gives the following dispersion relation

ω2“ k2c2´ 4πGρ0 (1.9)

and interpreting k as the inverse of the characteristic length of the collapsing system, in-stability condition ω2 ă 0 gives the same condition, but numerical coefficients, of equation 1.7. Obviously this is just an approximate analysis, since one assumes a lot of strong hy-pothesis (homogeneity at the 0th order for example), but this criterion is important because it also predicts the fact that while a molecular cloud is collapsing, instabilities can occur a much shorter scales while density increases, causing fragmentation and creation of different systems.

Going back to the collapse, as it was mentioned before that conservation of angular momentum causes the rotation to increase as the nebula radius decreases. This rotation

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causes the cloud to flatten out and to form of a disk. This occurs because centripetal acceleration from the orbital motion resists the gravitational attraction of the star only in the radial direction, but the cloud remains free to collapse in the vertical direction, i.e. parallel to the angular momentum vector. The outcome is the formation of a thin disk, known as circumstellar disk, supported by gas pressure in the vertical direction [11].

1.3 Protoplanetary disks

Protoplanetary disks, i.e. circumstellar disks where planets are forming, are believed to be generally quite thin, with a typical vertical height much smaller than the radius (that is called aspect ratio, i.e. h{r), and a typical mass much smaller than the central star [21, 22]. The heating of the disk is primarily caused by the viscous dissipation in the inner disk and by stellar irradiation in the outer part. Another heating mechanism that acts especially in the first evolution phase is the gas infall from the nebula [17]. Large disks are observed in many star-forming regions such as the Orion nebula [23] and the total lifetimes of such disks, also called accretion disks, are on average about 10 million years [24].

1.3.1 Disks observation and classification

Back in the 80’s astronomers have been detecting the presence of protoplanetary disks around forming stars using unresolved information from the Spectral Energy Distribution (SED), i.e. the distribution of flux as a function of frequency or wavelength [26].

These so called Young Stellar Objects (YSOs) are classified observationally according to the shape of their SED in the infrared. As shown in Figure 1.5, YSOs often display:

1. An infrared excess that is attributed to hot dust in the disk near the star.

2. A small ultraviolet excess, which is due to high temperature regions on the stellar surface where gas from the disk is accreting.

To quantify the magnitude of the IR excess, it is useful to define a measure of the slope of the SED between the near and the mid infrared

αIR“

∆logpλFλq

∆logλ (1.10)

since this parameter can be useful in classifying YSOs, as shown in Figure 1.6, as follows: • Class 0: SED peaks in the far-IR or mm part of the spectrum, with no flux being

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Figure 1.5: Schematic depiction of the SED of a YSO surrounded by a disk. The presence of a disk is deduced from an infrared excess, while an ultraviolet excess is also commonly detected, and this is attributed to gas accretion on to the stellar surface producing hot spots. Armitage (2007) [17]

• Class I: approximately flat or rising SED into mid-IR (αIR ě ´0.3).

• Class II: falling SED into mid-IR (´1.6 ă αIR ă ´0.3). These objects are also called

T Tauri stars.

• Class III: pre-main-sequence stars with little or no excess in the IR (αIRă ´1.6).

This observational classification scheme is considered, at least from a theoretical point of view, also as an evolutionary sequence [27]. For example in Class 0 YSOs are thought to be at the earliest stage of cloud collapse. At this stage the central protostar is embedded within an optically thick envelope and it is not visible yet. YSOs in Class I are the first in which a disk can be really detected, even if they are still embedded within an envelope of gas and dust, which filters radiation from the star and the disk, usually increasing its wavelength. Class II objects are in a later phase when the envelope has largely been accreted, or dissipated, and the SED can be considered as the sum of the light from the visible star together with the IR emission from the surrounding disk. Once disks have been totally dissipated, we have a Class III YSO, a pre main-sequence star with no evidence of a circumstellar disk around itself. These objects can be distinguished from ordinary stars by, for example, their position

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Figure 1.6: Classification scheme for Young Stellar Objects. Passive disks are disks in which the main energy source is stellar irradiation, in active disks the main energy source is viscous heating.

Isella (2006) [25]

in the HR diagram.

Back to observations, today sub-millimeter radio telescopes like the the James Clerk Maxwell Telescope, the Submillimeter Array (SMA), and the Combined Array for Research in Millimeter Astronomy (CARMA) have greatly improved the study of protoplanetary disks, because they can resolve the emission at these wavelengths. Observations of protoplanetary disks have also taken another big step thanks to the introduction of the Atacama Large Millimeter Array (ALMA), a submillimeter array located in the Atacama Desert in Chile [29].

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Figure 1.7: ALMA image of the protoplanetary disc around HL Tauri. ALMA observations reveal substructures within the disc that have never been seen before and show also the possible positions of planets forming in the dark patches within the system.

Alma collaboration (2014) [28]

For instance ALMA produced the image in Figure 1.7 of a protoplanetary disk surrounding HL Tauri (a very young T Tauri star in the constellation Taurus), that was published in 2014, showing a series of concentric bright rings separated by gaps, indicating, probably, protoplanet formation [28].

1.3.2 Minimum Mass Solar Nebula

According again to the nebular model (section 1.2.1) our Solar System formed from a pro-toplanetary disk, but mass and density of gas and dust in this particular disk are not known today. However, as a first step, it is possible to use the observed masses, orbital radii and compositions of the planets to derive a lower limit for the amount of material that must have been present, neglecting for example migration of planets and changes in dust and gas profiles in time. This is called Minimum Mass Solar Nebula model [30].

The procedure is simple:

1. Start from the known mass of heavy elements in each planet, and increase this mass with enough hydrogen and helium to bring the composition similar to Solar one. This

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is of course a moderate change for Jupiter, but it is much more relevant for the Earth. 2. Divide the Solar System into annuli, with one planet for each annulus, then distribute the mass computed before uniformly across the annuli, to get a typical gas surface density at the location of each planet.

For the gas component the result is that at least between Venus and Neptune Σ9r´3{2_.

The precise normalization is a matter of convention, but the most common value used is [10]

Σprq “ 1.7 ˆ 103

´ r

1 AU ¯´3{2

g cm´2 _(1.11)

Integrating this expression it turns out that the total gas mass should be be around 0.01M@,

comparable to the typical mass estimated for protoplanetary disks around other stars (be-tween 0.001M@and 0.1M@) [29].

Starting from planets composition it is also possible to estimate the surface density of dust or, in general, of solid materials, as

Σs“ $ ’ & ’ % 7.1`_{1 AU}r ˘´3{2g cm´2 _{r ă 2.7 AU} 30`_{1 AU}r ˘´3{2g cm´2 _{r ą 2.7 AU} (1.12)

where for r ą 2.7 AU solid materials are mixture of ice and rocks, i.e. the snowline of the systems is located at about 2.7 AU . The snowline is the particular distance in the solar nebula from the central protostar where it is cold enough for volatile compounds (water for example) to condense into solid ice grains. Assuming a freezing temperature of 180 K [31] it turns out that the snowline is at 2.7AU for a MMSN model.

As the name of the model says, this is a minimum mass, it is not a correct and complete estimate of the disk mass at the time the Sun formed. Theoretical models of disks based on the α-viscosity prescription (to be discussed later) predict a smoother behavior (Σ9r´1₎

[32], while models based on direct calculations of disk angular momentum transport often suggest a more complex profile that is not describable by a single power-law.

1.3.3 1D symmetric model of disks

Another simple way to investigate the structure of a protoplanetary disk is to produce a 1D azimuthally symmetric model, then taking into account variations only in the r direction and averaging z direction, assuming a flat disk and using fluid dynamics equations [33, 11, 34, 35]. The vertical structure of a geometrically thin disk is derived by considering vertical hy-drostatic equilibrium. Starting from Euler equation in the z direction, the pressure gradient

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is, for z ! r BP pr, zq Bz “ ´ρpr, zqgzpr, zq “ ´ρpr, zq GM‹ r2 z r “ ρpr, z, qΩ 2 Kprqz (1.13)

where ρ is the gas density, ΩK is the keplerian angular velocity and we ignore any

con-tribution to the gravitational force from the disk itself (this is justified provided that the disk is not too massive). If we assume that the disk is vertically isothermal we can use an equation of state like P pr, zq “ kBT prq

µ ρpr, zq “ c 2

sprqρpr, zq where µ is the average molecular

mass of the gas and cs is the speed of sound. Solving equation 1.13 we finally get

ρpr, zq “ ρpr, z “ 0qexp ˆ ´ z 2 2h2_prq ˙ (1.14) where hprq “ csprq{ΩKprq is the scale-height of the disk. Starting from this definition we

also have that having a flat disk (i.e. h ! r) is equivalent to having a highly supersonic azimuthal velocity in the disk itself, i.e. ΩKr " cs. In a disk around a Solar mass star with

a temperature of about 100K this condition often occurs, for instance at r “ 1 AU we have h{r “ 0.02. Notice also that surface density in this case can be defined by

Σprq “ ż`8

´8

ρpr, zqdz “?2πρpr, z “ 0qhprq (1.15)

and this definition is used to reduce the system’s dimensions from 2 (radial and vertical) to only 1 (radial) when dealing to Euler (or Navier-Stokes) equation.

If we do not want to use observational constraints (section 1.3.2) it is necessary to consider the nature of angular momentum transport in order to compute the radial structure of the disk. To begin with I can write the r component of the stationary Euler equation in the mid-plane, that gives

v_φ2prq r “ GM‹ r2 ` 1 ρprq BP prq Br “ v2_Kprq r ` 1 ρprq BP prq Br (1.16)

Assuming again a flat disk, i.e. that c2_s ! v_K2 and estimating pressure via P » c2_sρ, we have that v2_φ“ v2K „ 1 ´ Oˆ h 2 r2 ˙ » vK2 (1.17)

i.e. the azimuthal velocity in a thin disk is approximately equal to the keplerian velocity, that simplifies the treatment.

To conclude this simple 1D model it is possible to use all these approximations to solve the continuity equation and the azimuthal component of Euler equation (in this case Navier-Stokes equation) together. Integrating along z direction and following viscous stress

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deriva-tion we get $ ’ & ’ % rBΣ Bt ` B BrpΣvrrq “ 0 rBpΣr_Bt2Ωq `_BrBpr3ΣvrΩq “ _2π1 BG_Br ñ BΣ Bt “ 3 r B Br „ r1{2 B Br ´ νΣr1{2 ¯ (1.18)

where G “ 2πνΣr3 dΩ_dr is the azimuthal viscous force due to viscosity ν [11].

This model is interesting because setting the time derivative to 0 and using the fact that the accretion rate onto the central star is 9M “ 2πrΣvr we are able to find an equilibrium

relation between ν and Σ (i.e. νΣ “ 9M {3π), and, as a consequence, it is possible to compute viscous heating in the disk and a value for radial velocity, that in general is

vr “ ´ 3 Σr1{2 B Br ´ νΣr1{2 ¯ (1.19) but at the equilibrium we have vr“ ´3₂ν_r.

The last thing one need to clarify is the origin of the viscosity ν. Estimating mean free paths of gas particles one can notice that molecular viscosity is negligible in protoplanetary disks, but turbulence within the disk can provide an effective viscosity that greatly overtakes it [36]. For isotropic turbulence, the maximum scale of turbulence within the disk will be of the same order as the vertical scale height h, while the maximum velocity of turbulent motions relative to the mean flow should be comparable to the sound speed cs. The most

natural way to model this viscosity is consequently

ν “ αcsh (1.20)

where α is a dimensionless parameter that measures the efficiency of the turbulence in creat-ing viscosity (and angular momentum transport). In the standard theory of so-called α-disks, α is usually treated as a constant.

Magnetorotational instability (MRI) could be another source of viscosity. The MRI is a linear instability that leads to turbulence in sufficiently ionized accretion disks [37], since it is caused by the concurrent effects of rotation and MHD dynamics in the disk. It is particularly interesting because it usually leads to angular momentum transport outward in the disk. The magnitude of the effective α, generated by the MRI under ideal MHD conditions, has been estimated from local simulations to be of the order of α „ 10´2 _{in protoplanetary disks [38].}

I introduced this 1D model because its prescriptions are still used in recent approaches, such as fluid and MHD simulations, especially in semianalytical models, and because it can be successfully used to describe disks around planets, as it will be explained in the next chapter.

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1.4 Planet formation in disk

It was already pointed out that it is believed that planets form in a protoplanetary disk during the evolution of the central star. As a first step, dust particles tend to stick to each other while embedded in the disk gaseous environment, causing the formation of larger particles up to several centimeters in size. This coagulation process leaves signs that are observed in the IR spectra of YSOs. Following stages of planetesimal formation in a protoplanetary disk are another unsolved problem of disk physics, as simple sticking becomes ineffective as dust particles grow larger. Furthermore computing the effect of the drag force on dust particles, one notices that it grows going from mm-size to cm-size, because smaller objects are still coupled with the gas, and then decrease again for meter-size particles, because drag is less effective against bigger bodies. It is possible to estimate that in its minimum, for cm-size particles, radial drift timescale ! accretion time for meter-scale bodies [30] via collisions, i.e. it does not seem possible to form meter-size bodies before migrating into the central star. In the next section we will go through these stages in planet formation, while showing alternative theories to bypass this issue.

1.4.1 Coagulation

The growth of micron-sized dust particles up to millimeter-size is driven by collisions that lead to sticking and particle growth, but a complete coagulation model for planetesimal formation from dust grains has to face two important challenges. The first one is related to the concurrence of sticking and fragmentation, i.e. the physical properties of colliding bodies with a realistic velocity distribution must permit growth rather than bouncing or fragmentation, in the full range of sizes between dust particles and planetesimals5. The outcomes of the process can be thought as a sort of equilibrium of sticking and fragmentation processes. Second, the rate of growth must be high enough to form planetesimals before the material is lost into the star via gas drift. These constraints are not easy to satisfy, but in order to build an end-to-end model one need to assume them. The most simple and used hypothesis for planetesimal formation is that the same sticking process continues uninterrupted up to the planetesimal scales [17].

The outcome of collisions between micron to cm-sized bodies can be studied today ex-perimentally, ideally under low-pressure microgravity conditions [39]. Most of the current experiments have been performed using silicates particles, then one has to be careful in

5

A planetesimal is one of the smallest celestial bodies that, according to current planet formation theories were fused together to form the planets of the solar system.

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extrapolating the results to conditions of the protoplanetary disk, where there is also an important quantity of icy and organic materials. With that caveat, using experimental re-sults into coagulation models leads to the fact that a priori there are not reasons preventing the rapid growth of dust into mm-sized particles, but for typical collision velocities in disks growth beyond mm scales is difficult [40].

Given this other issue, two mechanisms, very similar mathematically to giant planets formation scenarios, have been basically developed in order to explain how to bypass the step between dust and planetesimal: Goldreich-Ward mechanism and streaming instability.

1.4.2 Goldreich-Ward mechanism & gravitational instability

Figure 1.8: Illustration of the Goldreich-Ward mechanism for planetesimal formation. The disk becomes thin enough to be gravitationally unstable, leading to fragmentation into plan-etesimals. The same mechanism is valid for gas gravitational instability that leads to frag-mentation.

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A first hypothesis for planetesimal formation considers that planetesimals form from gravitational instability and fragmentation of a dense layer of solid particles near the disk mid-plane (Figure 1.8). Generally this idea, known as the Goldreich-Ward (G-W) mechanism [41], seems not to work, because it turns out that settling a layer of dust particles to densities high enough for gravitational instability should be very difficult. Then the idea remains (and has remained) pretty interesting just because it forms planetesimals bypassing the size scales in which radial drift is strongest, but, since it has similarity to gas gravitational instability for giant planets formation, it useful to discuss the mechanism before proceeding with other approaches.

First of all dust or gas gravitational instability requires the dust or gas layer to be massive and dynamically cold enough, in order to be thin and dense. Proceeding with a classic first order perturbation analysis of the conditions for gravitational instability [42], let me first consider the stability of a rotating fluid sheet of negligible thickness in the z “ 0 plane, with constant surface density Σ0 and angular velocity ~Ω “ Ω0z at the 0ˆ th order. The goal is to

calculate the stability of the sheet to planar perturbations, i.e. without considering the z direction. Working in a corotating frame with the (unperturbed) angular velocity, the fluid equations, together with Poisson equation, are

$ ’ ’ ’ ’ ’ & ’ ’ ’ ’ ’ % BΣ Bt ` ~∇ ¨ pΣ~vq “ 0 B~v Bt ` p~v ¨ ~∇q~v “ ´ ~ ∇P Σ ´ ~∇Φ ´ 2~Ω ˆ ~v ` Ω 2 pxˆx ` y ˆyq ∇2_{Φ “ 4πGΣδpzq} (1.21)

The first order perturbative analysis bring to the Goldreich-Ward dispersion relation

ω2 “ c2_sk2´ 2πGΣ0|k| ` 4Ω2 (1.22)

in which we see that rotation and pressure tend to stabilize the disk, while gravity tends to create an instability. Taking the first derivative we see what the most instable scale is kcrit“ πGΣ_c2 0

s , then replacing this value in equation 1.22 we get the Toomre stability criterion

Q “ csΩ πGΣ0

ą Qcrit„ 1 (1.23)

where Q is the Toomre parameter. In this case one can derive stability of a fluid disk in uniform rotation. Differential rotation and global effects alter the value of Qcrit, but do

not fundamentally change the result. For a collisionless disk a comparable result applies replacing the sound speed cs by the one-dimensional velocity dispersion σ.

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In a standard protoplanetary disk (MMSN) the most unstable wavelength is predicted to be about λ „ 3 ˆ 108 cm, which corresponds to a final planetesimal, using ρ „ 3 g cm´2

(silicate), of about R „ 6 km. The collapse time is equal to the free-fall time, that in this conditions is computed to be shorter than a year [43]. This shows how this mechanism can completely bypass the size regime most vulnerable to radial drift and how quick it can do this. However, in its simplest form, the mechanism fails, as anticipated before, because the formation of a dense solid layer leads to turbulence ignition and associated vertical stirring that anticipate to gravitational instability.

Goldreich-Ward mechanism was derived to a thin fluid layer in order to find instabilities, but an interesting thing is that it is applicable exactly in the same way to a dust or a gas layer [1, 2]. Applying the Toomre criterion to the gas component of a Minimum Mass Solar Nebula model we find for example Q ! 1, then gravitational instabilities in the gas component is likely to happen only in the first phases of protoplanetary disk evolution, when the disk is still quite massive, and cause gas disk fragmentation in objects of about M „ 5MJ, using

again the values for a MMSN model. These order of magnitude estimates for the clump mass could be enough to indicate that gravitational instability could form gas giants thanks to gravitational fragmentation.

The issue now is that also in gas layers there is a critical point for gravitational instabil-ities, i.e. we know that as Q decreases, non-axisymmetric instabilities show up and they do not necessarily cause fragmentation [44]. Instead of fragmentation, the instabilities can gen-erate spiral arms that both transport angular momentum and cause dissipation and heating. With analytic arguments and also numerical simulations the cooling rate has been identified as the control parameter determining whether a gravitationally unstable disk will fragment or having spiral arms forming in it [45].

For an annulus of the disk we can define the equivalent of the Kelvin-Helmholtz time6 for a star, that is tcool “ _2σTU4

disk

where U is the thermal energy of the disk per unit surface area. Then for an ideal gas equation of state we have that the condition for fragmentation is:

• t_coolď 3Ω´1 : the disk fragments

• t_cool ě 3Ω´1 : disk reaches a steady state in which heating due to dissipation of gravitational turbulence balances cooling, leading to spiral formation.

6

It is the approximate time it takes for a star to radiate away its total kinetic energy with its current luminosity.

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1.4.3 Streaming instability and core accretion

Figure 1.9: Illustration of the main stages of the core accretion model for giant planet formation.

solarsystemorigins.wordpress.com

In the classical Goldreich-Ward scenario turbulence in the gas disk prevents the mech-anism to work. Recently, however, some related models which share with Goldreich-Ward the central idea that planetesimals form via gravitational collapse have been developed. The point is the idea that protoplanetary disks may present regions in which turbulence could increase the ratio of solids to gas locally. As a result, parts of the disk may become dense enough to collapse into planetesimals via a gravitational instability similar to that was explained before, even if the global disk has a dust to gas ratio too small to support Goldreich-Ward mechanism in all its structure.

The main idea behind this instability is that the gas does not just embeds particles, but also receives a feedback reaction which becomes significant at particle densities that are much smaller than those needed for the ignition of gravitational collapse in the previous G-W model. A detailed analysis of this system shows that it can support this so called linear streaming instabilities, that is computable analytically [46], that can, under certain

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condi-tions, cause strong local particle concentration. The non-linear following phase of streaming instability is, on the other hand, an outcome of numerical simulations and unfortunately it does not have an intuitive physical description, but it appears to be very strong once dust density reaches the value of gas density. This numerical simulations suggest that the insta-bility can cause a gravitational collapse that quickly forms planetesimals up to 100 km in radius, once previous collisional growth manages to form bodies of cm-scale, bypassing the critical meter-scale [47].

Once planetesimals have formed, growth beyond them to terrestrial planets is believed to occur via direct collisions, with the important contribution of the gravitational focusing factor, that is simply the enhancement of the collision cross ratio, as masses become larger. Dynamical friction keeps the velocity dispersion of the most massive bodies low and this allows a phase of runaway growth to occur, in which the largest bodies grow rapidly at the expense of the smallest. This phase ends once the largest bodies become massive enough to sweep all the planetesimals in their neighborhood and a so called oligarchic growth phase takes place, in which the largest objects grow slowlier than they did during runaway growth, but still more rapidly than small bodies [48]. The formation of rocky planets is the first outcome of this model.

Again, we can link this mechanism to giant planet formation, starting from the last stages of planets formation via streaming instability, i.e. from the formation of a rocky or icy core as a result of oligarchic growth. Initially, this core does not have an atmosphere at all or in case this atmosphere is dynamically insignificant, but, as the core grows, it could become massive enough to attract and keep a significant gas envelope.

At first, the envelope is at the hydrostatic equilibrium, until the core reaches a critical mass[49], that we can estimate fitting numerical results with the expression

Mcrit MC » 12 ˜ 9 Mcore 10´6_M Cyr´1 ¸1{4_ˆ κR 1 cm2_g´1 ˙1{4 (1.24)

where κRis the Rosseland mean opacity and 9Mcoreis the accretion rate onto the central core.

Once the critical mass is reached, the envelope can no longer be in hydrostatic equilibrium and a phase of rapid gas accretion occurs. This process continues until the planet becomes massive enough to open up a gap (see section 2.3.2 for details) in the protoplanetary disk, slowing down the rate of gas accretion, or the gas disk itself is dispersed. This mechanism is what we call Core Accretion model [3] and it is the natural second result of planetesimals formation via streaming instability.

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goal of this thesis would be to produce a model about formation of moons around Jupiter, then it is crucial to know how Jupiter (and giant planets in general) forms first.

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Chapter 2

Jupiter system and its formation

Jupiter is the fifth (by distance from the Sun) and the largest planet in the Solar System, with a mass of „ 2 ˆ 1027Kg “ 10´3_M

@. Jupiter is mainly composed of hydrogen (75%)

and helium (25%) in a gaseous state, but it may also have a rocky core of heavier elements. Given its composition Jupiter does not have a well-defined solid surface and it does not have a clear separation between the planet and the atmosphere, just like the other giants in the Solar System (and beyond). The outer atmosphere is clearly divided into several bands and stripes at different latitudes that present turbulence (because of Kelvin-Helmoltz instability) and storms along their boundaries (see for instance the Great Red Spot).

2.1 The moon system of Jupiter

Jupiter has officially 69 moons in stable orbits and it is the planet with the largest num-ber of moons Solar System, followed by Saturn. The most massive moons are the four so called Galilean satellites (Io, Europa, Ganymede and Callisto), that were named after their discoverer Galileo Galilei (17th century). Until today a lot of smaller satellites have been discovered, together with a small system of rings (definitely smaller that Saturn’s one), but the Galilean moons are still the largest and most massive objects around Jupiter, with about 99.99 % of the total mass of the system.

Apart from other 4 regular, but not spherical, satellites with prograde1, circular and low inclined orbits as well, the remaining Jupiter’s moons are irregular satellites which orbit much farther away from Jupiter and have high inclinations and eccentricities. These orbital and physical features suggest that those moons were probably captured by the gravitational field of Jupiter only after its formation.

1

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Figure 2.1: Distances from Jupiter and masses of the 4 Galilean satellites. Units are planet (Jupiter in this case) radius Rp and planet mass Mp.

The orbits of the first three satellites, i.e. Io, Europa, and Ganymede, are locked in a 1:2:4 resonant configuration, that means that their orbits are kept stable by their mutual gravitational interactions that excite their eccentricities and by tidal energy dissipation that releases this energy [50]. To end with, spacecrafts’ observations (Voyager2, Galileo3) of surfaces, compositions, internal structures (via gravity and magnetic field measurements) led to the idea that the satellites compositions range from completely rocky (Io) to 60% silicate rock and 40% ices (in mass) for the outer satellites, Ganymede and Callisto.

2.1.1 Io

Io is the inner and the third largest moon of Jupiter, it also has the highest average density. Io is mainly composed by silicates and and it is often considered as a 100% rocky satellite, that makes it more similar in composition to a terrestrial planet than to other Galilean satellites, which are usually composed of a mixture of water ice and rocks [50].

Models based on spacecrafts’ (Voyager and Galileo) measurements of Io’s mass, radius, and quadrupole gravitational coefficients suggest that its interior is divided in three layers

2

https://voyager.jpl.nasa.gov/

3

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(differentiated4), i.e. a rocky and silicate crust, a mantle and an iron core [51].

2.1.2 Europa

Europa is the smallest of the four Galilean moons orbiting Jupiter an it is mainly, but not completely, made of silicates and it shows a water ice crust on its surface and, probably, an iron core. It also has a (faint) oxygen atmosphere. The idea of a water ice crust is supported by the fact that Europa has the smoothest surface among the Galilean moons and actually among any other known object in the Solar System and this fact leads to the hypothesis that there should be a water ocean just below the icy crust that in a way absorbs collisions with other minor bodies. This possible ocean is very interesting in the astrobiology field because it could perhaps host (extraterrestrial) life.

Another fact that supports the idea of the presence of a water oceans is one of the results of the Galileo mission, which showed that Europa has an induced magnetic field, caused by the interaction with Jupiter’s strong one, which suggests the presence of an internal conductive layer, then possibly a salty water ocean below the observed ice crust [52].

This three-layers model (ice and water crust, silicate mantle and iron core) is also con-firmed by Doppler tracking data acquired again by Galileo. It revealed that the axial moment of inertia (MOI) is I{M R2 “ 0.346˘0.005 [53], that is lower than the MOI for a homogeneous body (0.4) and then it implies a partial condensation of material toward its center.

2.1.3 Ganymede

Ganymede is the largest and most massive moon of Jupiter and also in the Solar System, but it is also the largest object in the Solar System without a proper atmosphere. It has the lowest moment of inertia factor of the system (I{M R2“ 0.3105 ˘ 0.0028) [53] and it is the only one having a magnetic field, then it is thought to have a big metallic and conductive core. It is the third of the Galilean moons and, as said before, it is the last satellite in the 1:2:4 orbital resonance.

Thanks to gravity field measurements, Ganymede seems to be fully differentiated, with an internal structure basically consisting of three layers again: an iron core, a silicate mantle and outer layers of water ice and liquid water [54]. The thickness of the different layers depends on the exact composition of the mantle and the core, that is not exactly known yet, but it is computable thanks to internal structure models in literature.

4 _{Planetary differentiation is the process of separating out different constitutes of a planetary body as a}

consequence of their physical or chemical behavior, where the body develops into compositionally distinct layers.

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2.1.4 Callisto

Callisto is the second biggest moon of Jupiter, after Ganymede, but one of its most char-acteristic features, that has important effects on satellites formation theories, is that it is almost completely undifferentiated and it is the biggest undifferentiated object in the Solar System. It is not in an orbital resonance like the three other Galilean satellites and therefore it is not appreciably tidally heated.

Below the icy crust and a hypothetic ocean similar to Europa’s one, Callisto’s internal structure seems to be entirely uniform. Galileo orbiter data (especially the moment of inertia factor I{M R2 “ 0.3549 ˘ 0.0042 [53]) suggest that its interior is composed of mixed and compressed rock and ice (almost 50%-50%), with the amount of rock increasing toeard the center due to partial settling [55].

2.2 Circumplanetary disks and their evolution

Jupiter’s satellites are believed to have formed in a circumplanetary disk (CPD), i.e. an accretion disk made by gas and dust analogous to a protoplanetary disk [7], but around a planet. The formation of the circumplanetary disk is basically an effect of angular momentum conservation, because when a forming giant planet is accreting mass, for example in the CA scenario, this gas coming from the protoplanetary disk has a certain angular momentum relative to the planet itself, then, in order to conserve it, it has to form a rotating disk and afterwards to transport this angular momentum outwards while accreting most of the mass onto the central planet.

We also know that all giant (regular and icy) planets in the Solar System have regular satellites, which should have formed in such circumplanetary disks [56], similarly to what happens for planet formation in protoplanetary disks. This suggests that during the evolution of these massive planets within the protoplanetary disk, it should be often possible, if not always, to form a circumplanetary disk around them. Furthermore these CPDs are thought to be continuously fed by an influx from the circumstellar disk due to gas accretion onto the central giant planet. Since such massive planets should open a gap in the gas profile (see section 2.3.2) simulation showed that most of this influx should come from the vertical direction [57].

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Figure 2.2: Representation of gas density around a giant planet forming in a protoplanetary disk. It is clearly visible a quite flared circumplanetary disk around the central planet. Szulagyi (2017) [58]

2.2.1 Circumplanetary disks structure

Today, there is no observational evidence of a circumplanetary disk, therefore, we have to rely mainly on numerical simulations to explore its properties and to extrapolation from protoplanetary disks observations. Making theoretical predictions for such observations, for example from hydro and/or MHD simulations, is fundamental, since the attempts to detect a CPD for the first time have already began, for example with the Atacama Large Millimeter Array. Furthermore, the features of circumplanetary disks that one can understand from models are also very important for satellite formation theories, because the timescales and other details of the formation mechanism itself are still quite unknown. This is why so many efforts has been and are done in this field.

Knowing the mass of a CPD is really important to have more realistic satellite formation models, as for example the model that is discussed in this thesis, and to have better con-straints about detection of CPDs themselves, since for example a more massive disk should be more easily detectable. The mass of the CPD depends on whether the central planet formed via core accretion (CA) or gravitational instability (GI), that are the 2 formation scenarios explained in section 1.4. In the GI models, a gas clump collapses into a gas planet together with its circumplanetary disk, therefore the CPD itself must have a large

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exten-sion and a large mass at the beginning of its evolution, possibly about 60% of the planet mass initially, but it can be up to 100% at later stages [4, 59]. On the other hand, if the CPD formed through CA, its mass, according to hydrodynamic simulations and analytical estimates, should (almost) linearly depend on the circumstellar disk’s mass, from where the feeding influx comes from, and for a PPD of about 0.6M@ we have a CPD’s mass that is

about 12% of the planet’s mass, 5 times lower than the GI case. In my satellites formation and evolution model a gas giant formed via CA was considered, i.e. this second less massive case [4].

CPDs are also much colder in the core accretion scenario, since CA and GI simulations often show a difference of more than an order of magnitude in temperatures. CA simulations generally predict that the maximum temperature, that is reached in the inner region of CPDs, is quite high, also up to 13000 K [60], depending for example on viscosity, while GI simulations show lower temperatures in central regions, with a bulk temperature down to only 40 K [5].

As one can understand hydrodynamic simulations are the best instruments we have so far to investigate CPD formation and evolution, since observations are especially challenging, because they require extremely good angular resolution, better than ever before, and the CPDs are also embedded in circumstellar disks, that makes even more difficult to resolve them. Just to show an example, Figure 2.2 shows how a circumplanetary disk looks like around a giant planet in a hydrodynamics simulation, both embedded in a protoplanetary disk , as the result of a CA simulation for CPDs’ formation.

2.2.2 Circumplanetary disks evolution

First of all we know that CPDs form because of angular momentum conservation during accretion (in both core accretion and gravitational instability scenarios) from the protoplan-etary disk to the central giant planet and they are fed by this mechanism during all their existence [57]. As a consequence it is thought that the lifetime and the density of the CPD should be studied together with the evolution of its protoplanetary disk .

In studying the evolution of a protoplanetary disk it is possible use the 1D model described by equation 1.18, where the sink terms due to photoevaporation [61] and accretion onto the core are neglected. Solving the equation one finds that Σ9pt{tdisp` 1q´3{2 as a

self-similar solution. Taking into account photoevaporation, density decays more rapidly, then an exponential decay seems more realistic to fit simulated results [13], i.e.

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Figure 2.3: Accreting stars occurrences as a function of clusters age. New data (2010) are shown as (red) dots, literature data as (green) squares.

Fedele et al. (2010) [24]

where tdisp is the dispersion timescale of the disk. Since CPDs are fed by an influx from

PPDs we can assume that density in the CDP decays with the same exponential law in time. The dispersion timescale of the disk is known within a certain range given by observations. Recent surveys of young stellar clusters identified stars with a protoplanetary disk around them analyzing the width of Hα line5 at 10% of the line peak, because objects onto which

gas is accreting have a larger value of this width compared to non-accreting stars [24, 62]. The result is that the fraction of accreting stars in a cluster goes exponentially from 100% to 0% going from a cluster age of 1 Myr to a cluster age of 10 Myr and that the characteristic timescale for the exponential law is τ “ 2.3 Myr (Figure 2.3). We can deduce that the lifetime of a protoplanetary disk and then of a circumplanetary disk is distributed exponentially between 1 Myr and 10 Myr.

The last step in this analysis is to link lifetime of the disk with dispersion time tdisp, that

is the parameter I would like to use in my model to compute disk evolution. First of all these 5

The Hαline is an accretion tracer spectral line that is often used in the astronomy community to measure

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surveys consider a star as an accreting star when the accretion rate is ą 10´11_M

@yr´1, which

is their sensitivity limit. Since we know that at the beginning of their evolution protostars have an accretion rate from the protoplanetary disk „ 10´7_M

@yr´1 [63] and considering the

exponential evolution of disk density (and mass) it is possible to conclude that tlif etime“ ´tdispln

ˆ 10´11_M @yr´1 10´7_M @yr´1 ˙ » 10tdisp (2.2)

then we know that dispersion timescales are distributed exponentially between 0.1 Myr and 1.0 Myr, with a mean timescale at 0.23 Myr.

As CPDs evolve, their temperature is also changing. This is obviously affecting the forming satellite composition, for instance whether the body formed in a disk that was above the water freezing point or it formed in a colder disk, in which case water could be accreted (like the outer 3 Galilean satellites). One way to investigate cooling in disks, and in particular in circumplanetary disks, is for example via radiative cooling, that are also used for example in SPH (Smoothed Particle Hydrodynamics) methods [14]. This methods give an analytical prescription applicable to other simulations and also to a semianalytical model, which is 9 U “ ´σ T 4 ´ T₀4 Σpτ ` τ´1_q (2.3)

where U is the energy per unit mass and τ is the optical depth of the disk.

Optical depth can be derived from opacity as τ “şh₀ρκdz that, if κ, that is the opacity, does not vary significantly with height, can be estimated as τ » κΣ. Opacity is computed in literature in tables as a function of P and T or as a function of ρ and T in the midplane of the disk [64, 65].

2.3 Satellite formation

Satellite formation in a circumplanetary disk can be studied with the same approach as in section 1.4. The scenario is in fact really similar. We have a dust and gas disk around a central object, we have spherical satellites that show a lot of features in common with terrestrial planets (composition, geology, ...) and we also believe a very similar dynamical behavior (resonances, migration, ...). Of course there are also differences, such as dynamical timescales of the disks, with satellites rotating much faster than planets do around the central object, and higher bulk temperatures, at least in the core accretion scenario. Starting from these considerations, it is convenient to have an overview about the state of the art in the field of satellites formation also to see what processes are in common between satellites evolution in CPDs and planets evolution in PPDs (basically migration and accretion).

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2.3.1 Current knowledge of satellites formation

Assuming a Minimum Mass Subnebula Model (analogous of the MMSN for a protoplanetary disk) using the total mass of the Galilean satellites would predict a circumplanetary disk of about 0.02MJ, that is basically a minimum mass required to produce Galilean moons. Since

it is known that the CPD is fed by the influx from the PPD during its entire evolution, it is believed that at least 1MJ of gas should pass through the disk, that is about 50 times

the lower limit computed before. The idea to solve this issue is that several generations of Galilean-mass satellites may have formed in Jupiter’s history. Each generation of moons might have migrated into Jupiter, because of gas drag from the disk, while new moons have been forming from the new dust fallen from the protoplanetary disk. This mechanism is usually called sequential formation of satellites [6].

It is also believed that migration mechanism (sections 2.3.2 and 3.2.4) does not allow the formation of satellites larger than a certain critical mass, and therefore causes the total satellite system mass to have a relatively constant ratio with the mass of the central planet. This happens because, as I show later in this thesis, there should be a threshold mass at which accretion of protosatellites becomes slower than migration and this avoids the satellites to grow further [7].

Another important fact that has been taken into account in satellites formation inves-tigation is the internal structure of Callisto, that is not differentiated. This result implies that Callisto’s formation did not involve melting of ice. Taking into account only the release of gravitational binding energy during accretion, an accretion timescale of at least 105 yr is required to have a surface temperature increase low enough not to allow melting [66].

In contrast, Ganymede, that has similar size and composition, appears fully differentiated. There are basically three possibilities to solve this issue [6]:

1. Callisto and Ganymede both formed rapidly, but conditions during accretion produced a differentiated Ganymede but only a partially differentiated Callisto.

2. Callisto’s accretion interval was substantially longer then those of the other Galilean satellites.

3. The accretion interval of all of the satellites was quite long, but observed evidence of melting and heating on the inner three satellites has been influenced by subsequent heating due to tidal interactions.

A semi-analytical model for the formation of Jupiter's satellites in circumplanetary disks

Dipartimento di Fisica ”Enrico Fermi”

Tesi di Laurea Magistrale in Fisica