Modelling planet-debris disks interactions

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Eccentric planets and debris disks

At least 20% of the extrasolar planetary systems are known to harbor debris disks (Marshall et al. 2014). The first debris disk was discovered in 1984, when the InfraRed Astronomical Satellite (IRAS) found a strong IR excess around Vega, revealing the presence of micron-sized dust grains (Aumann et al. 1984). Because of collisions and stellar radiation effects, these grains have a limited lifetime, which is shorter than the system’s age. Consequently, this dust is assumed to be replenished by collisional grinding of much larger parent bodies, which are at least kilometre-sized for this collisional cascade to be sustained over the system’s age (Backman & Paresce 1993; Löhne et al. 2008). Therefore, these disks are proof that the accretion of solid material around a star has permitted the formation of at least kilometer-sized bodies. It is thus not surprising that several of these disks are accompanied by planets.
Spatially resolved structures in debris disks can provide clues to the invisible planetary compo-nents of those systems. Indeed, planets may be responsible for sculpting these disks and may leave their signature through various asymmetries such as wing asymmetries, resonant clumpy structures, warps, spirals, gaps, or eccentric ring structures (see, e.g., Wyatt 1999).
The diversity of these asymmetries is to be compared with the variety of exoplanetary systems discovered around Main Sequence stars since 1995 (51 Peg b, Mayor & Queloz 1995). In particular, the common discovery of significantly eccentric planets is in complete contrast with the circular planetary orbits of our Solar System. According to Udry & Santos (2007), the median eccentricity of planets with orbital period greater than 6 days is 0:3. This has revealed that our own Solar System is far from being a reference, and that our current planetary systems formation and evolution models, which were naturally built from its study, require refinements. Therefore, the study of systems containing eccentric perturbers and their dynamical history is crucial to achieve these refinements.
This thesis will firstly focus on the eccentric debris disk resolved around 2 Reticuli by the Herschel space telescope and its PACS instrument (Eiroa et al. 2010). This case is particularly interesting because 2 Reticuli is a Gyr-old system. These systems are rarely accessible to observations, because debris disks tend to lose luminosity on long-term periods: the dust grains emitting at infrared wavelengths are continuously blown away by stellar radiations effects (see e.g. Thébault & Augereau 2007) while replenished via collisional processes among the km-sized parent bodies (Backman & Paresce 1993). Since the parent bodies population is not replenished, the amounts of dust, and thus the disk luminosity in mid-far IR decreases adiabatically (Krivov 2010), until instrument sensitivity does not allow us to detect them anymore.
Therefore, the debris disk of 2 Reticuli illustrates one aspect of mature systems as ours, and is a particularly useful example to understand the long-term history of exoplanetary systems, especially as its disk bears signs for the presence of an eccentric massive body in this system. The first question that arises then is what type of perturber creates this eccentric pattern, and what constraints one might set on it. In addition to retrieve constraints on this companion, one might also question whether the disk asymmetry can be sustained on Gyr timescales, or whether the dynamical history of this system would rather involve a recent setting of the belt-shaping massive perturber on its eccentric orbit. One of the goals of this thesis is to provide answers to these questions through a detailed modelling of the structure of this debris disk, which consists in performing extensive N-body simulations with trial eccentric perturbers, exploring their dynamical influence on massless planetesimals on Gyr timescales, in order to determine which of these perturbers can produce the corresponding observed eccentric pattern, and finally clarify whether this pattern can be sustained on Gyr timescales (Chapter 3).
Another system of major interest is that of Fomalhaut (See Figure 1.5 and Kalas et al. 2005). The eccentric-ring shape of its debris disk was quickly attributed to the dynamic action of a massive and eccentric perturber orbiting near the inner edge of the ring (Quillen 2006; Chiang et al. 2009). This hypothesis was apparently confirmed by the direct detection of a companion near the inner edge of the belt, as predicted, called Fomalhaut b (hereafter Fom b) (Kalas et al. 2008), but new constraints on its orbit revealed that it is belt-crossing, highly eccentric (e 0:69 0:98), and can hardly account for the shape of the belt (Graham et al. 2013; Beust et al. 2014). The best scenario to explain this paradox is that there is another massive body in this system, Fom c, which drives the debris disk shape. The resulting planetary system is highly unstable, which hints at a dynamical scenario involving a recent scattering of Fom b on its current orbit, potentially with the putative Fom c.
One of the goals of this thesis is to investigate the dynamics of this hypothetical two planets system, and in particular, to give insights on the probability for Fom b to have been set on its highly eccentric orbit by a close-encounter with the putative Fom c (Chapter 4).
The two systems Fomalhaut and 2 Reticuli possess eccentric perturbers, and thus offer a wider picture of exoplanetary systems than our Solar System has so far. The study of their dynamical history is crucial to refine the models of formation and evolution of planetary systems, which were built from the study of our Solar System.

Planetary migration and stellar binary companion

Finally, this thesis will discuss the impact of a binary companion on planetary migration (Chap-ter 5). Planetary migration is a phenomenon that can significantly alter the distance of a planet to its star, and is therefore a key element of the morphology of planetary systems. One of its most famous manifestations is certainly the class of so-called « hot » planets that orbit too close to their star to have formed in-situ, which includes 51 Peg b (Lin et al. 1996). Migration results from interactions between the planet and the material of the disk in which it has formed, that is, gas and/or solids depending on the age of the system.
In a protoplanetary disk, interactions with the gaseous material will be predominant and gener-ate migration. This type of migration has been extensively studied in the past years. However, in several Myr, a protoplanetary disk is emptied of most of its gas, and it is ultimately the in-teractions with the remaining km-sized solids, asteroid- or comet-like, and called planetesimals, which may generate migration in systems typically older than ten million years. It is on this late migration, called Planestesimal-driven migration (PDM), and which gives its final archi-tecture to planetary system, that this thesis will focus on. More specifically, it is the impact of a stellar binary companion on this migration process that will be explored numerically. It is obviously expected that a secondary star perturbs the material orbiting the primary star, and therefore any migration process. Therefore, one can expect that a binary companion influences the final architecture of a planetary system. In addition, the impact of a stellar companion on the formation and evolution of planetary systems is by no means negligible, since more than half of the stars possess one.


Five centuries ago, Copernicus revolutionised astronomy by extricating it from geocentrism. Today, thanks to a collective effort involving actors from the whole world in search for new planetary worlds, we know that our Solar System is not a generic model, and witness another major revolution, one that will lead astronomy out of heliomorphism.
The diversity of exoplanetary systems often questions our formation and evolution models, however, these models have been primarily built from the study of our Solar System, which turns out to be quite exceptional, because it contains nearly circular orbits and involves a single star. Highlighting all stages of formation and evolution of planetary systems, being able to describe all the processes at work from their birth to their death, and explain their diversity, is a large and ambitious project of modern astronomy, to which this thesis aims to contribute.
By focusing on the dynamical history of systems containing eccentric perturbers, and the impact of a second star on the architecture of exoplanetary systems, this thesis opens the way to a more appropriate view of exoplanetary systems.

Modelling planet-debris disks interactions

I will review here essential features of the dynamics of planetary systems with particular focus on the models and methods that are classically used to investigate planet-debris disk interactions, and which were used throughout this thesis. I will describe the possible gravitational effects of a massive planet on much less massive bodies, therefore named « test-particles », and which include all the component of debris disks, from micron-sized dust grains to km-sized planetesimals. I will also describe the radiative effects of the central star on these solid components, and will explain how planet-debris disks interactions can lead to observable features in debris disks.

Keplerian motion

A single test-particle of mass m and a star of mass M? will mutually attract each other with ~ 3 , ~r being the position vector of the test-particle in the a gravitational force Fgrav = GmM?~r=r frame centered on the star. It follows, from the application of Newton’s second law, that the differential equation which describes the motion of the test-particle is given by :
~r (2.1)
~r + r3 = 0 ;
where = G(m + M?).
The test-particle will thus adopt a Keplerian motion, that is, it will describe an ellipse which one of the foci is occupied by the star. This motion is defined by six parameters (a; e; i; ; !; ), called orbital elements (see Figure 2.1).
• a is the semi-major axis. It represents the size of the orbit ;
• e is the eccentricity, with value between 0 and 1, and gives an idea of the shape of the orbit : the greater e is, the more elliptic the orbit is ;
• ~ ~ is the inclination of the orbit ; i = (OZ; k)
• ~ ~
= (OX; ON) is the longitude of the ascending node. This permits, with the inclination, to specify the position of the plane of the orbit in space ;
• ~ ~ ! = (ON ; OP ) is the argument of periastron, and specifies the position of the orbit in its plane ;
• is the time of passage of the test-particle at its periastron.
The position of the test-particle on its orbit is specified by the true anomaly , i.e., the angle between the position vector ~r of the test-particle and the direction of its periastron. The trajectory itself, that is, the equation that defines the orbit in its plane and relates the distance r of the test-particle to the star, and its angular position on its orbit , reads :
r( ) = a(1 e2) : (2.2)
1 + e cos
The test-particle will have an orbital period T which depends only on a, m, and M?, as defined with Kepler’s third law by : T 2 = 4 a3 : (2.3)
Finally, the equation which gives the position of the test-particle on its orbit as a function of the time is the Kepler’s equation, and reads : M = E e sin E; (2.4) where E and M are respectively the eccentric and mean anomalies, being defined by: tan 2 = r 2 ; (2.5) M = n(t ) :
This last equation contains the mean-motion n, simply defined by 2 =T .
If the test-particle was to be effectively alone in the system, the shape and orientation of its orbit would be constant. Of course, this situation is ideal, and Keplerian orbits are in fact perturbed by various phenomena. In particular, it will be affected by the presence of a massive body such as a planet. If effectively acted upon by a planet, a whole population of small solids such as the components of a debris disk may bear the imprint of this planet in its spatial distribution, and lead to observable features in debris disks.

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Planetary patterns in debris disks

Spatially resolved structures in debris disks can provide clues to the invisible planetary com-ponent of those systems. Such planets may be responsible for sculpting these disks and may leave their signature through various asymmetries such as wing asymmetries, resonant clumpy structures, warps, spirals, gaps, or eccentric ring structures (see, e.g., Wyatt 1999). Dynami-cal modelling of such asymmetries is the only method to place constraints on the masses and orbital parameters of planets in systems where direct observations are not possible (see, e.g., Mouillet et al. 1997b; Wyatt et al. 1999; Augereau et al. 2001; Moro-Martín & Malhotra 2002; Wyatt 2004; Kalas et al. 2005; Quillen 2006; Stark & Kuchner 2008; Chiang et al. 2009; Ertel et al. 2011; Boley et al. 2012; Ertel et al. 2012; Thebault et al. 2012).
Most images of resolved debris disks have been obtained so far in the visible or near-IR. At these wavelengths, the emission is dominated by sub-micron to micron-sized grains, which are released by larger km-sized bodies as these suffer collisions. The smallest a solid component is, the more sensitive it is to stellar radiation effects. Radiation pressure tends to blow material out of the system. As the gravitational force, it is a radial force, that scales as the inverse square of the radius, but it is directed outwards and therefore counteracts gravitational attraction. Using the parameter , which is the radiation pressure to gravity ratio, a particle is submitted to a total force: ~ GM?(1 )m (2.7)
One can see that this is equivalent to a gravitational force ponderated by a factor (1 ).
Therefore if > 1, the particle will no longer be bound to the system and be expelled from it. In fact, a particle becomes unbound as soon as > 0:5 (Augereau & Beust 2006). This means that a planetesimal on a circular orbit will release bound dust grains on eccentric orbits, as long as the value of for these dust grains does not exceed 0:5 (See for instance Figure 1 of Krivov 2010).
An expression for was given by Burns et al. (1979) in the ideal case where the body subject to radiation pressure is spherical with radius s and density , and absorbs the totality of the radiation it is exposed to: = 0:574 L? M 1g:cm 3 1 m ; (2.8) L M? s where L? and M? are the star luminosity and mass, respectively.
As one can see from Eq. (2.8), depends on the size of the particle : the larger it is, the smaller
is, which characterizes the fact that the largest components of a debris disk, that is, km-sized planetesimals, are unsensitive to radiation pressure and suffer gravitational effects only, while smaller grains tend to be blown out of the system.
Therefore, the grains observed in the visible or near-IR are close to the blow-out limit imposed by stellar radiation. However, since these grains are the product of collisions of larger bodies which endure gravitational effects only, planetary perturbations among a collisionally active population of planetesimals will thus infer on the zone of production of dust grains, which explains why large-scale asymmetries can be visible among a population of short-lived dust grains, which will eventually leave the system. The effect of radiation pressure is that it may strongly alter or even mask the dynamical structures imparted by a massive perturber on a debris disk (See for instance Figure 1 of Krivov 2010).
Observations of larger bodies, less affected by radiation pressure effects, that is, observations at longer wavelegths (mid-IR to sub-mm), allow one to obtain better constraints on the structure of debris disks and its potential large-scale asymmetries. These observations were made possible by the Herschel Space Telescope and the interferometer ALMA.
In any case, by using constraints derived from resolved observations of large-scale asymmetries in debris disks, one can then proceed to a dynamical modelling work: using analytical and/or numerical tools, and more precisely N-body simulations, it is possible to explore a given space of parameters for any disturbing planet within a system and study its gravitational effect on a population of planetesimals. The goal is to try to reproduce the asymmetries observed, and thus constrain the parameters of the planetary system. It is precisely this type of work that I have conducted in my thesis for the case of the debris disk of 2 Reticuli and that I will present in Chapter 3. In the following sections, I will give more details on the general modelling assumptions, the analytical methods, and the numerical methods used throughout this thesis.

Perturbed Keplerian motion : introduction

The usual assumption that is made when trying to relate a large-scale asymmetry among an observed dust population to a planetary perturber, is that this asymmetry already ex-ists amongst the parent planetesimal population that produces the observed dust and result from pure gravitational perturbations. Another usual assumption is that at the end of the protoplanetary phase, the planetesimals start from almost circular orbits because of orbital eccentricity-damping by primordial gas, and that any perturbing planet in the system is fully formed by the time the gas disappears. Thus, one can consider the disappearance of the gas as time zero for the onset of planetesimal perturbations by a planetary companion.
In this context, one can study the influence of different perturbers in a simplified way, neglecting the effect of radiation pressure and considering initially cold parent planetesimals as mass-less and collision-less particles in orbit around their host star and perturbed by a companion.
When a particle suffers perturbations due to the presence of a planet in the system, the context is this of the three-body problem, which has unfortunately no exact solution. As shown by Burns (1976), perturbing forces in the plane of the orbit of the particle will induce changes on its semi-major axis a, eccentricity e, and argument of periastron !, while forces normal to the plane of the orbit will induce changes in the inclination i and longitude of ascending node . The orbital elements of the particle are no longer constant, and the goal is then to find the equations describing their behaviour. This is best achieved using a Hamiltonian formulation of the problem, which is a global and energetic approach.
Indeed, the Hamiltonian H is a function that describes the system as a whole, and is the sum of the kinetic and potential energies at play. In the case the system studied is conservative, the total energy and thus the Hamiltonian of this system are constants. This function can be defined with any set of variables (qi; pi) with i = 1:::N, as soon as these variables are conjugated, which means that they allow the equations of motion to be written in a very simple form, that is, a system of 2N first order equations

Table of contents :

1 Context and thesis outline 
1.1 Introduction
1.2 Eccentric planets and debris disks
1.3 Planetary migration and stellar binary companion
1.4 Summary
2 Modelling planet-debris disks interactions 
2.1 Keplerian motion
2.2 Planetary patterns in debris disks
2.3 Perturbed Keplerian motion : introduction
2.4 Secular perturbations
2.5 Resonant interactions
2.6 Close-encounters
2.7 Numerical methods : N-body symplectic codes
3 Picture of a mature Gyr-old system, Faramaz et al. (2014b) 
3.1 The eccentric debris disk of 2 Reticuli
3.2 Modelling approach
3.2.1 Analytical model
3.2.2 N-body simulations
3.2.3 Synthetic images
3.3 New constraints on the 2 Reticuli system
3.3.1 Constraints on perturbers : N-body simulations
3.3.2 Constraints on the disk : Synthetic images
3.4 Conclusions and perspectives
3.4.1 Conclusion
3.4.2 Towards better knowledge of the 2 Reticuli system with ALMA
Article : Can eccentric debris disks be long-lived? –
A first numerical investigation and application to 2 Reticuli
4 On the dynamical history of the Fomalhaut system, Faramaz et al., in prep 
4.1 The Fomalhaut system
4.2 Modelling approach
4.2.1 Expected routes to form Fom b-like orbits
4.2.2 N-body simulations
4.3 A resonant origin for Fom b
4.3.1 A first two-step scenario
4.3.2 On the mass and eccentricity of Fom c
4.4 Apsidal alignement and refinement of the scenario
4.4.1 An unexpected feature : apsidal alignment
4.4.2 Close-encounters with Fom c
4.4.3 Further secular evolution with Fom c
4.4.4 A three-step dynamical scenario
4.5 Conclusions and perspectives
Article : Insights on the dynamical history of the Fomalhaut system – Investigating the Fom c hypothesis
5 Planetesimal-driven migration in binary systems 
5.1 Why planetesimal-driven migration?
5.1.1 Migration around a single star
5.1.2 Dedicated numerical tools
5.1.3 Influence of a stellar binary companion
5.2 PDM in single star systems: the theory
5.3 Influence of a binary companion: first results, Faramaz et al. (2014a)
5.4 Conclusion and Perspectives
6 Perspectives 
6.1 Architecture of exoplanetary systems in close binaries
6.2 Exoplanetary systems in time
6.3 An answer to the origin of exozodiacal dust?
6.4 Conclusion
A The 2 Reticuli system
A.1 Inclination of 2 Reticuli
A.2 Constraints on 2 Ret set by direct imaging
B Article : An independent determination of Fomalhaut b’s orbit and the dynamical effects on the outer dust belt, Beust et al. (2014)


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