Formation and destruction of hot Earths during the migration of a hot Jupiter

Transcription

1 FORMATION Sciences de la Matière École Normale Supérieure de Lyon Université Claude Bernard Lyon 1 STAGE BOLMONT, Emeline M2 Option Physique Formation and destruction of hot Earths during the migration of a hot Jupiter Abstract About 2% of stars host a hot Jupiter, a gas giant with an orbital period shorter than 10 days. Previous simulation works have shown that hot super Earths, terrestrial planets with also a short period, could be created during the hot Jupiter migration process. My internship consisted in determining the parameters influencing the survival or the destruction of hot super Earths when the hot Jupiter migrates to the inner edge of the disk. Key words: Hot Jupiter, hot Earths, N-body code Mercury, hydro-code FARGO Laboratoire d astrophysique de Bordeaux Université Bordeaux 1 ( Supervisor: Sean Raymond (Sean.Raymond@obs.u-bordeaux1.fr) July 26, 2010

2 Acknowledgments I wish to thank Sean Raymond, my supervisor, thanks to whom I discovered the curiosities of planetary dynamics. I also wish to thank Franck Selsis who helped me a lot when Sean was absent. Many thanks to Arnaud Pierens, who helped me understand the dynamics of gas disks and the subtleties of the FARGO code and to Franck Hersant, who helped me through some fortran computation difficulties. I also wish to thank Aurélien Crida with whom I had interesting discussions about the physics of gas disks and the use of the hydrocode FARGO. A great ambience prevailed in the laboratory, one could feel at ease among other students, technicians and researchers.

3 Contents 1 Introduction General picture Hot Jupiters systems The birth of giant gaseous planets Interaction with the gas disk Hot Super Earths Basic knowledge on gas disk evolution Simulations Dynamical simulations of a planetary system Mercury Code Taking into account the gaseous disk Improvements of the gas disk description The FARGO code Simulations details Discussion Results Initial conditions Resonance shepherding phenomenon Survival or destruction of hot embryos Preliminary results of the improved version Conclusion Limits and improvements Conclusion 23 Appendices 27

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5 3 Chapter 1 Introduction 1.1 General picture My internship took place in the Laboratoire d Astrophysique de Bordeaux (LAB) in Floirac. I worked in a team named [EXO]TERRES[1], which studies the: Origin and evolution of planets and their atmosphere; Characterisation of exoplanets and of their atmosphere; Evolution of the Earth atmosphere; Photo-chemistry of primordial atmosphere and pre-biotic chemistry; Habitability; Spectral bio-signature. Recently, a new aspect of the study of exoplanets was added with the arrival of Sean Raymond: the dynamics and formation of planetary systems. Climates and conditions of habitability will be linked with the dynamical aspects and the formation of the planets. Under the supervision of Sean Raymond, I studied the effect of the migration of a giant planet on the survival of planetary embryos. First, I studied the original N-body code, then I endeavoured to improve the description of the gas disk used in this code. 1.2 Hot Jupiters systems The question of the apparition of life on Earth stimulates a great number of researchers over the world, and it is intimately linked with the question of extraterrestrial life in the Universe. As of July 26, 2010, 464 exoplanets have been detected, among them 20% are hot Jupiters[2]. About 2% of stars host a hot Jupiter. A hot Jupiter is a giant gaseous planet, that orbits very close to its star (orbital period smaller than 10 days), hence the adjective hot and, that has a mass bigger than 0.5 times the mass of Jupiter. Hot Jupiters puzzled researchers

6 1.2 Hot Jupiters systems 4 for many years, because it was unexpected to find such massive objects so close to their star The birth of giant gaseous planets The first stage in the creation of a planetary system is the proto planetary disk (PPD), where a protostar is embedded in a rotating gas disk. Figure 1.1: Artist picture of a PPD[3]. A PPD can be more than 100 AU wide. Its extent is limited by neighbouring stars or if it has a binary companion or if photo-evaporation 1 occurs in the disk. A disk has about a million-year lifetime. In such an environment, dust begins to accrete. The first step of accretion (from µm to cm) is due to surface forces such as electrostatic forces[5]. The second step (from cm to km) is not well understood and is thought to be due to turbulence in the disk. Once the dust grains reach a size of a few km and thus form planetesimals, they start accreting gravitationally and form planetary embryos[6]. The mass of the core depends on the distance to the parent star[7], cores growing beyond the snow line, where water condenses and more solids form, can be as massive as 10 M. These heavy icy cores can accrete a gaseous envelope in about 10 6 years thus becoming gas giants. This phenomenon is constrained by the presence of the gas disk, which after a time disperses via photo-evaporation for example. In this scenario, gas giants are supposed to form at a distance bigger than about 3 AU. Hence, it does not explain the existence of hot Jupiters Interaction with the gas disk Currently, the preferred explanation for the origin of hot Jupiters is that the core of these planets was formed at a distance of a few astronomical units from their star and then migrated towards the star via complex interactions with the gas disk. A planet can interact in different ways with the gas disk: 1 There is photo-evaporation of the disk by its central star if the stellar radiation warms up the gas on the surface of the disk enough for it to flow out of disk and escape the gravity of the star.

7 1.2 Hot Jupiters systems 5 Type I migration[8]: When a terrestrial mass planet is in the disk, it creates spiral density waves in the vicinity. The sum of the forces exerted by this structure on the planet results in an inward migration. This phenomenon is not well understood and is an active field of studies. These density waves have another effect, which is much better apprehended, they bring about a damping of the eccentricity and inclination of the orbit of the embedded planet[9]. Type II migration[10]: This migration occurs for more massive planets, like Jupiter-mass planets. Contrary to type I migration scenarios, where the disk density profile is mostly unperturbed, type II migration occurs when heavy planets clear a gap in the gas disk by accreting the nearby matter and gas (Figure 1.2). Forces are exerted on the planet both by the inner and outer part of the disk and depending on the viscosity of the gas, the mass of the planet, it will migrate either inward or outward. As the planet is no longer embedded in the gas disk, there is no eccentricity damping as long. Figure 1.2: Gas surface density in a disk with a Jupiter-mass planet. This is a result of a simulation I made with the FARGO-2D1D code, only the result obtained in the 2D-grid is represented. In blue is the gap and in red the density waves created by the planet (x = 1.5 AU, y = 1.8 AU). As the migration timescale (10 5 yrs) is compatible with a gas giant formation, inward type II migration of Jupiter-like planets is a possible explanation for the creation of hot Jupiters. To sum up, planets lighter than 100 M can undergo a type I migration and an eccentricity damping, while planets heavier than 100 M are likely to undergo a type II migration and an eccentricity pumping from other bodies. It is important to note that much smaller bodies, from planetesimals to protoplanets of a few 10 1 M, undergo a gas drag due to the difference of velocities between them and the gas[11]. For this internship I considered hot Jupiters formed via type II migration, and the question is whether planetary embryos can survive this migration.

8 1.2 Hot Jupiters systems Hot Super Earths Formation by resonant shepherding Let us consider a system initially composed of a gas giant (M JUP ), planetary embryos (about M ) and planetesimals (bodies of 10 3 M 0.3 M Moon ). During its migration, the gas giant will shepherd planetesimals and embryos at mean motion resonances. Resonant situations are common in planetary dynamics. There are several kinds of resonance, and the one considered here is defined as following: When planet A and planet B are in resonance, it means that the ratio of their orbital periods is given by: T A = p T B p + q, (1.1) where (p, q) N 2. q gives the order of the resonance. The more common resonances are: 2 : 1 (q = 1, p = 1) and 3 : 2 (q = 1, p = 2). These situations are stable, two bodies in a resonant configuration can stay that way for a while. Let us consider the migrating giant and a planetesimal. When the giant approaches the planetesimal and gets it locked in a resonance, the small body undergoes a resonant pumping, which increases its eccentricity. Bodies on eccentric orbits experience a greater friction with the gas than on circular orbits, making their semi-major axis decrease (see section 1.3 on the following page). Far from the gas giant and embedded in the gas disk its eccentricity will decrease via gas drag, and type I damping for embryos, and it will stay where it is until the gas giant comes closer and get it locked in a resonance once more. By this method, embryos can be shepherded to a very close distance from the star. On its way, it is likely that it accreted nearby planetesimals or collided with other embryos thereby increasing its mass, and this might result in a super hot earth[12]. Survival The probability of survival of super hot Earth depends on several factors: The presence of the gas, which triggers type I damping and causes a drag force on the bodies. The presence of planetesimals, which causes dynamical friction on embryos. The radius of the magnetospheric cavity 2, i.e. the inner radius of the gas disk. In this internship, I tried to determine the best parameters quantifying these factors to increase the probability of hot Earths surviving. 2 At this distance from the star the magnetic field lines reconnect, it is thought to be the corotation radius[13].

9 1.3 Basic knowledge on gas disk evolution Basic knowledge on gas disk evolution The fate of the planets in a planetary system is highly dependent on the gas disk properties, so it is crucial to understand the disk global evolution and main characteristics. In the next section, the assumptions used in the FARGO hydrocode will be described. The FARGO code is a Fast eulerian transport algorithm for differentially rotating disks[14][15], which will be discussed in section Model used in the modified Mercury version Let us consider a gas disk evolving around a star. A disk is characterised by its aspect ratio h = H/r, where H is the scale-height 3 of the disk at the radial coordinate r (figure 1.3), the radius of its inner and outer edge named R min and R max in the figure, its volume density ρ and its azimuthal velocity v gas. Figure 1.3: Schema of a PPD. An important parameter to describe a gas disk is also the viscosity ν of the disk. The model, which is used in the hydrodynamic code FARGO is based on the following hypothesis: The disk aspect ratio h is assumed constant; The viscosity ν is assumed constant in the disk 4 ; The pressure P of the gas is given by: P = ρc 2 s, where ρ is the volume density c s is the speed of sound, c s = hv Kep, where v Kep = GM /r is the Keplerian velocity, G the gravitational constant and r the radial distance from the star. It is the isothermal hypothesis. Gas azimuthal velocity Under these assumptions, the gas azimuthal velocity can be inferred from the following hydrodynamic equation: a = f grav m 1 ρ P, (1.2) 3 H is defined as: P (z) = P 0 e z/h, where P is the gas pressure and z is the height in the disk. 4 The viscosity of disks is not well known. Several models exist among which the α model, and they all give similar results.

10 1.3 Basic knowledge on gas disk evolution 8 where P is the gas pressure, m the mass of a gas particle, f grav the gravitational force exerted by the star on the particle. Let us assume the initial density profile of the form: Σ(r) = Σ 0 r α where Σ 0 and α are constant. The gas azimuthal velocity is thus given by: GM ( v gas = ) 1 (α + 1)h 2 = v Kep 1 (α + 1)h2, (1.3) r In our model, h is constant so v gas varies like 1/ r. The gas azimuthal velocity has an important influence on the planetesimals and embryos. Unlike the gas giant, these bodies evolve embedded in the gas disk. As the gas azimuthal velocity is slightly different from the Keplerian velocity, there is a viscous friction between the gas and a body, which tends to slow the latter. This gas drag tends to decrease the semi-major axis of the embedded bodies. This gas drag is modelled by the Stokes drag for high Reynolds number. Indeed, the Reynolds number, which characterise the flow around the planetesimals and the embryos is huge (R E 10 4 to 10 7 ), so the gas drag is given by: a drag = Kv rel v rel, where v rel = v gas v Kep and the drag parameter K is function of the density of both gas and object, the radius of the object and the drag coefficient C D. During the FARGO simulation with a migrating planet, the velocity remains approximatively the same. Gas surface density From an initial profile as given above, the gas density evolution is given by a complex diffusion equation, which takes into account the spreading of the disk, its accretion on the star and also the interactions of the giant planet with the gas. There are simple physics that explain how the gap created by the gas giant evolves. A criterion has been found by Crida and Morbidelli[14] to characterise gap opening. It involves the three main parameters of the problem, which are the mass of the planet, the height of the disk and its viscosity. A gap is opened if: 3 H (1.4) 4 R H qr where R = r2 Ω ν is the disk Reynolds number, which compares the viscosity effects and the inertial effects, R H is the Hill radius 5 and q = M P /M. The larger the viscosity, or the smaller the distance from the gas giant to the star, the harder it is to create a deep gap, and thus the slower the migration. It is then possible for the planet to stop at a small distance from the star and not fall into it. The probability of the planet stopping is increased if the disk aspect ratio is larger. 5 When a body is closer than one Hill radius from a planet the gravity it feels is dominated by that of the planet, and not of the central star.

11 9 Chapter 2 Simulations Simulations are necessary to study planetary systems. They help understanding the different physical phenomena taking place by comparison with observations. 2.1 Dynamical simulations of a planetary system To study the survival of embryos during the migration of a hot Jupiter, a N- body code is needed. The Mercury code, developed by J.E. Chambers[16] is widely used to do so. This code calculates the orbital elements of bodies, that revolve around a central object Mercury Code Mercury is a fortran N-body code, which contains five different integration algorithms among which the hybrid/bs algorithm (see the annex 4 page on page 28 for details). It calculates the orbital evolution of objects moving in the gravitational field of a large central body. This code s inputs include the simulation time, the time step and the initial conditions of the system. There are two initial condition files, one for the giant and the embryos and one for the planetesimals, which give the semi-major axis, the eccentricity and the position of the different bodies Taking into account the gaseous disk Some necessary improvements have been brought by Sean Raymond to take into account the gas disk and its interaction with the Jupiter, the embryos and the planetesimals. This modified version of the Mercury code (hereafter referred to as the original Mercury code), takes an initial gas surface density profile and gas radial velocity profile, which was provided by Crida and Morbidelli[17] with the FARGO code, and rescale it to follow the evolution of the Jupiter. The disk considered in the original Mercury simulations is flared, h r 1/4.

12 2.1 Dynamical simulations of a planetary system 10 Prescribed phenomena As the calculation of the disk evolution and of its effect on the gas giant is not easily obtained, a choice has been made to simulate the type II migration of the gas giant. The user can choose the planet migration time, its initial and final positions. A subroutine in the Mercury code takes this data and forces the gas giant migration artificially, with a method, which does not produce artificial changes in eccentricity and inclination[12]. Therefore the observed changes are purely physical. The gas density profile follows the forced evolution of the planet. The initial density is rescaled at each integrator step time making sure that the position of the planet coincides with the position of the gap. The big planet and the gas disk evolution are thus constrained. evolution of the planetesimals and the embryos is free. Free phenomena But the The embryos feel the gravitational attraction of the star, of the gas giant, of the other embryos and of the planetesimals. They also feel gas drag and type I damping. The planetesimals feel the gas drag, the gravitational attraction of the star, of the gas giant, of the embryos, but they do not interact with each other. A summary of the different effects the bodies feel can be seen in the following table2.1. Hot Embryos Plan. Gas Type I Type II Jupiter drag damping migration Hot Jupiter Embryos Plan. Table 2.1: Plan. means planetesimals. : feels the effect of, : does not feel the effect of. A subroutine linearly interpolates the gas surface density and the gas velocity in order to know these quantities for each embryo and planetesimal at each time step. The subroutine then calculates the accelerations due to the gas drag, and the type I damping. Planetesimals are complex to take into account. In a real disk, the total mass of planetesimals is important ( 100 M 10 4 M ) and they are thought to weigh M. This means that in a simulation, one should consider planetesimals, which would be unfeasible. The solution is to consider a smaller amount of heavier planetesimals (10 8 M ), in order to keep the dynamical properties such as the dynamical friction, and to define a sort of gas drag cross-section, which would allow us to compute the real effect of the gas on them.

13 2.2 Improvements of the gas disk description Improvements of the gas disk description Rescaling the initial gas density is not a consistent description of the gas disk evolution, so I chose to compute the gas surface density and gas azimuthal velocity profiles with the hydrocode FARGO to efficiently reproduce the real disk evolution. As only the gas giant has an non-negligeable effect on the gas disk, the simulations I ran with FARGO only take into account a giant planet, with the same characteristics as before The FARGO code The FARGO code[18] is a two-dimension hydrodynamic code. It solves the Navier-Stokes and continuity equations for a non self-graviting gas disk containing planets and evolving around a central star. It uses an isothermal equation of state with an arbitrary sound speed radial profile. This code can efficiently produce type II migration. The version FARGO-2D1D I used is slightly different, it uses a 2D-grid and a 1D-grid as shown in the figure 1 in the annex 4 on page 28. This modification allows to study the disk global evolution when planets are present, it speeds up the calculations. Usually, the giant planet is placed in the 2D-grid and at a distance of at least few Hill radii from the border of the 1D-grid. During the simulation, the planet should stay in this area. In this internship, where the topic is a planet migrating from a distance of a few AU to a distance of 0.1 AU 1, a run with the 2D version of FARGO would take several months. A compromise has been found in using the version FARGO-2D1D, for which a run lasts a few days to a few weeks. Inputs The FARGO code needs several pieces of information to compute the evolution of the system. It needs to know where the planet is and its mass. The user also has to give values to parameters describing the gas disk, the gas initial density profile parameters: α and Σ 0, the disk viscosity, its aspect ratio and the radius of its inner and outer edge. Computation parameters have to be chosen as well. The radial spacing, the number of sectors in the 2D-grid, the location of the 2D-grid and of course the time step Simulations details The aim of the simulation was to make a giant planet migrate from a distance of a few AU to 0.1 AU. I ran several types of simulations to test the different parameters. After several tests 2, the best solution found was to constrain the migration of the planet and to calculate at each time step the effect of the planet on the gas disk and thereby neglecting the effect of the gas on the planet. Even though it is not the best description possible, it is better than what was done previously. 1 It is the upper limit of the semi-major axis of the observed Hot Jupiters. 2 The 1D-grid simulations gave non-physical results, and the 2D1D-grid simulations were too long for an accurately described inner disk.

14 2.2 Improvements of the gas disk description 12 Two parameters allow us to fix the migration time and the final semi-major axis making the planet migrate linearly and stop at the desired distance. I ran simulations with these constraints and using only the 1D-grid to speed up the calculations. In my simulations, I placed a Jupiter at a distance of 2 AU from a 1 M star and I made it migrate to 0.1 AU in 10 5 yrs. I chose to place the initial position of the planet at 2 AU for time considerations. My simulations showed that the most interesting phenomena happen at the end of the migration, so the migration from 5 AU to 2 AU is not crucial for my study. I ran two simulations with the following disk parameters: Simulation h Σ 0 α ν R min R max Calculation (M /AU 2 ) (m 2 s 1 ) (AU) (AU) time (days) I II Table 2.2: Disk parameters for the two FARGO-2D1D simulations, which results I used in Mercury. I used a time step of dt = 1 yr for the simulation and a radial spacing of dr = AU so that the disk is correctly described. FARGO-2D1D calculated the gas density and velocity profiles with a linear radial spacing of dr, which corresponds to 1600 data points for each output files. Output files were created for every two thousand years. In figure 2.1 is plotted a snapshot of the gas density profile yrs after the beginning of the migration. To use these results in the Mercury code, I processed the data and modified the code. The details of these processes can be seen in the annex 4 on page 29. Figure 2.1: Gas density profile in g.cm 2 when the giant planet is at 1 AU.

15 13 Chapter 3 Discussion During the first two months of my internship I studied the influence of different parameters on the evolution of the planetary bodies inside a gas disk, thanks to the original Mercury code. Then, as the description of the gas disk is very important for the study of the survival of the planetesimals and the embryos I decided to improve it. In the following section 3.1 are exposed the results obtained with the original version and the modified version of the Mercury code. In section 3.2, I will discuss the worth of these results, and in section 3.3 I will present the limits of my study and offer some suggestions of improvement. 3.1 Results The resonance shepherding phenomenon is needed to push the embryos close to the star thus creating hot Earths. This phenomenon is discussed and illustrated in the section The parameters influencing the efficiency of the shepherding are discussed in section The initial conditions of the systems I studied are exposed in the next section Initial conditions In my simulations I considered a gas giant and a population of embryos and planetesimals. The different simulations characteristics are presented in table 3.1. Simulation Initial giant Nb. of Nb. of Migration Simulation time position embryos plan. time time step A 3 AU day B 2 AU day C 2 AU day Table 3.1: Initial conditions for the original version of the Mercury code simulations and their characteristics. In all simulations, the gas giant of mass M JUP migrates to 0.1 AU. Its initial eccentricity is zero and its initial inclination is zero.

16 3.1 Results 14 The embryos of mass M and the planetesimals of mass 10 3 M are initially placed between the gas giant and the star. Their initial eccentricities range between and and their inclination range between 0 and 2 degrees. The planetesimals are given an aerodynamic gas drag cross section corresponding to 150 km radius objects. All bodies evolve in a gas disk of mass 0.01 M to 0.12 M, with an aspect ratio h r 1/4. The aim was to study the general behaviour of such systems Resonance shepherding phenomenon In all simulations, the resonance shepherding phenomenon can be observed. The following illustrations come from a set A simulation. Figure 3.1: Semi-major axis of the gas giant (full black curve) and the planetesimals (coloured curves) as a function of time. In figure 3.1, the planetesimals are separated in two different resonant configurations before the end of the migration. The first one is the 3 : 2 resonance (The closest to the giant) and the second one is the 2 : 1 resonance. They are shepherded by the gas giant during its migration thereby approaching the star. After the end of the migration, most of them are in the 2 : 1 and some of them are in the 5 : 2 resonance, which is a third order resonance, or out of resonance. For times lesser than yrs, several planetesimals are ejected from a resonance, their semi-major axis stabilises while their eccentricity decreases until they are caught once more when the planet approaches. This can be seen in figure 3.2, where is plotted a single planetesimal with the gas giant. This case is interesting because several phenomena can be seen. Before yrs, the planetesimal semi-major axis is at 0.7 AU and its eccentricity is very low. As the giant approaches, at yrs the planetesimal crosses the 3 : 1 resonance and its eccentricity rises suddenly to decrease slowly afterwards. At time t 1, the planetesimal gets locked in the 2 : 1 resonance, at t 2 in the 5 : 3 resonance and at t 3 in the 3 : 2 resonance for less than 10 4 yrs each time.

17 3.1 Results 15 Figure 3.2: Top graph: semi-major axis versus time for the giant (in black), for a planetesimal (in blue) and the distance of its pericentre and epicentre (in turquoise and light blue). Bottom graph: Ratio of the orbital periods of the giant and the planetesimal versus time, also plotted the lines corresponding to the different resonances taken by the system. However, at t4 a different phenomenon happens. The gas giant also shepherded embryos and although there are not plotted in figure 3.2, they influence the planetesimals. At t4 and t5, close encounters occur between the planetesimal and two embryos consecutively, sending the planetesimal closer to the star. Finally at t6, the planetesimal gets locked in the 2 : 1 resonance and is shepherded until the end of the migration and after1. This same phenomenon can be observed for embryos. As shown by the previous curves, the resonances are very efficient at shepherding bodies closer to the star. It works very well until the final stages of the giant planet migration, when close encounters occur and the embryos are ejected or accreted by the giant or the star. Let us now see the parameters influencing the survival of the embryos during these final stages of the migration Survival or destruction of hot embryos The general behaviour stays the same in all simulations. The embryos remain at their initial position until they cross a resonance. They are shepherded for 1 The ratio of the orbital periods is not exactly 0.5 at the end. What is conclusive in the resonance phenomenon is the alignment of the orbits as measured by resonant angles. Here, this criteria shows that the planetesimal is in the 2 : 1 resonance.

18 3.1 Results 16 a while and then are either sent in the outer disk, or onto the star, or accreted by the giant. When the giant comes close to the end of the migration, usually 1 2 embryos remain, but they do not often survive. Sometimes one of them does -but never both- and as in its evolution it accreted planetesimals or other embryos, its mass can be bigger than Earth s, thereby resulting in a hot super Earth. The probability of an embryo surviving is linked with the number of planetesimals around it, the gas average density and how close it is from the inner edge of the gas disk. Influence of planetesimals The presence of planetesimals is significant for the survival of embryos. Their presence allows dynamical friction with the embryos. When an embryo evolves in a field of planetesimals it will focus them and increase their number in its wake, thereby slowing it down. In this section and the next, 80 simulations from set B are considered. A statistical study of these simulations showed that the surviving embryos are indeed surrounded by a few planetesimals (1 to 5) during the last years of the migration. Their presence dampen the eccentricity of the embryo and allows the shepherding to occur more smoothly. In contrast, the destabilised embryos all have the same late evolution. They are shepherded and leave the resonance for an external reason 2, the number of planetesimals around them drops from a few (2 to 20) to zero, and they are either ejected or accreted after a few hundred to a few thousand years. However, in a few simulations an embryo survived the end of the migration with no planetesimals, illustrating the fact that planetesimals are not a necessity. The survival of an embryo is then not directly linked with the total number of planetesimals present in the system, but with the number of planetesimals in the same area as the embryo. Of course, this number increases if the number of planetesimals is initially higher, but that explains the fact that even high resolution simulations do not always result in a hot embryo, and low resolution simulations result sometimes in a hot embryo. Influence of the disk total mass In the original Mercury code, a parameter f G allows the user to modify the total disk mass by multiplying the gas surface density by a constant. The initial profile as given by Crida and Morbidelli (f G = 1) corresponds to a 0.06 M disk. The results of the 80 simulations can be seen in figures 3.3 and 3.4. Increasing the gas density augments the number of hot bodies. Indeed, the denser the gas disk, the better the eccentricity damping and the more regular the shepherding. The fraction of accreted planetesimals rises with the gas density. The planetesimals, which experience high eccentricity pumping or a close encounter are 2 For example, a close encounter with another embryo.

19 3.1 Results 17 Figure 3.3: Statistics for planetesimals over 80 simulations with 10 embryos and 50 planetesimals for different gas disk total mass. f G = 0.2 corresponds to a disk 5 times lighter than the original one. more likely accreted by the giant planet than ejected because there is more friction on them if the disk is denser. If they are ejected from the inner disk, the gas drag is more efficient at trapping them in the outer disk, hence the decreasing of the fraction of ejected bodies. The embryos experience the same kind of phenomena but some tendencies are less apparent because the damping is weaker. However, what remains clear is the higher proportion of hot embryos and the lower proportion of ejected embryos when the gas density increases. Figure 3.4: Statistics for embryos over 80 simulations with 10 embryos and 50 planetesimals for different gas disk total mass. For f G = 2, 7% of embryos survive the migration. For f G = 0.5, 33% of simulations result in a hot embryo, for f G = 1, 63% and for f G = 2, 70%. The probability of having surviving hot embryos is therefore increased when the disk is denser. Influence of the radius of the inner edge R min The study of the influence of the radius of the disk inner edge has been made with high resolution simulations of set C. Three simulations were run with

20 3.1 Results 18 R min = 0.05 AU, R min = AU and R min = 0.10 AU As this kind of simulations take about 5 days to complete and the computing space is limited, only a few of these simulations were run. I ran three simulations, varying the radius of the inner edge for each one. Figure 3.5: Final stage of a simulation with 20 embryos, 1000 planetesimals and an inner edge at 0.05 AU. Here is plotted the eccentricity versus the semi-major axis in the final configuration. The size of the dots is related to the mass of the bodies. The first one was made with R min = 0.05 AU, and the results are presented in figures 3.5 and 3.6. In this simulation a hot super Earth has been created in the 2 : 1 resonance, its final semi-major axis is about 0.06 AU. Figure 3.6: 0.05 AU. Initial and final stage of the simulation with an inner edge at The super Earth s final eccentricity is low, so it is likely it will remain here for a while. Figure 3.6 shows the initial and the final mass distribution, we can see that the hot embryo has gained a lot of mass to reach 5 M. In the late stages of the migration, five embryos were shepherded by the giant and they all collided with each other in about a thousand years to create the super Earth. For the other simulations, I fixed R min = AU and R min = 0.1 AU keeping the exact same initial condition as the R min = 0.05 AU simulation, and in both cases, no embryo survived.

21 3.1 Results 19 In the first one, the embryo was ejected and in the second one it was accreted by the gas giant after a short incursion inside the inner cavity. However, all my simulations 3 showed that if an embryo survives, most of the time it will be either in the 2 : 1 or the 3 : 2 resonance. If the inner edge boundary is larger than the location of these resonances, it is very likely that the embryo will not survive. These three simulations might illustrate this phenomenon. More simulations would have been needed to draw with certainty any conclusions Preliminary results of the improved version Improving the code Mercury in order to treat the gas consistently with the giant planet evolution took most of the last two months of my internship because test simulations took several days to run. Twelve high resolution simulations were done with this version of the code. Initial conditions The initial conditions of these simulations can be seen in table 3.2. The mass, eccentricity and inclination distribution of the embryos and planetesimals is the same as in section Simulation Initial giant Nb. of Nb. of Migration Simulation time position embryos plan. time time step D 2 AU day Table 3.2: Initial conditions for the modified version of the Mercury code simulations. I ran four sets of three simulations each with the improved version of Mercury. In the following table is written the number of simulations for the different four different sets of parameters. The three simulations of each set only differ by the initial conditions. h = 0.04 h = 0.06 M disk = 0.02 M 3 3 M disk = 0.08 M 3 3 Table 3.3: Simulations parameters. The same phenomena observed previously can be seen in these new simulations. Influence of the disk total mass For the simulations of FARGO (h = 0.04), the disk mass is of approximatively 0.1 M. The disks are thought to weigh between 0.01 M and 0.1 M. I chose 3 Low resolution ones included.

22 3.1 Results 20 to run Mercury simulations for two different values of disk mass, 0.02 M and 0.08 M. The results obtained with this version of the code are not numerous enough to be representative of what really happens. For the 0.02 M disk, 16.6% of simulations resulted in hot Earths, while for the 0.08 M disk, 33.3% did. The tendency is visible, but more simulations are needed to improve the statistics. The formation rate of hot super Earths with the modified version of the code is lower than with the original version but this may be due to the lack of statistics. In figure 3.7, is plotted the final stage of a simulation with h = 0.06 and M disk = 0.02 M, that resulted in a hot super Earth of 2.4 M. In figure 3.8 is plotted the final moments of an embryo and the number of planetesimals that surrounds it, for h = 0.06 and M disk = 0.08 M. Figure 3.7: Final stage of a simulation done with the improved version of the code. Here is plotted the mass of the different bodies versus the semi-major axis in the final configuration. The surviving embryo is in the 2 : 1 resonance and weighs 2.4 M. In figure 3.8, the embryo is shepherded in the 2 : 1 resonance from t = yrs to t 1 = yrs. The giant and the embryo accrete surrounding planetesimals making their number fall to 15. Then a series of close encounters occur between the embryo and the giant resulting in the ejection of the embryo at t 2. The embryos more likely to survive the shepherding until the end of the migration are those that enter the resonance later. The earlier an embryo gets locked in a resonance, the higher the probability it will have a close encounter with the giant. In the beginning of its shepherding, the embryo is often surrounded by planetesimals but either it accretes them progressively or the planetesimals leave due to other bodies. The shepherding occurs less smoothly and the embryo is more likely to have a close encounter with the giant. The candidates for surviving embryos are those that are initially closer to the star, and their probability of surviving is increased if the other embryos disappeared earlier. The survival and destruction of an embryo highly depends on the initial conditions, and of the efficiency of shepherding of the other embryos.

23 3.2 Conclusion 21 Figure 3.8: Top graph: semi-major axis versus time for the giant (in black), for an embryo (in purple) and the distance of its pericentre and epicentre (in deep purple and pink). Bottom graph: the number of planetesimals surrounding the embryo. Influence of the aspect ratio of the disk On the one hand, the value of the aspect ratio of the disk only influences the evolution of bodies with high inclination orbits. For a higher aspect ratio, the damping phenomenon will occur higher above the disk midplane. On the other hand, for a constant mass disk, increasing the aspect ratio is equivalent to decreasing the gas surface density. So for a higher aspect ratio, the damping will be reduced. These two effects compensate making the influence of the aspect ratio value difficult to gauge. For h = 0.04, 33.3% of simulations resulted in a hot Earth while for h = 0.06, 16.6% did. Apparently, the effect of increasing the gas surface density by decreasing the aspect ratio prevailed. So a lower aspect ratio appears to increase the probability of a hot super Earth surviving. 3.2 Conclusion Hot Jupiters are an interesting subject of studies. Their presence so close to their star is unexpected in the context of the solar system s giant planets. Several theories exist to explain their presence, including the type II migration hypothesis[20]. I studied this formation phenomenon with a N-body code completed with a gas disk description, in which the main purpose was to analyse the different

24 3.3 Limits and improvements 22 parameters influencing the survival of planetary embryos during the migration. My study showed the importance of the disk properties and of the number of small bodies considered in the formation a hot super Earth. Moreover, about 40% of all my simulations resulted in the survival of a hot Earth, whose inclination was the same as the hot Jupiter. It means that if type II migration of gas giants is indeed a way of formation of hot Jupiters, one could observe eventually hot super Earths in hot Jupiters systems. Hot super Earths have already been detected, like COROT 7b[21] but never in a system hosting a hot Jupiter. The Kepler Mission will detect exoplanets by transit and radial velocity measurements, from gas giants to planets 30 to 600 times less massive than Jupiter[22]. Looking for hot super Earths in a hot Jupiter system would then be a good way to corroborate or contradict the type II migration hypothesis for the creation of hot Jupiters. Not finding hot super Earths, or finding them in a much lower proportion could mean that type II migration is not the main formation mechanism of hot Jupiters. Some theories consider gravitational instabilities and tidal evolution to explain the formation of hot Jupiters. That kind of theory can explain the retrograde motion of some hot Jupiters, which type II migration hypothesis cannot. It could also mean that some of the neglected phenomena were crucial. Indeed, some phenomena could have been described in a much better way, had it been less time consuming. 3.3 Limits and improvements In my attempt to improve the code I came across many problems. The first one was the time needed to generate good density and velocity profiles with FARGO. The first simulations I made were purely 1D-grid simulations to see if one dimension was enough to reproduce the general evolution of the disk. Unfortunately, the runs ended either with the planet falling on the star or with a strange effect that was not physical. So I decided to use the 2D-grid in the range of migration of my planet. Unfortunately, to have a correct description so close to the star would have required a step size and a time step very small. That kind of simulation would have taken a few months in itself. Consequently, a possible follow-up of this internship would be to compute correct gas density profiles with the 2D-grid of FARGO and use them in Mercury. Another solution would have been to use Fogg&Nelson s approach[19], which consists in solving the 1D diffusion equation of the gas directly in the Mercury simulation. This method is equivalent to mine as I also use a 1D calculation in FARGO. A few important phenomena have been totally neglected in this study, in particular the effects of tidal evolution and general relativity. A possible followup of this internship would be to study the influence of these effects on the survival of hot Earths.

25 23 Chapter 4 Conclusion About 90 Hot Jupiters have been detected as of July 26, 2010, and many more will be detected in the future with the Kepler mission. Trying to understand how they form, and if they may be accompanied by a hot super Earth or not, is an interesting scientific question. During my internship I studied hot Jupiters systems with N-body simulations in order to understand what are the physical processes influencing the survival or the destruction of embryos during the migration of a giant planet. During the first part of my internship, I studied the original Mercury code. After two months I realised that rescaling the initial gas density was not a consistent description of the phenomenon. I then decided to use the FARGO code to create density and velocity profiles for different times during the migration of the giant planet. Once I obtained a satisfying disk evolution and modified the Mercury code, I was able to run simulations with this improved version. The development took a few weeks or so to complete because of few-days-long simulations. This study allowed me to learn about typical dynamical phenomena such as resonance shepherding, resonance pumping and dynamical friction as well as a few elements of viscous gas disk evolution. It also allowed me to improve my computing skills with the fortran language and IDL. The presence of a gas disk allows the resonance shepherding phenomenon to push planetesimals and embryos towards the star. The final stages of the migration are dangerous for embryos, but their destruction can be prevented by increasing the gas density, the number of planetesimals and decreasing the disk inner edge radius. This study also offers some perspective because it can corroborate or contradict the type II migration hypothesis given the observations that will be made with Kepler. The environment at the LAB is very pleasant. It is located on the site of a 100-year old observatory, with old instruments such as a meridian circle and an equatorial refractor telescope, in the middle of a park inhabited by researchers and deers. I had the opportunity to attend seminars and team meetings, which was very constructive. I intend to do a PhD-thesis here under the supervision of Sean

26 24 Raymond, and seminars allowed me to meet people whom I might collaborate with in the future. This thesis will not be a continuation of this internship but about the formation, evolution and habitability of planets around low-mass stars such as M-stars.

27 BIBLIOGRAPHY 25 Bibliography [1] [2] [3] [4] Cameron, A., G., W., Truran, J., W., The Supernova Trigger for Formation of the Solar System. Icarus, 30, , [5] Blum, J., Wurm, G., The Growth Mechanisms of Macroscopic Bodies in Protoplanetary Disks. Annual Review of Astronomy and Astrophysics, 46, 21-56, [6] Kokubo, E., Ida, S., Formation of Protoplanets from Planetesimals in the Solar Nebula. Icarus, 143, 15-27, [7] Kokubo, E., Ida, S., Formation of Protoplanet Systems and Diversity of Planetary Systems. The Astrophysical Journal, 581, , [8] Ward, W., R., Density waves in the solar nebula: Differential Lindblad torque. Icarus, 67, , [9] Tanaka, H., Ward, W.,R., Three-dimensional Interaction between a Planet and an Isothermal Gaseous Disk. II. Eccentricity Waves and Bending Waves. The Astrophysical Journal, 602, , [10] Cresswell, P., Nelson, R. P., Three-dimensional simulations of multiple protoplanets emnedded on a protostellar disc. Astronomy & Astrophysics, 482, , [11] Adachi, I., Hayashi, C., Nakazawa, K., The gas drag effect on the elliptical motion of a solid body in the primordial solar nebula. Prog. Theor. Phys., 56, , [12] Mandell, A., Raymond, S., Sigurdsson, S., Formation of Earth-like Planets During and After Giant Planet Migration, The Astrophysical Journal, 660, , [13] Shu, F., H., Shang, H., Glassgold, E., Lee, T., X-rays and Fluctuating X-Winds form Protostars. SCIENCE, 277, , [14] Crida, A., Morbidelli, A., Masset, F., On the width and shape of gaps in protoplanetary disks. Icarus, 181, , 2006.

28 BIBLIOGRAPHY 26 [15] Masset, F., FARGO: A fast eulerian transport algorithm for differentially rotating disks. Astronomy & Astrophysics Supplement series, 141, , [16] Chambers, J., E., A hybrid symplectic integrator that permits close encounters between massive bodies. Monthly Notices of the Royal Astronomical Society, 304, , [17] Crida, A., Morbidelli, A., Cavity opening by a giant planet in a protoplanetary disc and effects on planetary migration. Monthly Notices of the Royal Astronomical Society, 377, , [18] [19] Fogg, M., J., Nelson, R., P., On the formation of terrestrial planets in hot-jupiter systems. Astronomy & Astrophysics, 461, , [20] Lin, D., N., C., Bodenheimer, P., Richardson, D., C., Orbital migration of the planetary companion of 51 Pegasi to its present location. Nature, 380, , [21] Léger, A., Transiting exoplanets from the CoRoT space mission. VIII. CoRoT-7b: the first super-earth with measured radius. Astronomy & Astrophysics, 506, , [22] Borucki, W., J., Kepler Planet-Detection Mission: Introduction and First Results. Science, 327, 977-, 2010.

29 Appendices 27

30 28 Codes details Mercury The hybrid/bs algorithm uses a second-order mixed-variables symplectic integrator, which is based on a conservative Hamiltonian calculation and therefore cannot compute close encounters. A close encounter between two bodies A and B is designed in the code to occur when A enters a sphere centred on B, whose radius is about three times the Hill radius. When this condition is fulfilled, the code switches to a Burlish-Stoer (BS) integrator, which then computes the interaction component of the Hamiltonian. Collisions between bodies are taken into account in a simple way. When a collision occurs, the heaviest body survives and its mass is increased by the mass of the lightest body. In this model, inelastic collisions are treated and there are no lost fragments. The strength of this algorithm is that it is much faster than just the BS integrator, although it is less accurate. As the use of the BS algorithm alone increases greatly the computation time due to the modifications and due to the large number of planetesimals often taken into account, I used the hybrid/bs algorithm. It was a good compromise between efficiency and time. FARGO The FARGO-2D1D code uses a 2D-grid in the area of the planet, where precise calculations are needed and a 1D-grid, which allows to compute the gas density and velocity profiles faster (see figure 1). Figure 1: In the version FARGO-2D1D a 1D-grid surrounds the usual 2Dgrid[18].