PLANET MIGRATION IN PLANETESIMAL DISKS Harold F. Levison Southwest Research Institute, Boulder Alessandro Morbidelli Observatoire de la Coteˆ d'Azur Rodney Gomes Observatorio´ Nacional, Rio de Janeiro Dana Backman USRA & SETI Institute, Moffett Field Planets embedded in a planetesimal disk will migrate as a result of angular momentum and energy conservation as the planets scatter the planetesimals that they encounter. A surprising variety of interesting and complex dynamics can arise from this apparently simple process. In this Chapter, we review the basic characteristics of planetesimal-driven migration. We discuss how the structure of a planetary system controls migration. We describe how this type of migration can cause planetary systems to become dynamically unstable and how a massive planetesimal disk can save planets from being ejected from the planetary system during this instability. We examine how the Solar System's small body reservoirs, particularly the Kuiper belt and Jupiter's Trojan asteroids, constrain what happened here. We also review a new model for the early dynamical evolution of the outer Solar System that quantitatively reproduces much of what we see. And finally, we briefly discuss how planetesimal driven migration could have affected some of the extra-solar systems that have recently been discovered. 1. INTRODUCTION planet, tend to lie on one side of the planet in question or another, there will be a net flux of material inward or out- Our understanding of the origin and evolution of planets ward. The planet must move in the opposite direction in has drastically transformed in the last decade. Perhaps the order to conserve energy and angular momentum. In the ab- most fundamental change was the realization that planets, in sence of strong gravitational perturbations from other plan- general, may not have formed in the orbits in which we see ets, the semi-major axis of planet in question will smoothly them. Indeed, planets may have migrated huge distances af- change with time. In the remainder of this chapter, we refer ter they were born, as many of the extra-solar planetary sys- to this type of migration as simple migration. Third, plan- tems show (Moorhead and Adams, 2005; Papaloizou and etary systems can suffer a dynamical instability (Levison et Terquem, 2006), al: 1998) that can lead to a short, but violent period of or- There are three main dynamical mechanisms that can bital evolution during which the planetary eccentricities are cause such a wholesale evolution in planetary orbits. First, increased to large values (Rasio and Ford, 1996). If the in- at early times when the natal proto-planetary gas disk is still stability can be damped by some process (like dynamical in existence, gravitational interactions between the disk and friction with a disk), the planets can once again evolve onto a planet could cause a significant amount of orbital evo- nearly circular orbits, but in very different locations from lution (see the chapter by Papaloizou et al: for a discus- where they formed. It has been suggested that this kind of sion). Second, after the gas disk is gone, if there still is instability occurred in the outer parts of our planetary sys- a significant number of planetesimals in the system, the tem (Thommes et al:, 1999; Levison et al:, 2001; Tsiganis planets can migrate as a result of gravitational encounters et al:, 2005). with these objects. In particular, if a planet is immersed in The primary foci of this chapter are the second and the a sea of small bodies, it will recoil every time it gravita- third mechanisms described above; namely migration in tionally scatters one of these objects. In general, since the planetesimal disks. Although we claimed at the opening of small objects can come in from any direction, this will force this chapter that the role that planetesimal-driven migration the planets to undergo a small random walk in semi-major played in the evolution of planetary systems was only ac- axis. However, since the sinks for these objects, for exam- knowledged within the last decade, the idea itself is over 20 ple ejection from the system or encountering a neighboring years old. The first discussion of this process was presented 1 by Fernandez´ and Ip, (1984). Their paper describes the re- 2.1. A Simple Model sponse of Jupiter, Saturn, Uranus, and Neptune to a rem- IBLT00 shows that the rate of change of a planet's semi- nant disk of planetesimals. Although much of the details major axis, a is have changed as our ability to perform orbital integrations has improved, these authors found the basic result which is da 2 = paH_ ×; (1) still held true today. dt −Mp The importance of the work by Fernandez´ and Ip was not really appreciated until the discovery of the Kuiper belt where Mp is the mass of the planet, and H_ × is the rate nearly a decade later, with its numerous objects on eccen- of transfer of angular momentum from the planetesimals tric orbits in mean motion resonances with Neptune. Mal- to the planet. This equation assumes that the eccentric- hotra (1993, 1995) first showed that these orbits could be ity of the planet is small and G 1. Using the particle- ≡ the natural result of Neptune's outward migration and con- in-a-box approximation, it is possible to show that H_ × = cluded that Neptune must have migrated about 7 AU in or- k¯M(t)a−1, where is a quantity that contains funda- der to explain the eccentricities that we see (c.f: Section 3). mental constants and information about the geometry of Much work as been done on this topic since Fernandez´ the planet-encountering region, M(t) is the total mass in and Ip's and Malhotra's groundbreaking papers. This liter- planet-encountering orbits, and k¯ is the average change of ature is the topic of the remainder of the chapter, although angular momentum per encounter, per unit mass planetesi- it should be noted that this chapter is intended more as dis- mal. Thus, cussion of the current state-of-the-art than a review paper. da M(t) 1 = 2k¯ : (2) In Section 2 and Section 3 we describe some of the basic dt − Mp pa physics that governs planet migration and resonant capture, The evolution of M(t) can be approximated by the equa- respectively. In Section 4, we examine simple planetesimal- tion driven migration in the Solar System. In Section 5 we look M_ (t) = M(t)/τ + 2πa a_ Σ(a) ; (3) at how instabilities in the orbits of the planets, coupled with − j j gravitational interactions with a massive planetesimal disk, where the first term in the r.h.s. represents the decay of the could lead to significant changes in the orbits of the plan- planetesimal population due to the planetesimal's finite dy- ets. We review a new model for the early dynamical evo- namical lifetime, and the second term stands for the plan- lution of the giant planets of the Solar System in Section 6, etesimals that, because of the change in the planet's posi- and planetesimal-driven migration in extra-solar systems in tion, enter the region where they can be scattered by the Section 7. In Section 8 we discuss some caveats and lim- planet for the first time. In (3) Σ(a) is the surface density itations of the N-body simulations on which most of the of the `virgin' (i.e. not yet scattered) planetesimal disk at content of this chapter is based. We present our concluding heliocentric distance a. Substituting (2) into (3) we get remarks in Section 9. −1 M_ (t) = τ + 4π k¯ paΣ(a)=Mp M(t) : (4) − j j 2. BASIC PRINCIPLES OF SIMPLE MIGRATION Let's assume for simplicity that the term α τ −1 + ≡ − The migration history for each individual planet is com- 4π k¯ paΣ(a)=Mp does not significantly change with time j j plicated because it is dependent on the details of how an- — an approximation that is clearly not true, but it allows us gular momentum flows through the system, namely on the to get the essence of the planet's behavior. Under this as- distribution and the evolution of the mass and the angular sumption, Eq. (4) becomes an exponential equation, which momentum of the planet crossing objects. These quantities, is solvable and thus will allow us to get some insight into in turn, are determined by the sources and sinks for these how the planet migrates. particles. As a result, attempts to develop analytic theories If α is negative, then M(t) decays exponentially to 0 for migration have studied very simple systems consisting and the planet (from Eq. 1) stops migrating. In this case, of a single planet in a dynamically cold disk (Murray et the loss of planetesimals due to their finite dynamical life- al:, 1998; Ida et al:, 2000, hereafter IBLT00). So, the goal time is not compensated by the acquisition of new planetes- of this section is not to develop a comprehensive analyti- imals into the scattering region. Therefore, the planet runs cal model, but to develop a toy model to help a qualitative `out of fuel'. GML04 called this migration mode damped understanding of some of the important physical processes migration. Conversely, if α is positive, M(t) grows expo- involved. Much of what we present is based on the work nentially and the planet's migration accelerates. In this case in Gomes et al:, (2004), hereafter GML04. We start with a the acquisition of new planetesimals due to the migration very simple model.