The Long-Term Dynamical Evolution of Planetary Systems Melvyn B. Davies Lund University Fred C. Adams University of Michigan Philip Armitage University of Colorado, Boulder John Chambers Carnegie Institution of Washington Eric Ford The Pennsylvania State University, University of Florida Alessandro Morbidelli University of Nice Sean N. Raymond University of Bordeaux Dimitri Veras University of Cambridge This chapter concerns the long-term dynamical evolution of planetary systems from both theoretical and observational perspectives. We begin by discussing the planet-planet interactions that take place within our own Solar System. We then describe such interactions in more tightly-packed planetary systems. As planet-planet interactions build up, some systems become dynamically unstable, leading to strong encounters and ultimately either ejections or collisions of planets. After discussing the basic physical processes involved, we consider how these interactions apply to extrasolar planetary systems and explore the constraints provided by observed systems. The presence of a residual planetesimal disc can lead to planetary migration and hence cause instabilities induced by resonance crossing; however, such discs can also stabilise planetary systems. The crowded birth environment of a planetary system can have a significant impact: close encounters and binary companions can act to destabilise systems, or sculpt their properties. In the case of binaries, the Kozai mechanism can place planets on extremely eccentric orbits which may later circularise to produce hot Jupiters. 1. INTRODUCTION views the application of theoretical models to observations of the solar system and extrasolar planetary systems. Currently observed planetary systems have typically Planetary systems evolve due to the exchange of angu- evolved between the time when the last gas in the protoplan- lar momentum and / or energy among multiple planets, be- etary disc was dispersed, and today. The clearest evidence tween planets and disks of numerous small bodies (“plan- for this assertion comes from the distribution of Kuiper belt etesimals”), between planets and other stars, and via tides objects in the outer solar system, and from the eccentrici- with the stellar host. A diverse array of dynamical evolution ties of massive extrasolar planets, but many other observed ensues. In the simplest cases, such as a well-separated two properties of planetary systems may also plausibly be the planet system, the mutual perturbations lead only to peri- consequence of dynamical evolution. This chapter sum- odic oscillations in the planets’ eccentricity and inclination. marizes the different types of gravitational interactions that Of greater interest are more complex multiple planet sys- lead to long-term evolution of planetary systems, and re- tems where the dynamics is chaotic. In different circum- 1 stances the chaos can lead to unpredictable (but bounded) with a summary of the key points of this chapter. excursions in planetary orbits, to large increases in eccen- tricity as the system explores the full region of phase space allowed by conservation laws, or to close approaches be- 2. THE SOLAR SYSTEM TODAY tween planets resulting in collisions or ejections. Qualita- A quick glance at our system, with the planets moving tive changes to the architecture of planetary systems can on quasi-circular and almost coplanar orbits, well separated likewise be caused by dynamical interactions in binary sys- from each other, suggests the idea of a perfect clockwork tems, by stellar encounters in clusters, or by changes to system, where the orbital frequencies tick the time with un- planetary orbits due to interactions with planetesimal discs. surpassable precision. But is it really so? In reality, due Theoretically, there has been substantial progress since to their mutual perturbations, the orbits of the planets must the last Protostars and Planets meeting in understanding vary over time. the dynamics that can reshape planetary systems. Observa- To a first approximation, these variations can be de- tional progress has been yet more dramatic. Radial velocity scribed by a secular theory developed by Lagrange and surveys and the Kepler mission have provided extensive cat- Laplace (see Murray and Dermott 1999) in which the or- alogues of single and multiple planet systems, that can be bital elements that describe a fixed Keplerian orbit change used to constrain the prior dynamical evolution of planetary slowly over time. The variations can be found using Hamil- systems (see the chapter by Fischer et al. for more details). ton’s equations, expanding the Hamiltonian in a power se- Routine measurements of the Rossiter-McLaughlin effect ries in terms of the eccentricity e and inclination i of each for transiting extrasolar planets have shown that a signifi- planet, and neglecting high-frequency terms that depend on cant fraction of hot Jupiters have orbits that are misaligned the mean longitudes. Only the lowest order terms are re- with respect to the stellar rotation axis, and have prompted tained since e and i are small for the planetary orbits. The new models for how hot Jupiters form. Despite this wealth variations for a system of planets j (ranging from 1 to N) of data, the relative importance of different dynamical pro- can then be expressed as cesses in producing what we see remains unclear, and we will discuss in this review what new data is needed to break N degeneracies in the predictions of theoretical models. Also ej sin ̟j = ekj sin(gkt + βk) uncertain is which observed properties of planetary systems Xk=1 reflect dynamical evolution taking place subsequent to the N dispersal of the gas disk (the subject of this chapter), and ej cos ̟j = ekj cos(gkt + βk) (1) which involve the coupled dynamics and hydrodynamics Xk=1 of planets, planetesimals and gas within the protoplanetary with similar expressions for i. Here ̟ is the longitude of disc. The chapter by Baruteau et al. (2013) reviews this perihelion, and the quantities ekj , gk, and βk are determined earlier phase of evolution. by the planet’s masses and initial orbits. We begin this chapter by considering the long term sta- In the Lagrange-Laplace theory, the orbits’ semi-major bility of the solar system. The solar system is chaotic, but axes a remain constant, while e and i undergo oscillations our four giant planets are fundamentally stable, and there with periods of hundreds of thousands of years. The or- is only a small probability that the terrestrial planets will bits change, but the variations are bounded, and there are experience instability during the remaining main-sequence no long-term trends. Even at peak values, the eccentricities lifetime of the Sun. We then compare the current solar sys- are small enough that the orbits do not come close to inter- tem to more tightly-packed planetary systems, which are secting. Therefore, the Lagrange-Laplace theory concludes hypothesized progenitors to both the solar system and ex- that the solar system is stable. trasolar planetary systems. We discuss the conditions, time The reality, however, is not so simple. The Lagrange- scales and outcomes of the dynamical instabilities that can Laplace theory has several drawbacks that limit its useful- be present in such systems, and compare theoretical models ness in real planetary systems. It is restricted to small values to the observed population of extrasolar planets. We then of e and i; theories based on higher order expansions exist, review how interactions between planets and residual plan- but they describe a much more complex time-dependence etesimal disks can lead to planetary migration, which de- of eccentricities and inclinations, whose Fourier expan- pending on the circumstances can either stabilize or desta- sions involve harmonics with argument νt where ν = N bilize a planetary system. Finally we discuss the outcome k=1 nkgk + mksk and nk,mk are integers, and g and s of dynamical interactions between planetary systems and Pare secular frequencies associated with e and i respectively. other stars, whether bound in binaries or interlopers that The coefficients of these harmonics are roughly inversely perturb planets around stars in stellar clusters. Dynamical proportional to ν, so that the Fourier Series representation evolution driven by inclined stellar-mass (and possibly sub- breaks down when ν 0, a situation called secular reso- stellar or planetary-mass) companions provides a route to nance. ∼ the formation of hot Jupiters whose orbits are misaligned to Moreover, the Lagrange-Laplace theory ignores the ef- the stellar equator, and we review the status of models for fects of mean-motion resonances or near resonances be- this process (often called the Kozai mechanism). We close tween the orbital periods of the planets. The existence of 2 mean-motion resonances can fundamentally change the dy- body integrations can never prove the stability of a system, namics of a planetary system and alter its stability in ways only its stability for the finite length of an integration. not predicted by Lagrange-Laplacetheory. In particular, the N-body integrations can be used to distinguish between terms dependent on the orbital frequencies, ignored in the regular and chaotic regions, and quantify the strength of Lagrange-Laplace theory, become important when the ratio chaos, by calculating the system’s Lyapunov exponent Γ, of two orbital periods is close to the ratio of two integers. given by This situation arises whenever the critical argument φ varies ln[d(t)/d(0)] Γ = lim (4) slowly over time, where t→∞ t where d is the separation between two initially neighboring φ = k1λi + k2λj + k3̟i + k4̟j + k5Ωi + k6Ωj (2) orbits. Regular orbits diverge from one another at a rate that is a power of time. Chaotic orbits diverge exponen- for planets i and j, where λ is the mean longitude, Ω is the tially over long timespans, although they can be “sticky”, longitude of the ascending node, and k1−6 are integers. The mimicking regular motion for extended time intervals.