Canup and Agnor: Accretion of the Terrestrial Planets 113 Accretion of the Terrestrial Planets and the Earth-Moon System Robin M. Canup Southwest Research Institute Craig B. Agnor University of Colorado Current models for the formation of the terrestrial planets suggest that the final stage of planetary accretion is characterized by collisions between tens to hundreds of lunar to Mars- sized planetary embryos. In this view, large impacts are an inevitable outcome as a system of embryos destabilizes to yield the final few planets. One such impact is believed to be respon- sible for the origin of the Moon. Improvements in numerical methods have recently allowed for the first direct orbit integrations of the final stage of accretion, which is believed to persist for ~108 yr. The planetary systems produced by these simulations bear a general resemblence to the terrestrial planets, but on average differ from our system in the final number of planets (fewer), their orbital spacings (wider) and their eccentricities and inclinations (larger). The dis- crepancy between these predictions and the nearly circular orbits of both Earth and Venus is significant, and is likely a result of the approximations inherent to the late-stage accretion simu- lations performed to date. Results from these works further highlight the important role of sto- chastic impact events in determining final planetary characteristics. In particular, impacts capable of supplying the angular momentum of the Earth-Moon system are predicted to be common. 1. INTRODUCTION In one of the first modern works to examine terrestrial planet formation, Safronov (1969) proposed that planets In the planetesimal hypothesis, the growth of terrestrial accreted in radially confined feeding zones, in a relatively planets is the result of the process of collisional accumula- quiescent manner through the accumulation of small bod- tion from initially small particles in the protoplanetary disk. ies. Developments in the past two decades suggest that the The accretion process is typically described in terms of three generally localized, runaway stage persists only until bod- stages of growth, which are distinguished by our basic un- ies grow to the size of the Moon or Mars, leaving many derstanding of the relevant physical processes involved in planetary embryos throughout the terrestrial region. Such a forming solid bodies in a particular size range. The first system of embryos is dynamically unstable on timescales stage involves the formation of kilometer-sized planetesi- that are short (~106 yr) compared to the age of the solar mals from an initial protoplanetary disk of gas and dust. By system, and highly energetic collisions between embryos the end of this stage of growth (discussed in chapter by then occur to yield the four terrestrial planets. In this sce- Ward, 2000), planetesimals have reached sizes large enough nario, the characteristics of the final planets are determined so that their dynamical evolution is determined primarily mainly by the specifics of the last few large impacts that by gravitational interactions with the central star and with each planet experiences. The stochastic nature of the final other planetesimals, rather than by surface, electromagnetic, stage thus yields an inherent degree of uncertainty for the or sticking forces. The middle stage consists of the accu- outcome of accretion in any given system. Indeed, a wide mulation of a swarm of kilometer-sized planetesimals into range of possible planetary architectures is found to arise lunar- to Mars-sized planetary embryos (see chapter by from even nearly identical initial conditions, suggestive of Kortenkamp et al., 2000). Numerous works have demon- the great variety of terrestrial planet systems that might exist strated that in this stage, dynamical friction acts to reduce in extrasolar systems. encounter velocities with the largest bodies, facilitating the The physical and dynamical environment in which the “runaway” growth of ~1025–1027 g (M = 5.98 × 1027 g) accretion of the terrestrial planets took place is directly rel- planetary embryos in as little as 105 yr (e.g., Greenberg et al., evant to models of lunar formation. In the giant impact sce- 1978; Wetherill and Stewart, 1993). The last stage then con- nario, the Moon forms as a result of a single impact with sists of the formation of the final few planets via the colli- Earth late in its formation history. While works to date have sion and merger of tens to hundreds of planetary embryos. generally considered the various stages of planet accretion Evolution during this period is thought to be driven by dis- and the formation of the Moon separately, a more holistic tant interactions between the embryos, and requires a few approach may be required. In particular, models of the pro- times ~108 yr (e.g., Wetherill, 1992). posed lunar-forming impact and the accretion of the Moon 113 114 Origin of the Earth and Moon have yet to identify a single impact that can yield the final The boundary between the middle and late stages is gen- masses of the Earth and Moon, together with the current erally believed to be representative of a dynamical transi- system angular momentum (Cameron and Canup, 1998; tion: from a stage in which growth is dominated by colli- Cameron, 2000). However, recent terrestrial accretion stud- sions with local material, to one in which distant interac- ies suggest that impacts subsequent to the lunar-forming tions among the embryos lead to collisions on much longer event may have contributed significantly to the final mass timescales. Recent simulations that model embryo forma- and/or angular momentum of the Earth-Moon system, offer- tion in the full terrestrial zone (0.5–1.5 AU) find that 90% ing a possible resolution to this dilemma (Agnor et al., 1999). of the system mass is contained in a few tens of embryos In this chapter, we review recent simulations of late-stage after about a million years, with the remaining ~10% of the accretion and discuss the successes and weaknesses of these mass contained in a swarm of much smaller planetesimals models (section 2). We then address issues especially rel- (Weidenschilling et al., 1997). These simulations have in- evant to the formation of the Moon via giant impact, includ- cluded effects due to gas drag and a parameterization of ing late-stage impact statistics and the potential role of distant perturbations between embryos, but to date have not multiple impacts in affecting planetary spin angular mo- included collisional fragmentation. At present, it is not clear menta (section 3). A brief discussion of open issues is in- to what extent the planetesimal swarm persists or is regen- cluded in section 4. erated via collisional erosion throughout the late stage. It is often assumed that all the small material in the disk would 2. ACCRETION OF THE be rapidly swept up by the embryos, since the timescale for TERRESTRIAL PLANETS embryo formation (105–106 yr) is much shorter than the timescale for the accumulation of the final planets (~2 × The environment in which the final accretion of terres- 108 yr). In this case, the dynamics of the final stage are trial-type planets takes place is dependent upon the outcome governed solely by gravitational interactions among and of the preceeding runaway growth stage. Midstage accre- collisions between the large embryos. Interactions among tion models utilizing both statistical treatments that model embryos initially on nearly circular orbits lead to eccentric- the entire terrestrial region, and N-body simulations of run- ity growth, and then to orbit crossing; once orbital isola- away growth within a local radial zone of the disk, yield tion is overcome, the embryos collide and merge. The accu- qualitatively similar results: embryos with masses ~0.01– mulation of embryos into larger bodies then proceeds until 0.1M , occupying nearly circular, low-inclination orbits the secular orbital oscillations (primarily the eccentricities) after 105–106 yr. The embryos have typical orbital spacings of the remaining bodies are insufficient to allow bodies to ∆ of a ~ 10 RHill, where RHill is the mutual Hill radius given encounter each other, and a few planets remain on well- by separated orbits. Until recently, simulations of this final stage were limited to statistical treatments due to the large num- 1/3 aamm++ ber of orbital times involved. However, numerical tech- R ≡ 12 1 2 (1) Hill 23M niques now exist that allow for direct integration of systems of N ~ 10–100 embryos for ~108 yr. Results from simula- where a1, a2, m1, and m2 are the semimajor axes and masses tions using both methods are reviewed in the next section. of adjacent embryos, and M is the mass of the Sun. A sys- tem of two bodies on initially circular orbits will be stable 2.1. Late-Stage Simulations ∆ against mutual collision so long as a > 3.5 RHill (e.g., Gladman, 1993). However, multiplanet systems (or planets Late-stage terrestrial accretion has been modeled using with nonzero initial eccentricities) require larger separations two basic (and complementary) techniques. Monte Carlo for stability. Chambers et al. (1996) performed numerical simulations follow the orbital evolution of embryos in a integrations of like-sized planets intially on circular orbits statistical manner based on two-body scattering events (e.g., ∆ ≈ and found a separation of a 8–10 RHill provided stabil- Wetherill, 1985), while N-body orbital integrations directly ity for ~106–107 yr, in fair agreement with the predicted track the trajectories of each embryo at all times (Cham- spacings from the midstage accretion simulations (e.g., bers and Wetherill, 1998, hereafter CW98; Agnor et al., Weidenschilling et al., 1997). Ito and Tanikawa (1999) con- 1999, hereafter ACL99). Under comparable sets of assump- ducted numerical stability analyses of planetary embryos, tions, both methods produce similar configurations of final including initial eccentricities and inclinations, a range of planets.