Models of terrestrial planet growth around single stars

The 'planetesimal hypothesis' states that planets grow within circumstellar disks via pairwise accretion of small solid bodies known as planetesimals (Chamberlin 1905; Safro-nov 1969; Hayashi et al. 1977). The process of planetary growth is generally divided for convenience and tractability into several distinct stages. Planets begin to grow when microscopic dust grains collide and agglomerate via sticking/local electromagnetic forces as they settle towards the midplane of the disk. The least well-understood phase of solid planet formation is the agglomeration from cm-sized pebbles to km-sized bodies that are referred to as planetesimals. The gaseous component of the protoplanetary disk plays an important role in this stage of planetary growth (Adachi et al. 1976; Weidenschilling 1977). Collective gravitational instabilities among the solid grains in the disk (Safronov 1969; Goldreich & Ward 1973) might be important, although turbulence of the gas may prevent the protoplanetary dust layer from becoming thin enough to be grav-itationally unstable (Weidenschilling & Cuzzi 1993). Recent calculations suggest that high-metallicity disks may form planetesimals via gravitational instabilities, but that dust in disks with fewer solids may not be able to overcome turbulence and settle into a subdisk that is dense enough to undergo gravitational instability (Youdin & Shu 2002). Planetesimal formation is a very active research area (Goodman & Pindor 2000; Ward 2000; Cuzzi et al. 2001), and results may have implications for our estimates of the abundance of both terrestrial and giant planets within our galaxy.

Once solid bodies reach kilometer-size (in the case of the terrestrial region of the proto-solar disk), gravitational interactions between pairs of solid planetesimals provide the dominant perturbation of their basic Keplerian orbits. Electromagnetic forces, collective gravitational effects, and in most circumstances, gas drag, play minor roles. Planetesi-mals agglomerate via pairwise mergers. The rate of solid body accretion by a planetesimal or planetary embryo is determined by the size and mass of the planetesimal/planetary embryo, the surface density of planetesimals, and the distribution of planetesimal velocities relative to the accreting body. The evolution of the planetesimal size distribution is determined by the gravitationally enhanced collision cross-section, which favors collisions between bodies having smaller relative velocities. Runaway growth of the largest planetesimal in each accretion zone appears to be a likely outcome. The subsequent accumulation of the resulting planetary embryos leads to a large degree of radial mixing in the terrestrial planet region, with giant impacts probable. Growth via binary collisions proceeds until the spacing of planetary orbits becomes sufficient for the configuration to be stable to gravitational interactions among the planets for the lifetime of the system (Safronov 1969; Wetherill 1990; Lissauer 1993, 1995; Chambers 2001; Laskar 2000).

Safronov (1969) developed analytical tools to predict planetary growth from small (~1 km) solid bodies, although he was required to make numerous physical assumptions/approximations, as his theory did not incorporate computer modeling. Numerous groups (e.g., Greenberg et al. 1978; Wetherill & Stewart 1989; Kolvoord & Greenberg 1992) have attempted to examine the accumulation and dynamics of 1 km planetesimals via numerical simulations, using different sets of approximations to make the problem numerically tractable. Recent advances in computer hardware and coding are allowing the direct modeling of these early phases of planetary growth within localized regions of the disk (Barnes 2004; Barnes et al. 2006). Assuming perfect accretion, i.e., that all physical collisions are completely elastic, the initial stages of growth are quite rapid, especially in the inner regions of a protoplanetary disk, and large particles form quickly (Fig. 1).

106 105 104 ^ 1000 100 10 1

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1 10 100 1000 Mass (rrij)

Figure 1. Growth of planetesimals within an accreting shearing patch of the protoplanetary disk that initially contained 106,130 planetesimals, each of radius 1 km, density 3 g/cc and mass m1 = 1.2 x 1016 g, located 0.4 AU from a 1 Mq star. The mass distribution is shown after 100 orbits (top panel) and after 900 orbits (bottom panel). After 100 orbits, all of the particles are in a distribution described by a power law. At 900 orbits, most particles are still in this type of distribution, but several particles appear to have broken away. The two most massive particles are very close in size, 1566 m1 and 1518 m1. Courtesy R. Barnes; see Barnes et al. (2006) for details.

Planetesimal growth regimes are sometimes characterized as either orderly or runaway. In orderly growth, the particles containing the most mass double their masses in about the same amount of time as the largest particle. The stochasticity of the system creates a small fraction of larger particles. Runaway growth involves a qualitative distinction between the particles of the system. When the relative velocity between planetesimals is comparable to or larger than the escape velocity, v > ve, the growth rate is approximately proportional to R2, where R is the radius of the growing planetesimal, and the evolutionary path of the planetesimals exhibits an orderly growth of the entire size distribution. When the relative velocity is small, v ^ ve, the growth rate is proportional to R4. In this situation, the planetary embryo rapidly grows much larger than any other planetesimal in its accretion zone, which can lead to runaway growth. By virtue of its large, gravitationally enhanced cross section, a runaway particle doubles its mass faster than smaller bodies, and detaches itself from the mass distribution (Wetherill & Stewart 1989; Ohtsuki, Stewart, & Ida 2002).

Local models break down when the impinging velocities no longer result from a homogeneous distribution. This occurs when a particle transitions from dispersion-dominated growth to shear-dominated growth (Lissauer 1987). Initially, growth is dispersion dom

100 Orbits

900 Orbits

Figure 1. Growth of planetesimals within an accreting shearing patch of the protoplanetary disk that initially contained 106,130 planetesimals, each of radius 1 km, density 3 g/cc and mass m1 = 1.2 x 1016 g, located 0.4 AU from a 1 Mq star. The mass distribution is shown after 100 orbits (top panel) and after 900 orbits (bottom panel). After 100 orbits, all of the particles are in a distribution described by a power law. At 900 orbits, most particles are still in this type of distribution, but several particles appear to have broken away. The two most massive particles are very close in size, 1566 m1 and 1518 m1. Courtesy R. Barnes; see Barnes et al. (2006) for details.

inated; the velocity distribution of incoming particles is simply that of the velocity dispersion of the planetesimal swarm. However, when a particle reaches a size such that the Keplerian shear across its Hill sphere (the volume of space over which the body's gravity dominates over the tidal force resulting from the gradient of the star's gravitational potential) is larger than the velocity dispersion, it enters the shear-dominated regime. At this point, larger embryos take longer to double in mass than do smaller ones, although embryos of all masses continue their runaway growth relative to surrounding planetesi-mals; this phase of rapid accretion of planetary embryos is known as oligarchic growth (Kokubo & Ida 1998).

The self-limiting nature of runaway/oligarchic growth implies that massive planetary embryos form at regular intervals in semi-major axis. The agglomeration of these embryos into a small number of widely spaced terrestrial planets necessarily requires a stage characterized by large orbital eccentricities, significant radial mixing, and giant impacts. At the end of the runaway phase, most of the original mass is contained in the large bodies, so their random velocities are no longer strongly damped by energy equipartition with the smaller planetesimals. Mutual gravitational scattering can pump up the relative velocities of the planetary embryos to values comparable to the surface escape velocity of the largest embryos, which is sufficient to ensure their mutual accumulation into planets. The large velocities imply small collision cross-sections and hence long accretion times.

Once the planetary embryos have perturbed one another into crossing orbits, their subsequent orbital evolution is governed by close gravitational encounters and violent, highly inelastic collisions. This process has been studied using W-body integrations of planetary embryo orbits, which include the gravitational effects of the giant planets, but neglect the population of numerous small bodies which must also have been present in the terrestrial zone; physical collisions are assumed to always lead to accretion (i.e., fragmentation is not considered). Few bodies initially in the terrestrial planet zone are lost; in contrast, most planetary embryos in the asteroid region are ejected from the system by a combination of Jovian perturbations and mutual gravitational scatterings. As the simulations endeavor to reproduce our Solar System, they generally begin with about 2 of material in the terrestrial planet zone, typically divided among several dozen or more protoplanets. The end result is the formation of two to five terrestrial planets on a timescale of about 1-2 x 108 years (Agnor, Canup, & Levison 1999; Chambers 2001). Some of these systems look quite similar to our Solar System, but most have fewer terrestrial planets which travel on more eccentric orbits. It is possible that the Solar System is, by chance, near the quiescent end of the distribution of terrestrial planets. Alternatively, processes such as fragmentation and gravitational interactions with a remaining population of small debris (or gas drag; Kominami & Ida 2004), thus far omitted from most calculations because of computational limitations, may lower the characteristic eccentricities and inclinations of the ensemble of terrestrial planets.

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