# Planetary formation and migration

Curator and Contributors

1.00 - Philip Armitage

Planets form from the protoplanetary disks of gas and dust that are observed to orbit young stars (the Nebula Hypothesis that was advanced by Kant, Laplace, and others in the 18th century). Once formed, planetary orbits may be modified as a result of interactions with the gas disk, or with other planets, stars or small bodies present in the system. Such modification can result in planetary migration.

## Planet formation

The formation of planets requires growth through at least 12 orders of magnitude in spatial scale, from micron-sized particles of dust and ice up to bodies with radii of thousands or tens of thousands of km. It is convenient to divide the process up into distinct stages in which different physical processes are dominant.

### Planetesimal formation

The initial reservoir of solid material for planet formation is micron-sized particles of rocky or icy dust, which makes up about 1% of the mass of a typical protoplanetary disk. Some dust grains travel with the gas when a portion of a molecular cloud collapses to form a star and a protoplanetary disk, while more dust condenses from the gas phase within the disk. The dynamics of dust within a disk is dominated by gravity from the star and aerodynamic forces from the gas, including turbulence. In contrast, gravitational interactions between small bodies are very weak (the escape velocity from a 1 meter diameter rock is less than 0.1 cm/s). Aerodynamic forces remain dominant until bodies grow to 1-100 km in size. Such bodies, referred to as planetesimals, are massive enough that their gravitational interactions are significant, while their small surface area to volume ratio means they are only weakly affected by aerodynamic forces.

Dust grains grow by colliding with one another and sticking together by electrostatic forces. Small particles also physically embed themselves in larger aggregates during high-speed collisions. The motion of small dust grains is closely coupled to that of the gas, and turbulence causes dust to diffuse over large distances leading to substantial radial and vertical mixing of material within the disk. Particles larger than 1 mm develop significant velocities relative to the gas because gas orbits the star somewhat more slowly than a solid body due to an outward pressure gradient in the disk. This velocity differential causes particles to migrate radially towards the star, and particles also settle vertically towards the midplane of the disk. The inward migration time scale is particularly rapid for cm to meter sized bodies - at 1 AU from the star it can be as short as $$10^2$$ years. This implies that growth through this size range must be rapid, or else much of the solid material in the disk would evaporate when it enters the hot regions close to the star. The required rapid growth might occur as a result of ongoing pairwise collisions, possibly aided by the concentration of particles into small regions due to turbulent eddies. Alternatively, planetesimals may form via the gravitational collapse of regions containing dense concentrations of solid particles (the Goldreich-Ward mechanism). Both models face substantial challenges. For pairwise collisions to work fast enough, meter sized objects need to efficiently stick together upon collision rather than breaking up. This has not been demonstrated in laboratory experiments, and theoretical arguments suggest that at the expected collision velocities it is very difficult for growth to occur. The Goldreich-Ward scheme avoids this problem, but at the expense of requiring a substantially larger than normal dust fraction, and/or larger disk mass, in order that collapse is not prevented by turbulence stirred up in the particle-laden flow.

### Terrestrial planet formation

In the absence of direct observations or suitable laboratory experiments, much of what we know about terrestrial-planet formation comes from computer simulations. Terrestrial planet formation has been studied extensively using statistical models based on the coagulation equation to study the early stages of growth, and N-body simulations to model later stages when the number of large bodies is small. Once planetesimals have formed, their subsequent evolution is dominated by mutual gravitational interactions and collisions as they orbit the central star. Colliding planetesimals typically merge to form a larger body with some mass escaping as small fragments. Planetesimals also undergo numerous close encounters with one another, which alter their orbits but not their masses. At an early stage, runaway growth takes place, in which large bodies typically grow more rapidly than small ones due to differences in their orbital eccentricities and inclinations. Typical time scales for the runaway growth phase are of the order of $$10^5$$ years. Runaway growth is followed by oligarchic growth, in which a relatively small number of large bodies grow at similar rates until they have swept up most of the smaller planetesimals. Collisions and radioactive decay heat the large bodies until they melt, causing dense elements such as iron to sink to the center to form a core overlain by a rocky mantle. Oligarchic growth generates a population of $$10^2 - 10^3$$ lunar-to-Mars-sized planetary embryos, probably in 1 million years or less. Subsequent collisions between these embryos lead to the final assembly of the terrestrial planets, on a time scale of up to 100 million years. The Moon is thought to have formed about 40 million years after the start of the Solar System from debris placed into orbit about the Earth when it collided with a Mars-sized planetary embryo. A substantial fraction of the Earth's mass is thought to have been accreted via large impacts, so requiring such a cataclysmic event to form the Moon is in principle not a problem, though only a small fraction of giant impacts would lead to the formation of a satellite with the properties of the Moon. Another puzzle is why the Moon has such a similar composition to the Earth - this is not an obvious consequence of the giant impact theory. The time scale for lunar formation, along with other time scales such as that for asteroids to become large enough to differentiate, is derived by applying radionuclide chronometers to samples of rock. Such cosmochemistry evidence is becoming increasingly important, and provides a growing number of constraints on the formation of the early Solar System. Planets typically acquire mass from a range of distances within a protoplanetary disk, although the mixture is different for each object, leading to a unique chemical composition. It is likely that Earth acquired most of its water and other volatile materials from relatively cold regions of the Sun's protoplanetary disk such as the asteroid belt.

Simulations of terrestrial planet formation are able to reproduce the basic architecture (a small number of terrestrial planets with low eccentricity orbits) of the inner Solar System from plausible initial conditions. The stochastic nature of planetary accretion, however, means that a precision comparison between the Solar System and theoretical models in not possible. The number and masses of terrestrial planets are predicted to vary from one planetary system to another due to differences in the amount of solid material available and the presence or absence of giant planets, as well as the highly stochastic nature of planet formation. The presence of a giant planet probably frustrates terrestrial-planet formation in neighboring regions of the disk, leading to the absence of terrestrial planets in these regions or the formation of an asteroid belt. These predictions will be tested by ongoing and future space missions designed to search for extrasolar terrestrial planets, such as COROT and Kepler.

### Giant planet formation

Figure 1: Schematic illustration showing how the core mass (blue line) and total mass (core + envelope: red line) grow in a calculation of giant planet formation via core accretion. The formation of a 10-20 Earth mass core is followed first by slow quasi-static growth of an envelope, before finally runaway gas accretion ensues. The time scale of the slow phase of growth is a few million years.

Giant planets are qualitatively distinct from terrestrial planets in that they possess significant gaseous envelopes. In the Solar System, the gas giants (Jupiter and Saturn) are predominantly composed of hydrogen and helium gas, although these planets are enriched in elements heavier than helium compared to the Sun. The ice giants (Uranus and Neptune) have lesser, but still substantial (several Earth masses) gas envelopes. The existence of these envelopes provides a critical constraint: giant planets must form relatively quickly, before the gas in the protoplanetary disk is dissipated. Observations of protoplanetary disks around stars in young clusters pin the gas disk lifetime in the 3-10 million year range.

The standard theory for the formation of gas giants, core accretion, is a two-stage process whose first stage closely resembles the formation of terrestrial planets. A core with a mass of the order of 10 Earth masses forms in the disk by numerous collisions between planetesimals. Typically, there is not enough solid material to form bodies this massive in the inner region of a protoplanetary disk. At larger orbital radii, beyond the snow line, the temperature is low enough that ices as well as rocky materials can condense. This extra solid material, together with the reduced gravity of the central star, allows large solid cores to form in the outer regions of a disk. Initially a core is surrounded by a low mass atmosphere, which grows steadily more massive as the gas cools and contracts onto the core. Eventually the core exceeds a "critical core mass", beyond which a hydrostatic envelope cannot be maintained. Determining an accurate time scale for reaching the critical core mass is very difficult, in part because the rate at which the gas cools depends upon how transparent the envelope is. The transparency varies dramatically with the amount of dust present, which is extremely uncertain. Once the core mass is exceeded gas begins to flow onto the core, slowly at first but increasingly rapidly as the planet becomes more massive. Growth ceases when the supply of gas is terminated, either because the planet opens a gap in the disk or because the disk gas dissipates.

A second theory for gas giant formation, gravitational disk instability, also remains under study. A gas disk with surface density $$\Sigma\ ,$$ sound speed $$c_s$$ and angular velocity $$\Omega$$ is said to be gravitationally unstable if Toomre's $$Q$$ parameter, defined such that,

$$Q = \frac{c_s \Omega}{\pi G \Sigma}$$

is less than unity. If, additionally, the disk is able to cool on an orbital timescale, then the instability leads to fragmentation of the disk into bound objects. In protoplanetary disks, these objects would have masses comparable to giant planets. A key feature of this mechanism for forming giant planets is that it works extremely rapidly. Unlike core accretion, solids play no direct role in the process.

Core accretion is generally considered to be a more plausible model for giant planet formation than gravitational instability for several reasons. First, theoretical calculations suggest that although young protoplanetary disks may be massive enough to be unstable, they are unlikely to cool rapidly enough to fragment (except perhaps at very large radius). Secondly, the core accretion model naturally explains the existence of ice giant planets like Neptune (although the time scale for formation of the ice giants is worryingly long if they formed at their present locations). Finally, the observed correlation between the frequency of extrasolar planets and the metallicity of their host stars is qualitatively explicable as a consequence of core accretion: if the disk is enriched in solids, a critical-mass core can form more readily. It is unclear whether this correlation can be explained by the gravitational instability model. Against this, the inferred core mass of Jupiter (which can be estimated by comparing the measured multipoles of the gravitational field with theoretical structure models) is lower than simple estimates based on core accretion. More subtle observational constraints - such as the abundance of different elements measured in Jupiter's atmosphere by the Galileo probe - are also in conflict with at least the simplest models of giant planet formation. These problems suggest that a full understanding of giant planet formation has yet to be attained. Observations of the frequency of giant planets in extrasolar planetary systems with very different properties to the Solar System promise to provide valuable new constraints. For example, the core accretion model predicts that giant planet formation is very difficult at large orbital radii (even though gas disks can be 100 AU or more in size), and that the probability of planet formation ought to scale quite strongly with the stellar mass (generally it is believed to be harder around lower mass stars).

## Planetary migration

The possibility that planetary orbits might evolve subsequent to planet formation was recognized early on, notably by Peter Goldreich and Scott Tremaine in a 1980 paper. Interest in mechanisms for planetary migration increased dramatically with the discovery in 1995 of 51 Peg, whose orbital period of just 4.2 days places it so close to the star that it is highly unlikely to have formed in situ. Three main mechanisms for planetary migration have been studied.

### Gas disk migration

Figure 2: The surface density from a numerical simulation of the interaction between a massive planet and the protoplanetary gas disk.

A planet orbiting within a protoplanetary disk gravitationally perturbs the gas in its vicinity, launching density waves at orbital radii where the gas is in resonance with the planet. Interactions with gas in the waves adds or removes energy and angular momentum from the planet's orbit changing the semi-major axis (planetary migration) and possibly the orbital eccentricity.

Two main regimes of gas disk migration have been identified. Low-mass planets undergo type-I migration, where the surface density profile of the gas disk is only weakly altered by the planet and the migration rate is proportional to the planet's mass. The planet remains entirely embedded within the gas. In this situation, the most important resonances are those located close to the planet (with a radial displacement comparable to the thickness of the gas disk). The interaction with the gas disk interior to the planet's orbit adds angular momentum to the planet, while the interaction with the exterior disk removes angular momentum. Whether the planet migrates inward or outward depends upon the balance of the two effects. Theoretical calculations suggest that the planet migrates inward in almost all circumstances, potentially on a short time scale (Tanaka, Takeuchi and Ward estimate a migration time scale for an Earth mass planet from 5 AU as only about 1 million years). In a highly turbulent disk, type-I migration may be closer to a random walk than a smooth inward migration. Type I migration may be a relatively minor effect for terrestrial planets due to their low masses and because their final assembly probably occurs after the gas disk has dispersed, but type-I migration is likely to affect the formation of giant planets in the core accretion model.

Massive planets strongly perturb the gas disk. The exchange of angular momentum between the planet and the disk tends to repel gas from the vicinity of the planet's orbit, creating an annular gap in which the surface density of gas is low. The direction and rate of orbital migration then depends upon how quickly the gas disk, evolving under the action of its own internal angular momentum transport processes, tries to flow back toward the gap. In this regime, described as Type II migration, the motion of the planet is locked to the viscous evolution of the disk. In regions of the disk where the gas is flowing inward, the planet also moves inward, and vice versa. Type II migration is typically slower than Type I migration. The boundary between Type I and Type II migration is not sharp. In between these regimes, non-linear effects become important, especially for gas moving on horseshoe orbits in the corotation resonance close to the planet's orbit. These effects are poorly quantified at present, but numerical models suggest planetary migration may slow down or even change direction for intermediate-mass planets. Planets that have formed a gap continue to accrete some gas via narrow streams of material that cross the gap. However, the rate of gas accretion declines as the planet grows more massive and the gap becomes deeper.

Although there is no direct observational evidence for gas disk migration, it is widely believed that this mechanism explains the existence of hot Jupiters - giant planets on very short-period orbits such as the planet orbiting 51 Pegasi. It has been suggested that gas disk migration may also excite planetary eccentricity (thereby providing a simultaneous explanation for the wide spread of eccentricity observed among extrasolar planets), but this question remains open.

### Planetesimal-driven migration

Figure 3: The distribution of known trans-Neptunian objects in semi-major axis a and eccentricity e. Note the concentration of bodies in 3:2 resonance with Neptune - the Plutinos.

Related physics allows planets to migrate due to interaction with smaller bodies in their vicinity. A planet that ejects a planetesimal from the planetary system must give up energy, and thereby moves closer toward the star (this occurs, to a negligible degree, when spacecraft make use of gravitational slingshots from the giant planets). Conversely a planet that scatters planetesimals into shorter period orbits gains energy, and migrates outward. To order of magnitude, a planet will suffer a substantial change to its orbit if it interacts with a mass of planetesimals that is comparable to its own mass. Since the ratio of solids to gas in typical protoplanetary disks is of the order of $$10^{-2}$$ this condition is easier to meet for ice giants, which have accreted relatively modest gaseous envelopes, than for very massive planets with near stellar composition.

The distribution of trans-Neptunian objects provides strong evidence for planetesimal migration having occurred early in Solar System history. In addition to Pluto itself, a large number of other bodies (called Plutinos) are observed to be trapped in 3:2 resonance with Neptune. Some of these bodies have eccentricities high enough that they cross Neptune's orbit. This unusual distribution is likely the result of the outward migration of Neptune, driven by the scattering of a disk of planetesimals inward into orbits that eventually led to encounters with Jupiter and ejection from the Solar System. Simultaneously, the slow outward motion of Neptune captured Pluto and other bodies into the 3:2 resonance (a process known as resonant capture) and excited their eccentricity.

Although the evidence is less direct, it is also possible that all of the giant planets in the Solar System originated in a more compact configuration, which then evolved under the action of planetesimal scattering to its current state. The Nice Model postulates that this evolution included a crossing of the 2:1 resonance between Jupiter and Saturn, and links this crossing to the Late Heavy Bombardment (a transient spike in the cratering rate) on the Moon. The full consequences of such large-scale rearrangements of the giant planets remain to be explored.

### Planet-planet scattering

Interactions between planets can also occur after both the gas and planetesimal disks have been lost (or depleted to a dynamically negligible level). No general stability criteria is known for a planetary system with $$N_{planets} > 2\ ,$$ so numerical N-body experiments are needed to study the evolution of such systems. An initially unstable planetary system can evolve via:

• Ejection of one or more planets (typically the lightest)
• An increase in the orbital separation of the planets, toward a more stable configuration
• Physical collisions between planets, or between a planet and the star

The relative probability of these channels depends upon the orbital radii and masses of the planets, and so no blanket statement about the outcome of planet-planet scattering is possible. However, typically the survivors after scattering has ceased have migrated modestly inward, and gained significant eccentricity. Numerical calculations have shown that planet-planet scattering can reproduce the observed eccentricity distribution of massive extrasolar planets, and as a result this mechanism is the leading candidate for explaining why extrasolar planets frequently have non-circular orbits.

## References

• Planet Formation, J.J. Lissauer, Annual Review of Astronomy and Astrophysics, 31, 129 (1993)

Internal references