Planetary formation and migration
| Philip Armitage (2008), Scholarpedia, 3(3):4479. | doi:10.4249/scholarpedia.4479 | revision #186389 [link to/cite this article] |
Dr. Philip Armitage accepted the invitation on 25 July 2007 (self-imposed deadline: 25 October 2007).
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.
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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 small particles ("dust") of rocky or icy material. These particles either fall in with the gas when the protoplanetary disk forms from the interstellar medium, or condense from the gas phase within the disk. The dynamics of dust embedded within the gas is dominated by gravity from the star and aerodynamic forces. These forces remain dominant until bodies grow to 10-100 km in size (planetesimals), after which the declining surface to volume ratio renders aerodynamic forces negligible.
The mutual gravitational interaction between small bodies is very weak (the escape velocity from a 1 meter diameter rock is less than 0.1 cm/s), so small particles grow by physically colliding and sticking together. The smallest micron-sized particles are very closely coupled to the gas, but larger particles of mm size or above develop significant velocities relative to the gas as a consequence of the gas feeling pressure gradient forces that are not experienced by solid particles. This velocity differential leads to particles settling toward the midplane of the gas disk, and migrating radially toward the star. The radial migration time scale is particularly rapid for cm to meter sized bodies - at 1 AU 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 be lost to the star. The required rapid growth may occur as a result of ongoing pairwise collisions, though a competing theoretical idea - gravitational collapse of a dense disk of solids (the Goldreich-Ward mechanism) remains under consideration.
Terrestrial Planet Formation
Once planetesimals have formed, the formation of terrestrial planets is largely an N-body problem in which gravity (both that of the star, and mutual gravitational forces between the bodies) is the dominant force. Theoretical work customarily employs a mixture of statistical techniques, based on the coagulation equation, to study the early phases of terrestrial planet formation, together with direct N-body simulations to model the later stages when the number of large bodies is small. There may be an initial phase of runaway growth, in which the largest bodies grow more rapidly than smaller ones, followed by a phase of oligarchic growth, in which a relatively small number of bodies grow in unison to dominate the population. This results in a population of \(10^2 - 10^3\) planetary embryos, whose subsequent collisions lead to the final assembly of the terrestrial planets, probably on a time scale of about 100 million years. The Moon is thought to have formed from debris ejected when the Earth collided with another large body in this last phase.
Simulations of terrestrial planet formation are able to reproduce the basic architecture of the inner Solar System from plausible initial conditions, although the predicted eccentricities of the planets are often somewhat larger than observed in the Solar System. This discrepancy may imply significant damping by remnant planetesimals or residual gas. The number and mass of terrestrial planets is expected to vary with the mass of available solid material and the presence or absence of giant planets in the same planetary systems - these predictions may be tested once extrasolar terrestrial planets have been discovered.
Giant Planet Formation
Giant planets are qualitatively distinct from terrestrial planets in that they possess significant gaseous envelopes. In the Solar System, the mass of the gas giants (Jupiter and Saturn) is predominantly gaseous, though even these planets are more heavily enriched in heavy elements than the Sun. The ice giants (Uranus and Neptune) have lesser, but still substantial (several Earth masses) gas envelopes. These 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 from the collision of rocky and icy planetesimals. It is difficult to form such massive bodies rapidly in the inner protoplanetary disk, but feasible at orbital radii beyond the snow line where the temperature is low enough that icy as well as rocky materials can condense into solid form. Initially the core is surrounded by a low mass atmosphere, which grows steadily more massive as the gas cools and contracts onto the core. Eventually a "critical core mass", beyond which no hydrostatic envelope solution exists, is exceeded, and a phase of rapid gas accretion ensues. It is in this final phase that the massive envelopes of fully fledged gas giants such as Jupiter are assembled.
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 in the case of 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 two main 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). Second, the observed correlation between the frequency of extrasolar planets and the metallicity of their host stars is readily explicable as a consequence of core accretion: if the disk is enriched with more solids, the required core can form more readily. It is unclear whether the same correlation is a consequence of 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. This suggests that a full understanding of giant planet formation has yet to be attained.
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
A planet orbiting within the protoplanetary disk perturbs the gravitational potential felt by the gas. At orbital radii where the gas is in resonance with the planet these perturbations launch waves within the disk, which can either add or remove energy and angular momentum from the planet's orbit. This results in a change to the planet's semi-major axis (planetary migration), and may lead to a change in the eccentricity of the orbit.
Two main regimes of gas disk migration have been identified. The first - called Type I migration - applies to low mass planets. For low mass planets, the interaction between the planet and the gas disk is weak enough that the surface density profile of the gas disk is almost unaltered by the presence of the planet. 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). Type I migration may not matter for the formation of terrestrial planets, whose final assembly probably occurred after the gas disk has dispersed, but 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 sense 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. At radii where the gas is flowing inward, so does the planet, and vice versa. The boundary between Type I and Type II migration is not sharp, and the mass at which it falls depends upon poorly known properties of the protoplanetary disk. Estimates place the boundary around the mass of Saturn. Type II migration is typically slower than Type I migration, and except for very massive planets is accompanied by ongoing mass accretion via streams of gas that penetrate the gap.
Although there is no direct observational evidence for gas disk migration, this mechanism is the standard explanation for how 51 Peg and the other members of the hot Jupiter class of extrasolar planets attained their current short-period orbits. 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
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 easy to meet for ice giants, which have accreted relatively modest gaseous envelopes, but harder 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 crattering 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)
Planet Formation and Migration, J.C.B. Papaloizou and C. Terquem, Reports on Progress in Physics, 69, 119 (2006)
Lecture Notes on the Formation and Early Evolution of Planetary Systems, P.J. Armitage, arXiv:astro-ph/0701485v1 (2007)
Growth of Dust as the Initial Step Toward Planet Formation, C. Dominik, J. Blum, J. Cuzzi, and G. Wurm, in Protostars and Planets V (editors B. Reipurth, D. Jewitt, and K. Keil), University of Arizona Press, Tuscon, p.783 (2007)
Gravitational Instabilities in Gaseous Protoplanetary Disks and Implications for Giant Planet Formation, R. Durisen, A. Boss, L. Mayer, A. Nelson, T. Quinn and W.K.M. Rice, in Protostars and Planets V (editors B. Reipurth, D. Jewitt, and K. Keil), University of Arizona Press, Tuscon, p.607 (2007)
