Planetary Migration
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Recent papers in Planetary Migration
The present contribution is devoted to some aspects of history and evolution of the early Solar System. The origin of the Sun, Earth, other planets and its satellites has long been a matter of great concern for people. Over the past few... more
The present contribution is devoted to some aspects of history and evolution of the early Solar System. The origin of the Sun, Earth, other planets and its satellites has long been a matter of great concern for people. Over the past few decades astronomers and cosmologists have considerably advanced in the perception of the structure, history, and evolution of the Solar System. However, one can hardly speak about a proper narrative here; we more often work with hypotheses. The present paper is structured as follows. First, it outlines the history of formation of the Solar System in the first billion years of its existence, when the most considerable changes took place. Then while describing certain formative processes we show the opportunities to define them in terms of evolutionary laws and rules. Of course, this paper presents only a few such laws and rules. We suppose that the present study will be of interest to a reader in two ways. First, there are quite a few consistent and brief surveys of the Solar System history accounting the latest achievements in astrophysics and cosmology. Meanwhile, they are very important and productive for theorizing part of Big History. Second, the discussion employing the general evolutionary laws and rules allows defining some common features in the formation of the Solar System and especially of its planetary system which are characteristic for every level and stage of Big History. This brings us to the idea of the integrity of Big History not only in historical and systemic terms but also with respect to its integrity in detecting general laws, patterns and mechanisms.
Planetary migration is the process by which a forming planet undergoes a drift of its semi-major axis caused by the tidal interaction with its parent protoplan-etary disc. One of the key quantities to assess the migration of embedded... more
Planetary migration is the process by which a forming planet undergoes a drift of its semi-major axis caused by the tidal interaction with its parent protoplan-etary disc. One of the key quantities to assess the migration of embedded planets is the tidal torque between the disc and planet, which has two components: the Lind-blad torque and the corotation torque. We review the latest results on both torque components for planets on circular orbits, with a special emphasis on the various processes that give rise to additional, large components of the corotation torque, and those contributing to the saturation of this torque. These additional components of the corotation torque could help address the shortcomings that have recently been exposed by models of planet population syntheses. We also review recent results concerning the migration of giant planets that carve gaps in the disc (type II migration) and the migration of sub-giant planets that open partial gaps in massive discs (type III migration).
- by Clément Baruteau and +1
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- Planet Formation, Planetary Migration
Context. The migration of low-mass planets, or type I migration, is driven by the differential Lindblad torque and the corotation torque in non-magnetic viscous models of protoplanetary discs. The corotation torque has recently received... more
Context. The migration of low-mass planets, or type I migration, is driven by the differential Lindblad torque and the corotation torque in non-magnetic viscous models of protoplanetary discs. The corotation torque has recently received detailed attention, because of its ability to slow down, stall, or reverse type I migration. In laminar viscous disc models, the long-term evolution of the corotation torque is intimately related to viscous and thermal diffusion processes in the planet's horseshoe region. It is unclear how the corotation torque behaves in turbulent discs, and whether its amplitude is correctly predicted by viscous disc models. Aims. This paper is aimed at examining the properties of the corotation torque in discs where magnetohydrodynamic (MHD) turbulence develops as a result of the magnetorotational instability (MRI), considering a weak initial toroidal magnetic field. Methods. We present results of 3D MHD simulations carried out with two different codes. Non-ideal MHD effects and the disc's vertical stratification are neglected, and locally isothermal disc models are considered. The running time-averaged tidal torque exerted by the disc on a fixed planet is evaluated in three different disc models. Results. We first present simulation results with an inner disc cavity (planet trap). As in viscous disc models, the planet is found to experience a positive running time-averaged torque over several hundred orbits, which highlights the existence of an unsaturated coro-tation torque maintained in the long term in MHD turbulent discs. Two disc models with initial power-law density and temperature profiles are also adopted, in which the time-averaged tidal torque is found to be in decent agreement with its counterpart in laminar viscous disc models with similar viscosity alpha parameter at the planet location. Detailed analysis of the averaged torque density distributions indicates that the differential Lindblad torque takes very similar values in MHD turbulent and laminar viscous discs, and there exists an unsaturated corotation torque in MHD turbulent discs. This analysis also reveals the existence of an additional corotation torque in weakly magnetized discs. Conclusions. Our results of 3D MHD simulations demonstrate the existence of horseshoe dynamics and an unsaturated corotation torque in weakly magnetized discs with fully developed MHD turbulence.
Context. The lower limit to the distribution of orbital periods P for the current population of close-in exoplanets shows a distinctive discontinuity located at approximately one Jovian mass. Most smaller planets have orbital periods... more
Context. The lower limit to the distribution of orbital periods P for the current population of close-in exoplanets shows a distinctive discontinuity located at approximately one Jovian mass. Most smaller planets have orbital periods longer than P ∼ 2.5 days, while higher masses are found down to P ∼ 1 day. Aims. We analyze whether this observed mass-period distribution could be explained in terms of the combined effects of stellar tides and the interactions of planets with an inner cavity in the gaseous disk. Methods. We performed a series of hydrodynamical simulations of the evolution of single-planet systems in a gaseous disk with an inner cavity mimicking the inner boundary of the disk. The subsequent tidal evolution is analyzed assuming that orbital eccentricities are small and stellar tides are dominant. Results. We find that most of the close-in exoplanet population is consistent with an inner edge of the protoplanetary disk being located at approximately P > ∼ 2 days for solar-type stars, in addition to orbital decay having been caused by stellar tides with a specific tidal parameter on the order of Q * 10 7. The data is broadly consistent with planets more massive than one Jupiter mass undergoing type II migration, crossing the gap, and finally halting at the interior 2/1 mean-motion resonance with the disk edge. Smaller planets do not open a gap in the disk and remain trapped in the cavity edge. CoRoT-7b appears detached from the remaining exoplanet population, apparently requiring additional evolutionary effects to explain its current mass and semimajor axis.
We investigate the gravitational interaction of a Jovian-mass protoplanet with a gaseous disc with aspect ratio and kinematic viscosity expected for the protoplanetary disc from which it formed. Different disc surface density... more
We investigate the gravitational interaction of a Jovian-mass protoplanet with a gaseous disc with aspect ratio and kinematic viscosity expected for the protoplanetary disc from which it formed. Different disc surface density distributions are investigated. We focus on the tidal interaction with the disc with the consequent gap formation and orbital migration of the protoplanet. Non-linear two-dimensional hydrodynamic simulations are employed using three independent numerical codes. A principal result is that the direction of the orbital migration is always inwards and such that the protoplanet reaches the central star in a near-circular orbit after a characteristic viscous timescale of 10 4 initial orbital periods. This is found to be independent of whether the protoplanet is allowed to accrete mass or not. Inward migration is helped by the disappearance of the inner disc, and therefore the positive torque it would exert, because of accretion on to the central star. Maximally accreting protoplanets reach about 4 Jovian masses on reaching the neighbourhood of the central star. Our results indicate that a realistic upper limit for the masses of closely orbiting giant planets is 5 Jupiter masses, if they originate in protoplanetary discs similar to the minimum-mass solar nebula. This is because of the reduced accretion rates obtained for planets of increasing mass. Assuming that some process such as termination of the inner disc through a magnetospheric cavity stops the migration, the range of masses estimated for a number of close orbiting giant planets as well as their inward orbital migration can be accounted for by consideration of disc–protoplanet interactions during the late stages of giant planet formation.
Context. Embedded planets disturb the density structure of the ambient disk, and gravitational back-reaction possibly will induce a change in the planet's orbital elements. Low-mass planets only have a weak impact on the disk, so their... more
Context. Embedded planets disturb the density structure of the ambient disk, and gravitational back-reaction possibly will induce a change in the planet's orbital elements. Low-mass planets only have a weak impact on the disk, so their wake's torque can be treated in linear theory. Larger planets will begin to open up a gap in the disk through nonlinear interaction. Accurate determination of the forces acting on the planet requires careful numerical analysis. Recently, the validity of the often used fast orbital advection algorithm (FARGO) has been put into question, and special numerical resolution and stability requirements have been suggested. Aims. We study the process of planet-disk interaction for low-mass planets of a few Earth masses, and reanalyze the numerical requirements to obtain converged and stable results. One focus lies on the applicability of the FARGO-algorithm. Additionally, we study the difference of two and three-dimensional simulations, compare global with local setups, as well as isothermal and adiabatic conditions. Methods. We study the influence of the planet on the disk through two-and three-dimensional hydrodynamical simulations. To strengthen our conclusions we perform a detailed numerical comparison where several upwind and Riemann-solver based codes are used with and without the FARGO-algorithm. Results. With respect to the wake structure and the torque density acting on the planet, we demonstrate that the FARGO-algorithm yields correct a correct and stable evolution for the planet-disk problem, and that at a fraction of the regular cpu-time. We find that the resolution requirements for achieving convergent results in unshocked regions are rather modest and depend on the pressure scale height H of the disk. By comparing the torque densities of two-and three-dimensional simulations we show that a suitable vertical averaging procedure for the force gives an excellent agreement between the two. We show that isothermal and adiabatic runs can differ considerably, even for adiabatic indices very close to unity.
The torque felt by a non-accreting protoplanet on a circular orbit embedded in a uniform surface density protoplanetary disk is analyzed by means of time-dependent numerical simulations. Varying the viscosity enables one to disentangle... more
The torque felt by a non-accreting protoplanet on a circular orbit embedded in a uniform surface density protoplanetary disk is analyzed by means of time-dependent numerical simulations. Varying the viscosity enables one to disentangle the Lindblad torque (which is independent of viscosity) from the corotation torque, which saturates at low viscosity and is unsaturated at high viscosity. The dependence of the corotation torque upon the viscosity and upon the width of the librating zone is compared with previous analytical expressions, and shown to be in agreement with those. The effect of the potential smoothing respectively on the Lindblad torque and on the corotation torque is investigated, and the question of whether 3D effects and their impact on the total torque sign and magnitude can be modeled by an adequate smoothing prescription in a 2D simulation is addressed. As a side result, this study shows that the total torque acting on a Neptune-sized protoplanet is positive in a sufficiently thin, viscous disk (H/r > ∼ 4%, α > ∼ 10 −2), but the inward migration time of smaller bodies is still very short, making it unlikely that they reach the torque reversal mass before having migrated all the way to the central object.
We present a mechanism related to the migration of giant protoplanets embedded in a protoplanetary disc whereby a giant protoplanet is caught up, before having migrated all the way to the central star, by a lighter outer giant... more
We present a mechanism related to the migration of giant protoplanets embedded in a protoplanetary disc whereby a giant protoplanet is caught up, before having migrated all the way to the central star, by a lighter outer giant protoplanet. This outer protoplanet may get captured into the 2:3 resonance with the more massive one, in which case the gaps that the two planets open in the disc overlap. Two effects arise, namely a squared mass-weighted torque imbalance and an increased mass flow through the overlapping gaps from the outer disc to the inner disc, which both play in favour of an outwards migration. Indeed, under the conditions presented here, which describe the evolution of a pair of protoplanets respectively Jupiter-and Saturn-sized, the migration is reversed, while the semimajor axis ratio of the planets is constant and the eccentricities are confined to small values by the disc material. The long-term behaviour of the system is briefly discussed, and could account for the high eccentricities observed for the extrasolar planets with semimajor axis > 0.2 au
We study the horseshoe dynamics of a low-mass planet in a three-dimensional, globally isothermal, inviscid disk. We find, as reported in previous work, that the boundaries of the horseshoe region (separatrix sheets) have cylindrical... more
We study the horseshoe dynamics of a low-mass planet in a three-dimensional, globally isothermal, inviscid disk. We find, as reported in previous work, that the boundaries of the horseshoe region (separatrix sheets) have cylindrical symmetry about the disk's rotation axis. We interpret this feature as arising from the fact that the whole separatrix sheets have a unique value of Bernoulli's constant, and that this constant does not depend on altitude, but only on the cylindrical radius, in barotropic disks. We next derive an expression for the torque exerted by the horseshoe region on the planet, or horseshoe drag. Potential vorticity is not materially conserved as in two-dimensional flows, but it obeys a slightly more general conservation law (Ertel's theorem) that allows an expression for the horseshoe drag identical to the expression in a two-dimensional disk to be obtained. Our results are illustrated and validated by three-dimensional numerical simulations. The horseshoe region is found to be slightly narrower than previously extrapolated from two-dimensional analyses with a suitable softening length of the potential. We discuss the implications of our results for the saturation of the corotation torque, and the possible connection to the flow at the Bondi scale, which the present analysis does not resolve.
We give an expression for the Lindblad torque acting on a low-mass planet embedded in a protoplanetary disk that is valid even at locations where the surface density or temperature profile cannot be approximated by a power law, such as an... more
We give an expression for the Lindblad torque acting on a low-mass planet embedded in a protoplanetary disk that is valid even at locations where the surface density or temperature profile cannot be approximated by a power law, such as an opacity transition. At such locations, the Lindblad torque is known to suffer strong deviation from its standard value, with potentially important implications for type I migration, but the full treatment of the tidal interaction is cumbersome and not well suited to models of planetary population synthesis. The expression that we propose retains the simplicity of the standard Lindblad torque formula and gives results that accurately reproduce those of numerical simulations, even at locations where the disk temperature undergoes abrupt changes. Our study is conducted by means of customized numerical simulations in the low-mass regime, in locally isothermal disks, and compared to linear torque estimates obtained by summing fully analytic torque estimates at each Lindblad resonance. The functional dependence of our modified Lindblad torque expression is suggested by an estimate of the shift of the Lindblad resonances that mostly contribute to the torque, in a disk with sharp gradients of temperature or surface density, while the numerical coefficients of the new terms are adjusted to seek agreement with numerics. As side results, we find that the vortensity related corotation torque undergoes a boost at an opacity transition that can counteract migration, and we find evidence from numerical simulations that the linear corotation torque has a non-negligible dependency upon the temperature gradient, in a locally isothermal disk.
We evaluate the horseshoe drag exerted on a low-mass planet embedded in a gaseous disk, assuming the disk's flow in the co-orbital region to be adiabatic. We restrict this analysis to the case of a planet on a circular orbit, and we... more
We evaluate the horseshoe drag exerted on a low-mass planet embedded in a gaseous disk, assuming the disk's flow in the co-orbital region to be adiabatic. We restrict this analysis to the case of a planet on a circular orbit, and we assume a steady flow in the corotating frame. We also assume that the corotational flow upstream of the U-turns is unperturbed, so that we discard saturation effects. In addition to the classical expression for the horseshoe drag in barotropic disks, which features the vortensity gradient across corotation, we find an additional term which scales with the entropy gradient, and whose amplitude depends on the perturbed pressure at the stagnation point of the horseshoe separatrices. This additional torque is exerted by evanescent waves launched at the horseshoe separatrices, as a consequence of an asymmetry of the horseshoe region. It has a steep dependence on the potential's softening length, suggesting that the effect can be extremely strong in the three-dimensional case. We describe the main properties of the co-orbital region (the production of vortensity during the U-turns, the appearance of vorticity sheets at the downstream separatrices, and the pressure response), and we give torque expressions suitable to this regime of migration. Side results include a weak, negative feedback on migration, due to the dependence of the location of the stagnation point on the migration rate, and a mild enhancement of the vortensity-related torque at a large entropy gradient.
The increase of computational resources has recently allowed high-resolution, three-dimensional calculations of planets embedded in gaseous protoplanetary disks. They provide estimates of the planet migration timescale that can be... more
The increase of computational resources has recently allowed high-resolution, three-dimensional calculations of planets embedded in gaseous protoplanetary disks. They provide estimates of the planet migration timescale that can be compared to analytical predictions. While these predictions can result in extremely short migration timescales for cores of a few Earth masses, recent numerical calculations have given an unexpected outcome: the torque acting on planets with masses between 5 and 20 M È is considerably smaller than the analytic, linear estimate. These findings motivated the present work, which investigates existence and origin of this discrepancy or ''offset,'' as we shall call it, by means of two-and three-dimensional numerical calculations. We show that the offset is indeed physical and arises from the co-orbital corotation torque, since (1) it scales with the disk vortensity gradient, (2) its asymptotic value depends on the disk viscosity, (3) it is associated to an excess of the horseshoe zone width. We show that the offset corresponds to the onset of nonlinearities of the flow around the planet, which alter the streamline topology as the planet mass increases: at low mass the flow nonlinearities are confined to the planet's Bondi sphere, whereas at larger mass the streamlines display a classical picture reminiscent of the restricted three-body problem, with a prograde circumplanetary disk inside a ''Roche lobe.'' This behavior is of particular importance for the subcritical solid cores (M P 15 M È) in thin (H /r P 0:06) protoplanetary disks. Their migration could be significantly slowed down, or reversed, in disks with shallow surface density profiles.
Planetary systems are born in the disks of gas, dust and rocky fragments that surround newly formed stars. Solid content assembles into ever-larger rocky fragments that eventually become planetary embryos. These then continue their growth... more
Planetary systems are born in the disks of gas, dust and rocky fragments that surround newly formed stars. Solid content assembles into ever-larger rocky fragments that eventually become planetary embryos. These then continue their growth by accreting leftover material in the disk. Concurrently, tidal effects in the disk cause a radial drift in the embryo orbits, a process known as migration. Fast inward migration is predicted by theory for embryos smaller than three to five Earth masses. With only inward migration, these embryos can only rarely become giant planets located at Earth’s distance from the Sun and beyond, in contrast with observations. Here we report that asymmetries in the temperature rise associated with accreting infalling material produce a force (which gives rise to an effect that we call ‘heating torque’) that counteracts inward migration. This provides a channel for the formation of giant planets and also explains the strong planet–metallicity correlation found between the incidence of giant planets and the heavy-element abundance of the host stars.
The great diversity of extrasolar planetary systems has challenged our understanding of how planets form, and how their orbits evolve as they form. Among the various processes that may account for this diversity, the gravitational... more
The great diversity of extrasolar planetary systems has challenged our understanding of how planets form, and how their orbits evolve as they form. Among the various processes that may account for this diversity, the gravitational interaction between planets and their parent protoplanetary disc plays a prominent role in shaping young planetary systems. Planet-disc forces are large, and the characteristic times for the evolution of planets orbital elements are much shorter than the lifetime of protoplanetary discs. The determination of such forces is challenging, because it involves many physical mechanisms and it requires a detailed knowledge of the disc structure. Yet, the intense research of the past few years, with the exploration of many new avenues, represents a very significant improvement on the state of the discipline. This chapter reviews current understanding of planet-disc interactions, and highlights their role in setting the properties and architecture of observed planetary systems.
- by Frederic S Masset and +2
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- Planet Formation, Planetary Migration