Caroline Terquem

Migration and the Formation of Systems of Hot Super-Earths and Neptunes

The existence of extrasolar planets with short orbital periods suggests that planetary migration induced by tidal interaction with the protoplanetary disk is important. Cores and terrestrial planets may undergo migration as they form. In this paper we investigate the evolution of a population of cores with initial masses in the range 0.1-1 M⊕ embedded in a disk. Mutual interactions lead to orbit crossing and mergers, so that the cores grow during their evolution. Interaction with the disk leads to orbital migration, which results in the cores capturing each other in mean motion resonances. As the cores migrate inside the disk inner edge, scatterings and mergers of planets on unstable orbits, together with orbital circularization, causes strict commensurability to be lost. Near commensurability however is usually maintained. All the simulations end with a population typically between two and five planets, with masses depending on the initial mass. These results indicate that if hot super-Earths or Neptunes form by mergers of inwardly migrating cores, then such planets are most likely not isolated. We would expect to always find at least one, more likely a few, companions on close and often near-commensurable orbits. To test this hypothesis, it would be of interest to look for planets of a few to about 10 M⊕ in systems where hot super-Earths or Neptunes have already been found.

Critical Protoplanetary Core Masses in Protoplanetary Disks and the Formation of Short-Period Giant Planets

We study a solid protoplanetary core undergoing radial migration in a protoplanetary disk. We con- sider cores in the mass range D1¨10 embedded in a gaseous protoplanetary disk at diUerent radial M ^ locations. We suppose that the core luminosity is generated as a result of planetesimal accretion and calculate the structure of the gaseous envelope assuming hydrostatic and thermal equilibrium. This is a good approximation during the early growth of the core, while its mass is less than the critical value, above which such static solutions can no longer be obtained and rapid gas accretion begins. The M crit , critical value corresponds to the crossover mass above which rapid gas accretion begins in time- dependent calculations. We model the structure and evolution of the protoplanetary nebula as an accre- tion disk with constant a. We present analytic —ts for the steady state relation between the disk surface density and the mass accretion rate as a function of radius. We calculate as a function of radial M crit location, gas accretion rate through the disk, and planetesimal accretion rate onto the core. For a —xed planetesimal accretion rate, is found to increase inward. On the other hand, it decreases with the M crit planetesimal accretion rate and hence with the core luminosity. We consider the planetesimal accretion rate onto cores migrating inward in a characteristic time of D103¨105 yr at 1 AU, as indicated by recent theoretical calculations. We —nd that the accretion rate is expected to be sufficient to prevent the attain- ment of during the migration process if the core starts oU signi—cantly below it. Only at those small M crit radii at which local conditions are such that dust, and accordingly planetesimals, no longer exist can be attained. At small radii, the runaway gas accretion phase may become longer than the disk M crit lifetime if the mass of the core is too small. However, within the context of our disk models, and if it is supposed that some process halts the migration, massive cores can be built up through the merger of additional incoming cores on a timescale shorter than for in situ formation. A rapid gas accretion phase may thus begin without an earlier prolonged phase in which planetesimal accretion occurs at a reduced rate because of feeding zone depletion in the neighborhood of a —xed orbit. Accordingly, we suggest that giant planets may begin to form through the above processes early in the life of the protostellar disk at small radii, on a timescale that may be signi—cantly shorter than that derived for in situ formation. Subject headings: accretion, accretion disksplanetary systemssolar system: formation

Planet formation and migration

We review the observations of extrasolar planets, ongoing developments in theories of planet formation, orbital migration and the evolution of multiplanet systems.

Dynamical relaxation and massive extrasolar planets

Following the suggestion of Black that some massive extrasolar planets may be associated with the tail of the distribution of stellar companions, we investigate a scenario in which 5 < N < 100 planetary mass objects are assumed to form rapidly through a fragmentation process occuring in a disc or protostellar envelope on a scale of 100 au. These are assumed to have formed rapidly enough through gravitational instability or fragmentation that their orbits can undergo dynamical relaxation on a time-scale of ,100 orbits. Under a wide range of initial conditions and assumptions, the relaxation process ends with either (i) one potential ‘hot Jupiter’ plus up to two ‘external’ companions, i.e. planets orbiting near the outer edge of the initial distribution; (ii) one or two ‘external’ planets or even none at all; (iii) one planet on an orbit with a semi-major axis of 10 to 100 times smaller than the outer boundary radius of the inital distribution together with an ‘external’ companion. Most of the other objects are ejected and could contribute to a population of free-floating planets. Apart from the potential ‘hot Jupiters’, all the bound objects are on orbits with high eccentricity, and also with a range of inclination with respect to the stellar equatorial plane. We found that, apart from the close orbiters, the probability of ending up with a planet orbiting at a given distance from the central star increases with the distance. This is because of the tendency of the relaxation process to lead to collisions with the central star. The scenario we envision here does not impose any upper limit on the mass of the planets. We discuss the application of these results to some of the more massive extrasolar planets.

Stopping inward planetary migration by a toroidal magnetic field

We calculate the linear torque exerted by a planet on a circular orbit on a disc containing a toroidal magnetic field. All fluid perturbations are singular at the so--called magnetic resonances, where the Doppler shifted frequency of the perturbation matches that of a slow MHD wave propagating along the field line. These lie on both sides of the corotation radius. Waves propagate outside the Lindblad resonances, and also in a restricted region around the magnetic resonances. The magnetic resonances contribute to a significant global torque which, like the Lindblad torque, is negative (positive) inside (outside) the planet\s orbit. Since these resonances are closer to the planet than the Lindblad resonances, the torque they contribute dominates over the Lindblad torque if the magnetic field is large enough. In addition, if beta=c^2/v_A^2 increases fast enough with radius, the outer magnetic resonance becomes less important and the total torque is then negative, dominated by the inner magnetic resonance. This leads to outward migration of the planet. Even for beta=100 at corotation, a negative torque may be obtained. A planet migrating inward through a nonmagnetized region of a disc would then stall when reaching a magnetized region. It would then be able to grow to become a terrestrial planet or the core of a giant planet. In a turbulent magnetized disc in which the large scale field structure changes sufficiently slowly, a planet may alternate between inward and outward migration, depending on the gradients of the field encountered. Its migration could then become diffusive, or be limited only to small scales.

Precession of Collimated Outflows from Young Stellar Objects

We consider several protostellar systems in which either a precessing jet or at least two misaligned jets have been observed. We assume that the precession of jets is caused by tidal interactions in noncoplanar binary systems. For Cep E, V1331 Cyg, and RNO 15-FIR, the inferred orbital separations and disk radii are in the range 4-160 AU and 1-80 AU, respectively, consistent with those expected for pre-main-sequence stars. Furthermore, we assume or use the fact that the source of misaligned outflows is a binary and evaluate the length scale over which the jets should precess as a result of tidal interactions. For T Tau, HH1 VLA 1/2, and HH 24 SVS63, it may be possible to detect a bending of the jets rather than wiggling. In HH 111 IRS and L1551 IRS5, wiggling may be detected on the current observed scale. Our results are consistent with the existence of noncoplanar binary systems in which tidal interactions induce jets to precess.

New composite models of partially ionized protoplanetary disks

We study an accretion disk in which three different regions can coexist: MHD turbulent regions, dead zones, and gravitationally unstable regions. Although the dead zones are stable, there is some transport due to the Reynolds stress associated with waves emitted from the turbulent layers. We model the transport in each of the different regions by its own α parameter, which is 10-103 times smaller in dead zones than in active layers. In gravitationally unstable regions, α is determined by the fact that the disk self-adjusts to a state of marginal stability. We construct steady-state models of such disks. We find that for uniform mass flow, the disk has to be more massive, hotter, and thicker at the radii where there is a dead zone. In disks in which the dead zone is very massive, gravitational instabilities are present. Whether such models are realistic or not depends on whether hydrodynamical fluctuations driven by the turbulent layers can penetrate all the way inside the dead zone. If the disk is not in a steady state at some stage of its evolution, then the surface density will evolve toward the steady-state solution. However, if the value of α in the dead zone is much smaller than that in the active zone, the timescale for the parts of the disk that are beyond a few AU to reach a steady state can become longer than the disk lifetime. Steady-state disks with dead zones are a more favorable environment for planet formation than are standard disks, since the dead zone is typically 10 times more massive than a corresponding turbulent zone at the same location.

On the dynamics of multiple systems of hot super-Earths and Neptunes: tidal circularization, resonance and the HD 40307 system

In this paper, we consider the dynamics of a system of hot super-Earths or Neptunes such as HD 40307. We show that, as tidal interaction with the central star leads to small eccentricities, the planets in this system could be undergoing resonant coupling even though the period ratios depart significantly from very precise commensurability. In a three-planet system, this is indicated by the fact that resonant angles librate or are associated with long-term changes to the orbital elements. In HD 40307, we expect that three resonant angles could be involved in this way. We propose that the planets in this system were in a strict Laplace resonance while they migrated through the disc. After entering the disc inner cavity, tidal interaction would cause the period ratios to increase from two but with the inner pair deviating less than the outer pair, counter to what occurs in HD 40307. However, the relationship between these pairs that occur in HD 40307 might be produced if the resonance is impulsively modified by an event like a close encounter shortly after the planetary system decouples from the disc. We find this to be in principle possible for a small relative perturbation on the order of a few ×10−3, but then we find that the evolution to the present system in a reasonable time is possible only if the masses are significantly larger than the minimum masses and the tidal dissipation is very effective. On the other hand, we found that a system like HD 40307 with minimum masses and more realistic tidal dissipation could be produced if the eccentricity of the outermost planet was impulsively increased to ∼0.15. We remark that the form of resonantly coupled tidal evolution we consider here is quite general and could be of greater significance for systems with inner planets on significantly shorter orbital periods characteristic of, for example, CoRoT 7 b.

Evolution of Self-Gravitating Magnetized Disks. II. Interaction between Magnetohydrodynamic Turbulence and Gravitational Instabilities

We present three-dimensional magnetohydrodynamic (MHD) numerical simulations of the evolution of self-gravitating and weakly magnetized disks with an adiabatic equation of state. Such disks are subject to the development of both the magnetorotational and gravitational instabilities, which transport angular momentum outward. As in previous studies, our hydrodynamic simulations show the growth of a strong m = 2 spiral structure. This spiral disturbance drives matter toward the central object and disappears when the Toomre parameter, Q, has increased well above unity. When a weak magnetic field is present as well, the magnetorotational instability grows and leads to turbulence. In that case, the strength of the gravitational stress tensor is lowered by a factor of ~2 compared with the hydrodynamic run and oscillates periodically, reaching very small values at its minimum. We attribute this behavior to the presence of a second spiral mode with higher pattern speed than the one that dominates in the hydrodynamic simulations. It is apparently excited by the high-frequency motions associated with MHD turbulence. The nonlinear coupling between these two spiral modes gives rise to a stress tensor that oscillates with a frequency that is a combination of the frequencies of each of the modes. This interaction between MHD turbulence and gravitational instabilities therefore results in a smaller mass accretion rate onto the central object.

Numerical simulations of type I planetary migration in non-turbulent magnetized discs

Using 2D magnetohydrodynamic (MHD) numerical simulations performed with two different finite-difference Eulerian codes, we analyse the effect that a toroidal magnetic field has on low-mass planet migration in non-turbulent protoplanetary discs. The presence of the magnetic field modifies the waves that can propagate in the disc. In agreement with a recent linear analysis, we find that two magnetic resonances develop on both sides of the planet orbit, which contribute to a significant global torque. In order to measure the torque exerted by the disc on the planet, we perform simulations in which the latter is either fixed on a circular orbit or allowed to migrate. For a 5-M ○+ planet, when the ratio β between the square of the sound speed and that of the Alfven speed at the location of the planet is equal to 2, we find inward migration when the magnetic field B Φ is uniform in the disc, reduced migration when B Φ decreases as r -1 and outward migration when B Φ decreases as r -2 . These results are in agreement with predictions from the linear analysis. Taken as a whole, our results confirm that even a subthermal stable field can stop inward migration of an earth-like planet.