J. C. B. Papaloizou

On the tidal interaction between protoplanets and the protoplanetary disk. III - Orbital migration of protoplanets

The tidal interaction between a protoplanet and a gaseous protoplanetary disk is investigated, and the dynamical evolution of the disk and the orbital migration of the protoplanet in a self-consistent manner is considered. It is shown that the orbital migration of a protoplanet does not suppress the tendency for tidal truncation in the vicinity of its orbit. If the necessary condition for tidal truncation is satisfied, the protoplanet induces a tidal feedback mechanism that regulates the rate of angular momentum transfer between the protoplanet and the disk. Significant orbital migration can only occur on the viscous evolution time scale of the disk.

Tidally Induced Gap Formation in Protostellar Disks: Gap Clearing and Suppression of Protoplanetary Growth

We present the results of numerical simulations of protostellar accretion disks that are perturbed by a protoplanetary companion that has a much smaller mass than the central object. We consider the limiting cases where the companion is in a coplanar, circular orbit and is initially embedded in the disk. Three independent numerical schemes are employed, and generic features of the flow are found in each case. In the first series of idealized models, the secondary companion is modeled as a massless, orbiting sink hole able to absorb all matter incident upon it without exerting any gravitational torque on the disk. In these simulations, accretion onto the companion induces surface density depression and gap formation centered on its orbital radius. After an initial transitory adjustment, the accretion rate onto the sink hole becomes regulated by the rate at which viscous evolution of the disk can cause matter to diffuse into the vicinity of the sink hole orbit, and thus the sink hole grows on a disk viscous timescale. In the second series of comprehensive models, the companions gravity is included. When the tidal torque exerted by the companion on the disk becomes important, the angular momentum exchange rate between the companion and the disk induces the protoplanetary accretion to reduce markedly below that in the idealized sink hole models. Whether this process is effective in inhibiting protoplanetary accretion depends on the equation of state and disk model parameters. For polytropic or isothermal equations of state, we find, in basic agreement with earlier work, that when the mean Roche lobe radius of the companion exceeds the disk thickness and when the mass ratio, q, between the companion and the central object exceeds ~40/, where is the effective Reynolds number, a clean deep gap forms. Although precise estimation is rendered difficult due to the limitation of numerical schemes in dealing with large density contrasts, the generic results of three series of simulations indicate that accretion onto sufficiently massive protoplanets can become ineffective over the expected disk lifetimes in their neighborhood. We suggest that such a process operates during planetary formation and is important in determining the final mass of giant planets.

The migration and growth of protoplanets in protostellar discs

ABSTRA C T 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 time-scale 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.

On the Dynamical Foundations of α Disks

The dynamical foundations of α disk models are described. At the heart of the viscous formalism of accretion disk models are correlations in the fluctuating components of the disk velocity, magnetic field, and gravitational potential. We relate these correlations to the large-scale mean flow dynamics used in phenomenological viscous disk models. MHD turbulence readily lends itself to the α formalism, but transport by self-gravity does not. Nonlocal transport is an intrinsic property of turbulent self-gravitating disks, which in general cannot be captured by an α model. Local energy dissipation and α-like behavior can be reestablished if the pattern speeds associated with the amplitudes of an azimuthal Fourier decomposition of the turbulence are everywhere close to the local rotation frequency. In this situation, global wave transport must be absent. Shearing box simulations, which employ boundary conditions forcing local behavior, are probably not an adequate tool for modeling the behavior of self-gravitating disks. As a matter of principle, it is possible that disks that hover near the edge of gravitational stability may behave in accord with a local α model, but global simulations performed to date suggest matters are not this simple.

A planet on an inclined orbit as an explanation of the warp in the β Pictoris disc

We consider the deformation that has recently been observed in the inner part of the circumstellar disc around f3 Pictoris with the HST. Our recent ground-based, adaptive optics coronographic observations confirm that the inner disc is warped. We investigate the hypothesis that a yet undetected planet is responsible for the observed warp, through simulations of the effect of the gravitational perturbation resulting from a massive companion on the disc. The physical processes assumed in the simulations are discussed: since the observed particles do not survive collisions, the apparent disc shape is driven by the underlying collisionless parent population. The resulting possible parameters for the planet that are consistent with the observed disc deformation are reviewed.

Runaway migration and the formation of hot jupiters

We evaluate the coorbital corotation torque on a migrating protoplanet. The coorbital torque is assumed to come from orbit crossing fluid elements that exchange angular momentum with the planet when they execute a U-turn at the end of horseshoe streamlines. When the planet migrates inward, the fluid elements of the inner disk undergo one such exchange as they pass to the outer disk. The angular momentum they gain is removed from the planet, and this corresponds to a negative contribution to the corotation torque, which scales with the drift rate. In addition, the material trapped in the coorbital region drifts radially with the planet, giving a positive contribution to the corotation torque, which also scales with the drift rate. These two contributions do not cancel out if the coorbital region is depleted, in which case there is a net corotation torque that scales with the drift rate and the mass deficit in the coorbital region and has the same sign as the drift rate. This leads to a positive feedback on the migrating planet. In particular, if the coorbital mass deficit is larger than the planet mass, the migration rate undergoes a runaway that can vary the protoplanet semimajor axis by 50% over a few tens of orbits. This can happen only if the planet mass is sufficient to create a dip or gap in its surrounding region and if the surrounding disk mass is larger than the planet mass. This typically corresponds to planet masses in the sub-Saturnian to Jovian mass range embedded in massive protoplanetary disks. Runaway migration is a good candidate to account for the orbital characteristics of close orbiting giant planets, most of which have sub-Jovian masses. These are known to cluster at short periods, whereas planets of greater than two Jovian masses are rare at short periods, indicating a different type of migration process operated for the two classes of object. Further, we show that in the runaway regime, migration can be directed outward, which makes this regime potentially rich in a variety of important effects in shaping a planetary system during the last stages of its formation.

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.

The interaction of giant planets with a disc with MHD turbulence – IV. Migration rates of embedded protoplanets

We present the results of global cylindrical disc simulations and local shearing box simulations of protoplanets interacting with a disc undergoing magnetohydrodynamic (MHD) turbulence. The specific emphasis of this paper is to examine and quantify the magnitude of the torque exerted by the disc on the embedded protoplanets as a function of the protoplanet mass, and thus to make a first study of the induced orbital migration of protoplanets resulting from their interaction with magnetic, turbulent discs. This issue is of crucial importance in understanding the formation of gas giant planets through the so-called core instability model, and the subsequent orbital evolution post-formation prior to the dispersal of the protostellar disc. Current estimates of the migration time of protoplanetary cores in the 3–30 Earth mass range in standard disc models are τmig≃ 104–105 yr, which is much shorter than the estimated gas accretion time-scale of Jupiter-type planets. The global simulations were carried out for a disc with constant aspect ratio H/r= 0.07 and protoplanet masses of Mp= 3, 10, 30 Earth masses, and 3 Jupiter masses. The local shearing box simulations were carried out for values of the dimensionless parameter (Mp/M*)/(H/R)3= 0.1, 0.3, 1.0 and 2.0, with M*, R and H being the central mass, the orbital radius and the local disc semithickness, respectively. These allow both embedded and gap-forming protoplanets for which the disc response is non-linear to be investigated. In all cases the instantaneous net torque experienced by a protoplanet showed strong fluctuations on an orbital time-scale, and in the low-mass embedded cases oscillated between negative and positive values. Consequently, in contrast to the laminar disc type I migration scenario, orbital migration would occur as a random walk. Running time averages for embedded protoplanets over typical run times of 20–25 orbital periods indicated that the averaged torques from the inner and outer disc took on values characteristic of type I migration. However, large fluctuations occurring on longer than orbital time-scales remained, preventing convergence of the average torque to well-defined values or even to a well-defined sign for these lower mass cases. Fluctuations became relatively smaller for larger masses indicating better convergence properties, to the extent that in the 30-M⊕ simulation consistently inward, albeit noisy, migration was indicated. Both the global and local simulations showed this trend with increasing protoplanet mass, which is due to its perturbation on the disc increasing to become comparable to and then dominate the turbulence in its neighbourhood. The turbulence then becomes unable to produce very large long-term fluctuations in the torques acting on the protoplanet. Eventually gap formation occurs and there is a transition to the usual type II migration at a rate determined by the angular momentum transport in the distant parts of the disc. The existence of significant fluctuations occurring in the turbulent discs on long time-scales is an important unexplored issue for the lower mass embedded protoplanets, which are unable to modify the turbulence in their neighbourhood, and which have been studied here. If significant fluctuations occur on the longest disc evolutionary time-scales, convergence of torque running averages for practical purposes will not occur and the migration behaviour of low-mass protoplanets considered as an ensemble will be very different from predictions of type I migration theory for laminar discs. The fact that noise levels were relatively smaller in the local simulations may indicate the presence of long-term global fluctuations, but the issue remains an important one for future investigation.

The evolution of a supermassive binary caused by an accretion disc

The interaction between a massive binary and a non-self-gravitating circumbinary accretion disc is considered. The shape of the stationary twisted disc produced by the binary is calculated. It is shown that the inner part of the disc must lie in the binary orbital plane for any value of the ivi@tfa.fy.chalmers.se viscosity. When the inner disc midplane is aligned with the binary orbital plane on the scales of interest and it rotates in the same sense as the binary, the modification to the disc structure and the rate of decay of the binary orbit, assumed circular, due to tidal exchange of angular momentum with the disc, are calculated. It is shown that the modified disc structure is well described by a self-similar solution of the non-linear diffusion equation governing the evolution of the disc surface density. The calculated time scale for decay of the binary orbit is always smaller than “accretion” time scale tacc = m/ u M (m is the mass of secondary component, and u M is the disc accretion rate), and is determined by ratio of secondary mass m, assumed to be much smaller than the primary mass, the disc mass inside the initial binary orbit, and the form of viscosity in the disc.

On the orbital evolution and growth of protoplanets embedded in a gaseous disc

ABSTRA C T We present a new computation of the linear tidal interaction of a protoplanetary core with a thin gaseous disc in which it is fully embedded. For the first time a discussion of the orbital evolution of cores on eccentric orbits with eccentricity (e) significantly larger than the gasdisc scaleheight-to-radius ratioOH=rU is given. We find that the direction of orbital migration reverses for e . 1:1H=r: This occurs as a result of the orbital crossing of resonances in the disc that do not overlap the orbit when the eccentricity is very small. In that case, resonances always give a net torque corresponding to inward migration. Simple expressions giving approximate fits to the eccentricity damping rate and the orbital migration rate are presented. We go on to calculate the rate of increase of the mean eccentricity for a system of protoplanetary cores caused by dynamical relaxation. By equating the eccentricity damping time-scale with the dynamical relaxation time-scale we deduce that, for parameters thought to be applicable to protoplanetary discs, an equilibrium between eccentricity damping and excitation through scattering is attained on a 10 3 ‐10 4 yr time-scale, at 1 au. This equilibrium is maintained during the further migrational and collisional evolution of the system, which occurs on much longer time-scales. The equilibrium thickness of the protoplanet distribution is related to the equilibrium eccentricity, and is such that it is generally well confined within the gas disc. By use of a three-dimensional direct summation N-body code we simulate the evolution of a system of protoplanetary cores, initialized with a uniform isolation mass of 0.1 M%, incorporating our eccentricity damping and migration rates. Assuming that collisions lead to agglomeration, we find that the vertical confinement of the protoplanet distribution permits cores to build up in mass by a factor of ,10 in only ,10 4 yr, within 1 au. The time-scale required to achieve this is comparable to the migration time-scale. In the context of our model and its particular initial conditions, we deduce that it is not possible to build up a massive enough core to form a gas giant planet, before orbital migration ultimately results in the preferential delivery of all such bodies to the neighbourhood of the central star. This problem could be overcome by allowing for the formation of massive cores at much larger radii than are usually considered. It remains to be investigated whether different disc models or initial planetesimal distributions might be more favourable for slowing or halting the migration, leading to possible giant planet formation at intermediate radii.