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

Abstract

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 Ñow are found in each case. In the Ðrst 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 di †use 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 companionÏs 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 e †ective in inhibiting protoplanetary accretion depends on the equation of state and disk model parameters. For polytropic or isothermal equations of state, we Ðnd, 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 D40/R, where R is the e †ective 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 ine †ective over the expected disk lifetimes in their neighborhood. We suggest that such a process operates during planetary formation and is important in determining the Ðnal mass of giant planets.

On the width and shape of gaps in protoplanetary disks

“…Thus, independently of the adopted definition of 'gap' as in the previous case, the threshold viscosity for gap opening in the infinitely thin disk limit, ν 0 , scales proportionally to q. This is consistent with the gap opening criterion given in Bryden et al (1999).…”

Orbital eccentricity growth through disc-companion tidal interaction

“…To reach the 1:2 resonance this has to be three times larger. From Lin & Papaloizou (1993a) and Bryden et al (1999), the tidal torque, which varies as the inverse cube of the gap width, is then reduced by a factor of 27. To prevent the gap filling the viscosity would have to be reduced by at least the same factor requiring ν ≤ 3 10 −7 .…”

“…Recent simulations of protoplanets with 0.001 < q < 0.01, interacting with discs with physical parameters similar to those adopted here, being thought to be appropriate to protoplanetary discs, have been carried out by Bryden et al (1999), Kley (1999), and Lubow et al (1999). Fixed circular orbits were assumed.…”

On the orbital evolution of a pair of giant planets in mean motion resonance

“…Many of our runs are carried out with planets migrating in discs with local kinematic viscosity ν < 10 −6 in dimensionless units. Accretion onto the planet has been found to be a small effect in this case (Bryden et al 1999, Kley 1999. In addition the time required for the accretion rate through the disc to double the planet's mass is significantly longer than the migration time when larger viscosities are considered (see below).…”