1 Introduction

The formation of planets is likely to occur in the protoplanetary disks surrounding the very young stars. One of the most widely accepted formation scenario is the core accretion one, whereby a solid core is slowly built up by the accretion of solid material (rocks and ice) and subsequently, if its mass becomes large enough, by the rapid accretion of the surrounding gas. As the core builds up, it gravitationally interacts with the surrounding nebula, and as a result of the exchange of angular momentum and energy with the nebula, it undergoes a change in its orbital parameters. The basis of the gravitational interaction between a disk and an orbiting point-like perturber was worked out about twenty years ago (Goldreich & Tremaine 1979, 1980; Lin & Papaloizou 1979) and since then refined in great detail (Ward 1986, 1988; Artymowicz 1993; Ward 1997). The most striking prediction of these works was that a protoplanet embedded in a protoplanetary disk would undergo a rapid orbital decay or inward migration, and this expectation has received strong observational support from the discovery of short-period giant planets (Mayor & Queloz 1995), the formation of which is unlikely to have occurred in situ. Most of the work done on the planet-disk tidal interaction has been done in the linear regime. Each azimuthal Fourier component of the planet perturbing potential can be shown to contribute sizably to the disk-planet tidal interaction wherever its frequency in the local frame either vanishes or matches $\pm\kappa$, the epicyclic frequency (which coincides with the Keplerian frequency in a Keplerian disk such as a protoplanetary disk, where self-gravity, at least in the central parts, in considered as unimportant). The angular momentum exchange rate through the terms for which the perturber frequency in the local frame is $+\kappa$ (resp. $-\kappa$) is called the outer Lindblad torque (resp. the inner Lindblad torque), while the terms corresponding to a vanishing perturbed frequency in the local frame correspond to the corotation torque. If one makes the assumption of a circular orbit (which is reasonable as it can be shown that the global effect of the resonances is an eccentricity damping in most cases, see e.g. Papaloizou et al. 2001 and references therein), then the Outer Lindblad Resonances (OLR) fall exterior to the orbit, the Inner Lindblad Resonances (ILR) fall interior to it, and the corotation resonances nearly coincide with the orbit (they actually lie slightly inside of this latter as the disk is slightly sub-Keplerian due to the partial support of gravity by a radial pressure gradient), and the corresponding torque is called for obvious reasons the co-orbital corotation torque. The outer Lindblad torque can be shown to be negative, while the inner Lindblad torque is positive. There exists a mismatch between both (almost always in favor of the outer Lindblad torque, Ward 1997) which scales as h, the disk aspect ratio, and which leads to the inward migration. The Lindblad torque has been shown to be independent of viscosity (Meyer-Vernet & Sicardy 1987; Papaloizou & Lin 1984). This is true as long as the surface density profile itself is not sizably modified under the action of the one-sided Lindblad torque, i.e. as long as the relative depth of the dip opened around the orbit by the planet is small. This assumption is fulfilled in the present work except for the highest masses and the lowest viscosities (see also discussion at Sect. 4.4). On the other hand, the corotation torque in the linear regime does not correspond to an angular momentum flux carried away by waves. The angular momentum rather accumulates at corotation, and the corresponding torque appears as a discontinuity in the angular momentum flux at corotation (Goldreich & Tremaine 1979). Ward (1992) has shown, by summing the corotation resonances after an adequate vertical averaging of the perturbing potential, that the corotation torque is at most comparable to the differential Lindblad torque, whereas Korycansky & Pollack (1993) have shown by solving numerically the linearized equations for a steady state perturbation of the disk that the actual solution was even smaller than the analytical estimates. The corotation torque scales as the gradient of specific vorticity across corotation $\partial(\Sigma/B)/\partial r$, where B is the second Oort's constant and is $\Omega/4$ in a Keplerian disk. Ward (1991) has shown that the corotation torque comes from the exchange of angular momentum between the planet and the near-by fluid elements which move from one side to the other of their horseshoe streamline, and that the dependency upon the specific vorticity gradient was due to an uneven mapping of the fluid elements between the outer and inner leg of their horseshoe streamline. In the linear regime, i.e. as the planet mass tends to zero, the horseshoe zone width tends to zero, while the libration time of the fluid elements on the horseshoe streamlines tends to infinity, which is in agreement with the picture of a localized and constant discontinuity of the angular momentum flux at corotation. In the finite mass regime the situation is quite different however. The libration time can be much shorter than the migration time across the horseshoe region, and also much shorter than the viscous diffusion time, in which case libration removes the specific vorticity gradient (Ward 1992), and the corotation torque therefore vanishes (saturates) after a few libration times. In a sufficiently viscous disk however, the viscous diffusion and the radial transport of material across the horseshoe region may inhibit the corotation torque saturation. A way of evaluating the co-orbital corotation torque as a function of viscosity has been given by Masset (2001), for a planet held on a circular orbit in a uniform surface density disk, and for a steady flow in the planet frame. As the material set in libration in a steady flow is trapped, the total torque on the librating fluid elements region vanishes. This torque is the sum of the viscous torque on the trapped region and the gravitational torque of the planet, which is opposite to one component of the corotation torque. As the viscous force is a contact action, the integral of the viscous torque on the trapped region reduces to the torque exerted on its boundary (the separatrix between circulating and librating streamlines) by the outer and inner disks. The second and last component of the corotation torque is given by the angular momentum lost by the fluid elements which participate in the global accretion of the disk material onto the central object (accretion of which are excluded only the fluid elements of the librating region), when they flow from the outer to the inner disk and undergo one horseshoe like close encounter with the planet. The corotation torque can therefore be expressed (in the steady state case) only with the knowledge of the flow properties at the separatrix. The flow properties in the librating region can be split into an even and an odd part of the perturbed density (w.r.t. the distance to corotation). The former comes from the perturbations of the surface density and azimuthal velocity profiles under the action of the one-sided Lindblad torque, while the latter comes from libration. Masset (2001) gives the following expression for the corotation torque:

$$\Gamma_{\rm C}=\Gamma_{\rm M}^{\rm C}+\Gamma_{\rm A}^{\rm C}$$ (1)

where $\Gamma_{\rm M}^{\rm C}$ is the main term, which comes from the odd part of the perturbed density and which therefore is linked to libration and exhibits a dependence on the viscosity:

$$\Gamma_{\rm M}^{\rm C}=\frac 98x_{\rm s}^4\Omega_{\rm p}^2\Sigma_0{\cal F}(z_{\rm s})$$ (2)

where ${\cal F}(z_{\rm s})$ is a function of the viscosity and the horseshoe zone width $x_{\rm s}$ (in which, contrary to Masset 2001, a factor 4 has been introduced for convenience, cf. infra), while $\Gamma_{\rm A}^{\rm C}$ is an additional term which comes from the even part of the perturbed density, and which introduces a coupling between the one-sided Lindblad torque and the corotation torque:

$$\Gamma_{\rm A}^{\rm C}=\frac{x_{\rm s}}{r_{\rm p}}{\cal G}(x_{\rm s})\Gamma_{\rm LR}$$ (3)

where ${\cal G}(x_{\rm s})=O(1)$ is a function which depends on the exact position of the separatrix, and where $\Gamma_{\rm LR}$is the one-sided Lindblad torque (the arithmetical average of the Outer and Inner Lindblad torques absolute values). The fact that this additional term does not exhibit any dependence on the viscosity may seem paradoxical, as one expects the co-orbital corotation torque to vanish in an inviscid disk. This paradox is only apparent however. This term comes from the even terms in the perturbed density (and odd in the perturbed azimuthal velocity), which scale as $\nu^{-1}$, therefore this dependence cancels out with the $\nu$ overall dependence of the viscous torque on the separatrix. This results holds as long as the even perturbed density does not saturate (i.e. as long as the dip is shallow enough and does not correspond to a gap). In the strictly inviscid limit (and provided that in this limit one can have a steady state situation), the planet (which in this artificial situation is held on a fixed circular orbit) opens a gap, and therefore the even perturbed density term saturates and the additional term given by Eq. (3) vanishes.

The purpose of this paper is to investigate this situation and to check by means of numerical simulations the validity of Eqs. (2) and (3). For that purpose, numerical simulations are performed over a large range of viscosities. Other questions addressed in this paper are the ability of 2D numerical codes to predict correctly the total torque acting on an embedded point-like object through the choice of a correct value of the potential smoothing length, and the possibility of migration reversal (i.e. outward) due to the corotation torque, although linear regime studies seem to conclude that the (positive) corotation torque is too weak to counteract the (negative) differential Lindblad torque. Although the additional term of Eq. (3) is too small to cancel out the differential Lindblad torque by itself, it is of interest to examine the situation for weakly embedded objects where the corotation torque is lift up by the additional term of Eq. (3), and where the differential Lindblad torque may be reduced with respect to its linear value (see e.g. Miyoshi et al. 1999; Lin & Papaloizou 1979).

It should be noted that global simulations are needed to describe properly the corotation torque. Any local description, such as the one provided by a box or a wedge centered on the planet, will prevent saturation if inflow/outflow boundary conditions are used in azimuth, which inject "fresh'' material with an unperturbed density and velocity profile, while the use of periodic boundary conditions in these configurations artificially shortens the libration time. Furthermore, the use of the shearing sheet formalism, which usually neglects the radial derivative of the unperturbed disk specific vorticity, leads to a dependence of the corotation torque on the radial derivative of the surface density, thus switching off the corotation torque in the convenient situation where the surface density is uniform.