7 Discussion

The migration regime corresponding to the range of masses, aspect ratios and viscosities investigated in this work corresponds mostly to type I migration regime, except the low aspect ratio, low viscosity and large mass runs, in which quite a significant gap is opened in the co-orbital region. As type I migration is known to be too fast (i.e. the migration time of a protoplanetary core is shorter than its build-up time), it is of interest to investigate the migration time from these results, as a function of aspect ratio, planet mass and viscosity. Figures 25 and 26 show the migration time, in years, of a protocore at 1 AU embedded respectively in a 3% and 4% aspect ratio disk. In either case the disk surface density is chosen to be that of the minimum mass solar nebula (1700 g cm^-2, i.e. $\Sigma_0=1.9\times 10^{-4}$, see Hayashi et al. 1985).

Figure 25: Migration time in years at 1 AU for a h=0.03 disk, as a function of planet mass and disk viscosity. The dotted contours, labeled with negative values, correspond to a positive torque and therefore to an outwards migration, in which case the label indicates the semi-major axis doubling time. The dashed line is the set of positions where the unperturbed viscous drift rate $-3\nu /2r$ has the same absolute value as the inferred migration rate. Therefore the migration rate estimates are reliable only to the right of the dashed line, i.e. wherever the material flow rate across the orbit is mostly accounted for by the viscous drift. The solid thick line corresponds to a vanishing total torque (limit of migration reversal). The contours close to this limit correspond to migration times which tend to infinity, and they have been blanked for numerical precision reasons.

Figure 26: Migration time in years at 1 AU for a h=0.04 disk, as a function of planet mass and disk viscosity. The contour line style obeys the same conventions as in Fig. 25.

As the results of the high resolution runs of Sect. 6 dealt only with restricted mass and viscosity ranges, the results shown at Figs. 25 and 26 correspond to the low resolution runs (i.e. to Figs. 9 and 10), for which the parameter coverage is much larger. As the results for the h=0.04 case in the high and low resolution case differ sensibly, the contours of Fig. 26 (and of Fig. 25) should not be taken too literally. They do however have the merit to show the behavior of the migration time as a function of mass and viscosity (even if the boundary of the domain of torque reversal should not be taken literally) and they also have the merit to give a correct order of magnitude of this migration time, defined as:

$$\tau_{\rm mig}=\frac{M_{\rm p}r_{\rm p}^2\Omega_{\rm p}}{2\Gamma}\cdot$$ (41)

This migration time has been evaluated at $r_{\rm p}=1$ AU, where the protoplanetary disk is likely to be subject to the magneto-rotational instability (Balbus & Hawley 1991), which endows it with a source of significant viscosity, which could be in the range of the values of $\alpha$ for which migration can reverse. Although this migration time is a few times larger than previous results (e.g. Miyoshi et al. 1999 and references therein), it is still much shorter than the disk lifetime and the core build-up time. It is therefore very unlikely that a unique protocore reaches the torque reversal mass before having migrated all the way to the central object. One reason is that the torque reversal occurs in very thin disks, for which the absolute value of the differential Lindblad torque is large (it scales as h^-2), and the associated migration time-scale is short. One can see that the most favorable situation in both cases is for $\alpha=10^{-2}$ (i.e. for the corresponding viscosity a given mass protocore has the largest migration time). For this value of $\alpha$ one gets bottleneck values of the migration time respectively 7- $8\times 10^4$ years (for h=3%) and 6- $7\times 10^4$ years (for h=4%).

It should also be noted that, as the corotation torque scales with the slope of the specific vorticity gradient, the torque reversal that was found here for very thin disks is not likely to subsist if one takes a significant negative surface density gradient (i.e. if $\Sigma \propto r^{-q}$ with $q\sim 1$).

On the other hand if the specific vorticity gradient is even steeper than the one envisaged here, then the corotation torque, depending on how steep this gradient is, may well unconditionally dominate the differential Lindblad torque. This might be the case in the very central parts of the disk, where the outer disk solution has to be connected to an inner cavity (e.g. magnetically cleared). If the transition region is larger than the co-orbital zone of an infalling body, then the corotation torque acting on it might be able to counteract the differential Lindblad torque, providing another way of stopping the inward migrating bodies, as the disk there is very likely to be viscous and therefore very likely to be able to sustain an unsaturated corotation torque.

As the analysis presented here is restricted to a planet held on a fixed circular orbit, it should be realized that the migration time-scale estimated above is only valid whenever the viscous drift rate of material across the orbit is much larger than the planet inferred drift rate $2\Gamma/(M_{\rm p}r_{\rm p}\Omega_{\rm p})$, otherwise the corotation torque expression has to explicitly include a term proportional to $\dot a$, the migration rate, with a delay which scales as the outermost horseshoe libration time. The domain for which the migration rate is negligible compared to the viscous drift radial velocity is shown in Figs. 25 and 26. Therefore one can see that at very low viscosities [ $\alpha < (1$- $3)\times 10^{-3}$] the expression for the corotation torque needs to be reconsidered, as most of the specific vorticity drift across the co-orbital region is not accounted for by the viscous accretion but by the migration itself. In these circumstances the total torque felt by the planet depends on its migration rate. The analyze and consequences of this feed-back will be presented elsewhere.