Stochastic migration in turbulent disks

To a first approximation, the efficiency of angular-momentum transport (unless it is very low) has little impact on the predicted Type I migration rate. This assumes, however, that the disk is laminar. More realistically, angular-momentum transport itself derives from turbulence, which is accompanied by a spatially and temporally varying pattern of density fluctuations in the protoplanetary disk. These fluctuations will exert random torques on planets of any mass embedded within the disk, in much the same way as transient spiral features in the Galactic disk act to increase the velocity dispersion of stellar populations (Carlberg & Sellwood 1985). If we assume that the random torques are uncorrelated with the presence of a planet, then the random torques' linear scaling with planet mass will dominate over the usual Type I torque (scaling as Mp2 ) for sufficiently low masses. The turbulence will then act to increase the velocity dispersion of collisionless bodies, or, in the presence of damping, to drive a random walk in the semi-major axis of low mass planets.

To go beyond such generalities, and in particular to estimate the crossover mass between stochastic and Type I migration, we need to specify the source of turbulence in the protoplanetary disk. MHD disk turbulence (see Figure 3 for an illustration) driven

Figure 3. Structure in a turbulent disk accretion flow, from a global cylindrical MHD simulation (after Armitage 1998). The flow is visualized here using the square root of the vertically-averaged Maxwell stress (BrB^ ) as a tracer. In magnetically active disks, transient density and magnetic field fluctuations are present across a wide range of spatial scales.

Figure 3. Structure in a turbulent disk accretion flow, from a global cylindrical MHD simulation (after Armitage 1998). The flow is visualized here using the square root of the vertically-averaged Maxwell stress (BrB^ ) as a tracer. In magnetically active disks, transient density and magnetic field fluctuations are present across a wide range of spatial scales.

by the magnetorotational instability (Balbus & Hawley 1998) provides a well-understood source of outward angular-momentum transport in sufficiently well-ionized disks, and has been used as a model system for studying stochastic migration by Nelson & Papaloizou (2004) and by Laughlin, Steinacker, & Adams (2004). Density fluctuations in MHD disk turbulence have a typical coherence time of the order of the orbital period, and, as a consequence, are able to exchange angular momentum with an embedded planet across a range of disk radii (not only at narrow resonances). The study by Nelson & Papaloizou (2004) was based on both global ideal MHD disk simulations, with an aspect ratio of h/r = 0.07, and local shearing box calculations. The global runs realized an effective Shakura-Sunyaev a = 7 x 10~3, which, if replicated in a real disk, would be consistent with observational measures of T Tauri disk evolution (Hartmann et al. 1998). For all masses considered in the range 3 M® ^ Mp ^ 30 M®, the instantaneous torque on the planet from the MHD turbulent disk exhibited large fluctuations in both magnitude and sign. Averaging over « 20 orbital periods, the mean torque showed signs of converging to the Type I rate, although the rate of convergence was slow, especially for the lowest-mass planets in the global runs. These properties are generally in accord with other studies of the variability of MHD disk turbulence (Hawley 2001; Winters, Balbus & Hawley 2003a). Very roughly, the Nelson & Papaloizou (2004) simulations suggest that up to Mp ~ 10 the random walk component dominates steady Type I drift over time scales that substantially exceed the orbital period.

We caution that existing studies of the stochastic migration regime are unrealistic. Ideal MHD is not a good approximation for the protoplanetary disk at those radii where planet formation occurs, and there may be dead zones in which MHD turbulence and angular-momentum transport is highly suppressed (Gammie 1996; Sano et al. 2000; Fro-mang, Terquem, & Balbus 2002; Salmeron & Wardle 2005). We also note that a significant random migration component, if it indeed adds to rather than supplanting steady Type I migration, does nothing (on average) to help the survival prospects of low-mass planets. Nevertheless, if turbulent fluctuations (whatever their origin) do occur in the disk, the resulting random walk migration could be important for planet formation. In the terrestrial-planet region, stochastic migration might deplete low-mass planetary embryos that would be relatively immune to ordinary Type I migration, while simultaneously promoting radial mixing and collision of planetesimals. For giant-planet formation, a significant random component to core migration would have the effect of creating large fluctuations in the planetesimal accretion rate, while also potentially acting to diffuse the planetesimal surface density. Large fluctuations in the planetesimal accretion rate favor the early onset of rapid gas accretion, and allow for the final core mass to be substantially smaller than would be expected in the case of a static core (Rice & Armitage 2003).

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