Type I migration

In the Type I regime, the perturbation induced by the planet in the gas disk remains small, and the net torque has surprisingly little dependence on the microphysics of the protoplanetary disk (e.g., Goldreich & Tremaine 1978; Lin & Papaloizou 1979). In particular, viscosity—normally the most uncertain element of a protoplanetary disk model—enters only indirectly via its influence on the magnitude and radial gradient of the surface density and sound speed. Generically, the net torque scales with the planet mass as T x M2, so that the migration time scale at a given radius scales as t x M-1. Type I migration, therefore, becomes increasingly important as the planet mass increases, and is fastest just prior to gap opening (the onset of which does depend on the disk viscosity). Despite its attractive lack of dependence on uncertain disk physics, the actual calculation of the net torque is technically demanding, and substantial improvements have been made only recently. Here, we summarize a few key results—the reader is directed to the original papers (primarily by Artymowicz, Ward, and their collaborators) for full details of the calculations.

2.1. Analytic calculations

The simplest calculation of the Type I torque (Goldreich & Tremaine 1978, 1979, 1980) neglects significant pressure effects in the disk close to the planet, and is therefore valid for low m resonances. In this approximation, Lindblad resonances occur at radii in the disk where the epicyclic frequency k is an integral multiple m of the angular velocity in a frame rotating with the planet at angular velocity Qp. For a Keplerian disk, this condition,

D(r) = k2 - m2 (Q - Qp)2 = 0 , can be simplified using the fact that k = Q. The resonances lie at radii,

rL =1 ± — rp m where rp is the planet's orbital radius. The lowest order resonances lie at r = 1.587rp and r = 0.630rp, but an increasingly dense array of high m resonances lie closer to the planet. Resonances at r < rp add angular momentum to the planet, while those at r > rp remove angular momentum. The torque at each resonance Tm can be evaluated in terms of a forcing function as,

Tm = - mE m m rdD/dr where E is the gas surface density. Explicit expressions for are given by Goldreich & Tremaine (1979). The net torque is obtained by evaluating the torque at each resonance, and then summing over all m.

For Type I migration the behavior of gas close to the planet, where is largest, is critical. Accurately treating this regime requires elaborations of the basic Goldreich & Tremaine (1979) approach. For a razor-thin two-dimensional disk model—the effects of radial pressure and density gradients—the calculation is described in Ward (1997), and references therein (especially Ward 1988; Artymowicz 1993a, 199b; Korycansky& Pollack 1993). These papers include the shifts in the location of Lindblad resonances due to both radial and azimuthal pressure gradients, which become significant effects at high m. For Lindblad resonances, the result is that the dominant torque arises from wavenumbers m ~ rp/h, where h, the vertical disk scale height, is given in terms of the local sound speed cs by h = cs/ttp. The fractional net torque 2|T-nner + Tauter|/(|i|nner| + |Touter|) can be as large as 50% (Ward 1997), with the outer resonances dominating and driving rapid inward migration. Moreover, the small shifts in the locations of resonances that occur in disks with different radial surface density profiles conspire so that the net torque is only weakly dependent on the surface density profile. This means that the predicted rapid inward migration occurs for essentially any disk model in which the sound speed decreases with increasing radius (Ward 1997). Corotation torques—which vanish in the oft-considered disk models with E x r-3/2—can alter the magnitude of the torque but are not sufficient to reverse the sign (Korycansky & Pollack 1993; Ward 1997).

The observation that the dominant contribution to the total torque comes from gas that is only Ar ~ h away from the planet immediately implies that a two-dimensional representation of the disk is inadequate, even for protoplanetary disks which are geometrically thin by the usual definition (h/r < 0.1, so that pressure gradients, which scale as order (h/r)2, are sub-percent level effects). Several new physical effects come into play in a three-dimensional disk:

1. The perturbing potential has to be averaged over the vertical thickness of the disk, effectively reducing its strength for high m resonances (Miyoshi et al. 1999).

2. The variation of the scale height with radius decouples the radial profile of the midplane density from that of the surface density.

3. Wave propagation in three-dimensional disks is fundamentally different from that in two dimensions, if the vertical structure of the disk departs from isothermality (Lubow & Ogilvie 1998; Bate et al. 2002).

Tanaka, Takeuchi, & Ward (2002) have computed the interaction between a planet and a three-dimensional isothermal disk, including the first two of the above effects. Both Lindblad and corotation torques were evaluated. They find that the net torque is reduced by a factor of 2-3 as compared to a corresponding two-dimensional model, but that migration remains inward and is typically rapid. Specifically, defining a local migration time scale via,

Tanaka, Takeuchi, & Ward (2002) find that for a disk in which E x , t=<—)-* M (^)v.

As expected, the time scale is inversely proportional to the planet mass and the local surface density. Since the bracket is ~ (h/rp)2, the time scale also decreases quite rapidly for thinner disks, reflecting the fact that the peak torque arises from closer to the planet as the sound speed drops.

Although still limited by the assumption of isothermality, the above expression represents the current 'standard' estimate of the Type I migration rate. Although slower than two-dimensional estimates, it is still rapid enough to pose a potential problem for planet formation via core accretion. There is, therefore, interest in studying additional physical effects that might reduce the rate further. The influence of disk turbulence is discussed more fully in the next section; here we list some of the other effects that might play a role:

1. Realistic disk structure models. The run of density and temperature in the midplane of the protoplanetary disk is not a smooth power-law due to sharp changes in opacity and, potentially, the efficiency of angular momentum transport (Gammie 1996). Menou & Goodman (2004) have calculated Type I rates in Shakura-Sunyaev type disk models, and find that even using the standard Lindblad torque formula there exist regions of the disk where the migration rate is locally slow. Such zones could be preferred sites of planet formation.

2. Thermal effects. Jang-Condell & Sasselov (2005) find that the dominant non-axisymmetric thermal effect arises from changes to the stellar illumination of the disk surface in the vicinity of the planet. This effect is most important at large radii, and can increase the migration time scale by up to a factor of two at distances of a few AU.

3. Wave reflection. The standard analysis assumes that waves propagate away from the planet, and are dissipated before they reach boundaries or discontinuities in disk properties that might reflect them back toward the planet. Tanaka, Takeuchi, & Ward (2002) observe that reflection off boundaries has the potential to substantially reduce the migration rate. We note, however, that relaxation of vertical isothermality will probably lead to wave dissipation in the disk atmosphere within a limited radial distance (Lubow & Ogilvie 1998), and thereby reduce the possibility for reflection.

4. Accretion. Growth of a planet during Type I migration is accompanied by a non-resonant torque, which has been evaluated by Nelson & Benz (2003a). If mass is able to accrete freely onto the planet, Bate et al. (2003) find from three-dimensional simulations that Mp a: Mp for Mp < 10 Mq, with a mass-doubling time that is extremely short (less than 103 yr). In reality, it seems likely the planet will be unable to accept mass at such a rapid rate, so the mass-accretion rate and resulting torque will then depend on the planet structure.

5. Magnetic fields. The dominant field component in magnetohydrodynamic disk turbulence initiated by the magnetorotational instability is toroidal (Balbus & Hawley 1998). Terquem (2003) finds that gradients in plausible toroidal magnetic fields can significantly alter the Type I rate, and in some circumstances (when the field decreases rapidly with r) stop migration. More generally, a patchy and variable toroidal field might lead to rapid variations in the migration rate. Whether this, or density fluctuations induced by turbulence, is the primary influence of disk fields is unclear.

6. Multiple planets. The interaction between multiple planets has not been studied in detail. Thommes (2005) notes that low-mass planets, which would ordinarily suffer rapid Type I migration, can become captured into resonance with more massive bodies that are themselves stabilized against rapid decay by a gap. This may be important for understanding multiple planet formation (and we have already noted that there is circumstantial evidence that multiple massive-planet formation may be common), though it does not explain how the first planet to form can avoid rapid Type I inspiral.

7. Disk Eccentricity. Type I migration in an axisymmetric disk is likely to damp planetary eccentricity. However, it remains possible that the protoplanetary disk itself might be spontaneously unstable to development of eccentricity (Ogilvie 2001). Pa-paloizou (2002) has shown that Type I migration can be qualitatively altered, and even reversed, if the background flow is eccentric.

2.2. Numerical simulations Hydrodynamic simulations of the Type I regime within a shearing-sheet geometry have been presented by Miyoshi et al. (1999), and in cylindrical geometry by D'Angelo, Hen-

ning, & Kley (2002), D'Angelo, Kley, & Henning (2003), and Nelson & Benz (2003b), with the latter paper focusing on the transition between Type I and Type II behavior. The most comprehensive work to date is probably that of Bate et al. (2003), who simulated in three dimensions the interaction with the disk of planets in the mass range 1 M® < Mp < 1 MJ. The disk model had h/r = 0.05, a Shakura-Sunyaev (1973) a parameter a = 4 x 10-3, and a fixed locally isothermal equation of state (i.e., cs = cs(r) only). This setup is closely comparable to that assumed in the calculations of Tanaka, Takeuchi, & Ward (2002), and very good agreement was obtained between the simulation results and the analytic migration time scale. Based on this, it seems reasonable to conclude that within the known limitations imposed by the restricted range of included physics, current calculations of the Type I rate are technically reliable. Given this, it is interesting to explore the consequences of rapid Type I migration for planet formation itself.

2.3. Consequences for planet formation The inverse scaling of the Type I migration time scale with planet mass means that the most dramatic effects for planet formation occur during the growth of giant plant cores via core accretion (Mizuno 1980). In the baseline calculation of Pollack et al. (1996), which does not incorporate migration, runaway accretion of Jupiter's envelope is catalyzed by the slow formation of an « 20 M® core over a period of almost 10 Myr. This is in conflict with models by Guillot (2004), which show that although Saturn has a core of around 15 M®, Jupiter's core is observationally limited to at most « 10 Mq, leading to discussion at this meeting of several ways to reduce the theoretically predicted core mass. Irrespective of the uncertainties, however, it seems inevitable that planets forming via core accretion pass through a relatively slow stage in which the growing planet has a mass of 5-10 M®. This stage is vulnerable to Type I drift.

Figure 2 shows the migration time scale for a 10 M® planet within gas disks with surface-density profiles of E a: r(very roughly that suggested by theoretical models, e.g., Bell et al. 1997) and E a: r-3/2 (the minimum mass Solar Nebula profile of Weidenschilling 1977). We consider disks with integrated gas masses (out to 30 AU) of 0.01 Mq and 1 MJ. The latter evidently represents the absolute minimum gas mass required to build Jupiter or a typical extrasolar giant planet. The torque formula of Tanaka, Takeuchi, & Ward (2002) is used to calculate the migration time scale t. As is obvious from the figure, migration from 5 AU on a time scale of 1 Myr—significantly less than either the typical disk lifetime (Haisch, Lada & Lada 2001) or the duration of the slow phase of core accretion—is inevitable for a core of mass 10 M®, even if there is only a trace of gas remaining at the time when the envelope is accreted. For more reasonable gas masses, the typical migration time scale at radii of a few AU is of the order of 105 yr. Another representation of this is to note that in the giant planet forming region, we predict significant migration (t = 10 Myr) for masses Mp > 0.1 M®, and rapid migration (t = 1 Myr) for Mp > 1 M®. We can also plot, for the same disk model, an estimate of the isolation mass (Lissauer 1993; using the gas to planetesimal surface density scaling of Ida & Lin 2004a). The isolation mass represents the mass a growing planet can attain by consuming only those planetesimals within its feeding zone—as such, it is reached relatively rapidly in planet formation models. Outside the snow line, the migration time scale for a planet (or growing core) at the isolation mass is typically a few Myr, reinforcing the conclusion that migration is inevitable in the early stage of giant planet formation. By contrast, in the terrestrial planet region, interior to the snow line, planets need to grow significantly beyond isolation before rapid migration ensues. It is therefore possible for the early stages of terrestrial planet formation to occur in the

Figure 2. Left hand panel: planet mass for which the Type I migration time scale at different radii equals 1 Myr (upper solid curve) and 10 Myr (lower curve). The gas disk is assumed to have a surface density profile £ x r-3/2, and a mass within 30 AU of 0.01 Mq. The dotted line shows an estimate of the isolation mass in the same disk model, assuming Solar metallicity and a snow line at 2.7 AU. Right hand panel: the migration time scale for a 10 M® core in protoplanetary disks with surface density profiles of £ x r-3/2 (solid lines) and £ x r-1 (dashed lines). For each model, the lower curve shows the time scale for a disk with a gas mass of 0.01 Mq within 30 AU, while the upper curve shows results for an absolute minimum mass gas disk with only 1 Mj within 30 AU.

Figure 2. Left hand panel: planet mass for which the Type I migration time scale at different radii equals 1 Myr (upper solid curve) and 10 Myr (lower curve). The gas disk is assumed to have a surface density profile £ x r-3/2, and a mass within 30 AU of 0.01 Mq. The dotted line shows an estimate of the isolation mass in the same disk model, assuming Solar metallicity and a snow line at 2.7 AU. Right hand panel: the migration time scale for a 10 M® core in protoplanetary disks with surface density profiles of £ x r-3/2 (solid lines) and £ x r-1 (dashed lines). For each model, the lower curve shows the time scale for a disk with a gas mass of 0.01 Mq within 30 AU, while the upper curve shows results for an absolute minimum mass gas disk with only 1 Mj within 30 AU.

presence of gas with only limited Type I migration, while the final assembly of terrestrial planets happens subsequently in a gas-poor environment.

Does Type I migration help or hinder the growth of giant planets via core accretion? This question remains open, despite a history of investigations stretching back at least as far as papers by Hourigan & Ward (1984) and Ward (1989). Two competing effects are at work:

1. A migrating core continually moves into planetesimal-rich regions of the disk that have not been depleted by the core's prior growth. This depletion is, in part, responsible for the slow growth of Jupiter in static core calculations. Calculations suggest that a rapidly migrating core can capture of the order of 10% of the planetesimals it encounters (Tanaka & Ida 1999), with the collision fraction increasing with migration velocity. Slow migration velocities allow for inward shepherding of the planetesimals rather than capture, and a low accretion rate (Ward & Hahn 1995; Tanaka & Ida 1999).

2. A migrating core must reach the critical core mass before it is lost to the star, on a time scale that, as we noted above, can be an order of magnitude or more smaller than the gas disk lifetime. Unfortunately, a high accretion rate of planetesimals increases the critical core mass needed before runaway gas accretion starts and, at a fixed accretion rate, the critical mass also increases as the core moves inward (Papaloizou & Terquem 1999). Migration, therefore, favors a high rate of planetesimal accretion, but often hinders attaining the critical core mass needed for envelope accretion.

Calculations of giant planet formation including steady core migration have been presented by several groups, including Papaloizou & Terquem (1999), Papaloizou & Larwood

(2000), Alibert, Mordasini, & Benz (2004) and Alibert et al. (2005). The results suggest that, for a single growing core, Type I migration at the standard rate of Tanaka, Takeuchi, & Ward (2002) is simply too fast to allow giant planet formation to occur across a reasonable range of radii in the protoplanetary disk. Most cores are lost to the star or, if they manage to accrete envelopes at all, do so at such small radii that their ultimate survival is doubtful. More leisurely migration, on the other hand, at a rate suppressed from the Tanaka, Takeuchi, & Ward (2002) value by a factor of 10 to 100, helps core accretion by mitigating the depletion effect that acts as a bottleneck for a static core (Alibert et al. 2005).

This difficulty in reconciling our best estimates of the Type I migration rate with core accretion signals that something is probably wrong with one or both of these theories. Three possibilities suggest themselves. First, the Type I migration rate may be a substantial overestimate, by an order of magnitude or more. If so, there is no need for substantial changes to core-accretion theory or to protoplanetary-disk models. We have already enumerated a list of candidate physical reasons for why the Type I rate may be wrong, though achieving a sufficiently large suppression does not seem to be straightforward. Second, angular-momentum transport may be strongly suppressed in the giant-planet formation region by the low ionization fraction, which suppresses MHD instabilities that rely on coupling between the gas and the magnetic field (Gammie 1996; Sano et al. 2000). An almost inviscid disk could lower the threshold for gap opening sufficiently far that the slow stage of core accretion occurred in a gas-poor environment (elements of such a model have been explored by Rafikov 2002; Matsumara & Pudritz 2005). Finally, and perhaps most attractively, it may be possible to find a variant of core accretion that is compatible with undiluted Type I migration. For a single core, we have studied simple models in which random walk migration leads to large fluctuations in the planetesimal accretion rate and an early onset of criticality (Rice & Armitage 2003). In the more realistic situation where multiple cores are present in the disk, it is possible that the early loss of the first cores to form (at small radii just outside the snow line) could evacuate the inner disk of planetesimals, allowing subsequent cores to reach their critical mass and accrete envelopes as they migrate inward. Further work is needed to explore such scenarios quantitatively.

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