4.1. Conditions for the onset of Type II migration
In a viscous disk, the threshold between Type I and Type II migration can be derived by equating the time scale for Type I torques to open a gap (in the absence of viscosity) with the time scale for viscous diffusion to fill it in (Goldreich & Tremaine 1980; Papaloizou & Lin 1984). For a gap of width Ar around a planet with mass ratio q = Mp/Mt, orbiting at distance rp, Type I torques can open the gap on a timescale (Takeuchi, Miyama, & Lin 1996)
Viscous diffusion will close the gap on a time scale,
where v, the kinematic viscosity,is usually written as v = ac2/qp (Shakura & Sunyaev 1973). Equating Topen and Tclose, and noting that waves with m « rp/h dominate the Type I torque, the condition for gap opening becomes, q > (h V a1/2 .
For h/r ~ 0.05 and a = 10~2, the transition (which simulations show is not very sharp) occurs at qcrit ~ 2.5 x 10~4, i.e., close to a Saturn mass for a Solar-mass star. Since the most rapid Type I migration occurs when q « qcrit, we can also estimate a minimum migration time scale by combining the above expression with the time scale formula of Tanaka, Takeuchi, & Ward (2002). This yields,
Tmin ~ (2.7+1.1^)-1EMia-1/2Q-1 , and is almost independent of disk properties other than the local mass.
4.2. The rate of Type II migration Once a planet has becomes massive enough to open a gap, orbital evolution is predicted to occur on the same local time scale as the protoplanetary disk. The radial velocity of gas in the disk is,
which for a steady disk away from the boundaries can be written as,
If the planet enforces a rigid tidal barrier at the outer edge of the gap, then evolution of the disk will force the orbit to shrink at a rate rp ~ vr, provided that the local disk mass exceeds the planet mass, i.e. nrpE > Mp. This implies a nominal Type II migration time scale, valid for disk dominated migration only,
For h/r = 0.05 and a =10 2, the migration time scale at 5 AU is of the order of 0.5 Myr.
In practice, the assumption that the local disk mass exceeds that of the planet often fails. For example, a ¡3 =1 disk with a mass of 0.01 Mq within 30 AU has a surface density profile,
The condition that nrpE = Mp gives an estimate of the radius within which disk domination ceases of, r = 6(g) AU .
Interior to this radius, the planet acts as a slowly moving barrier which impedes the inflow of disk gas. Gas piles up behind the barrier—increasing the torque—but this process does not continue without limit because the interaction also deposits angular momentum into the disk, causing it to expand (Pringle 1991). The end result is to slow migration compared to the nominal rate quoted above. For a disk in which the surface density can be written as a power-law in accretion rate and radius,
Syer & Clarke (1995) define a measure of the degree of disk dominance,
Then for B < 1 (the planet dominated case appropriate to small radii) the actual Type II migration rate is (Syer & Clarke 1995),
Note that with this definition of B, disk dominance extends inward a factor of a few further than would be predicted based on the simple estimate given above.
Evaluating how the surface density depends upon the accretion rate—and thereby determining the a which enters into the suppression term—requires a full model for the protoplanetary disk (not just knowledge of the slope of the steady-state surface density profile). For the disk models of Bell et al. (1997), we find that a ~ 0.5 at 1 AU for M ~ 10~8 Mq yr_1. At this radius, the model with a = 10~2 has a surface density of about 200 gcm~2. For a Jupiter-mass planet we then have B = 0.3 and tjj = 1.5t0. This is a modest suppression of the Type II rate, but the effect becomes larger at smaller radii (or for more massive planets). It slows the inward drift of massive planets, and allows a greater chance for them to be stranded at sub-AU scales as the gas disk is dissipated.
These estimates of the Type II migration velocity assume that once a gap has been opened, the planet maintains an impermeable tidal barrier to gas inflow. In fact, simulations show that planets are able to accrete gas via tidal streams that bridge the gap (Lubow, Seibert & Artymowicz 1999). The effect is particularly pronounced for planets only just massive enough to open a gap in the first place. If the accreted gas does not have the same specific angular momentum as the planet, this constitutes an additional accretion torque in addition to the resonant torque. It is likely to reduce the Type II migration rate further.
Simulations of gap opening and Type II migration have been presented by a large number of authors, with recent examples including Bryden et al. (1999), Lubow, Seibert, & Artymowicz (1999), Nelson et al. (2000), Kley, D'Angelo, & Henning (2001), Papaloizou, Nelson, & Masset (2001), D'Angelo, Henning, & Kley (2002), D'Angelo, Kley, & Henning (2003), D'Angelo, Henning, & Kley (2003), Bate et al. (2003), Schafer et al. (2004) and Lufkin et al. (2004). These authors all assumed, for simplicity, that angular momentum transport in the protoplanetary disk could be represented using a microscopic viscosity. Only a few recent simulations, by Winters, Balbus, & Hawley (2003b), Nelson & Pa-paloizou (2004) and Papaloizou, Nelson, & Snellgrove (2004), have directly simulated the interaction of a planet with a turbulent disk. For planet masses significantly above the gap opening threshold, simulations support the general scenario outlined above, while also finding:
1. Significant mass accretion across the gap. For planet masses close to the gap opening threshold, accretion is surprisingly efficient, with tidal streams delivering gas at a rate that is comparable to the disk accretion rate in the absence of a planet (Lubow, Seibert, & Artymowicz 1999). It is also observed that accretion cuts off rapidly as the planet mass increases toward 10 Mj, giving additional physical motivation to the standard dividing line between massive planets and brown dwarfs.
2. Damping of eccentricity. Goldreich & Tremaine (1980) noted that the evolution of the eccentricity of a planet embedded within a disk depends upon a balance between Lindblad torques, which tend to excite eccentricity, and corotation torques, which damp it. Recent analytic work (Ogilvie & Lubow 2003; Goldreich & Sari 2003; and references therein) has emphasized that if the corotation resonances are even partially saturated, the overall balance tips to eccentricity excitation. To date, however, numerical simulations (e.g., those by Papaloizou, Nelson & Masset 2001) exhibit damping for planetary mass bodies, while eccentricity growth is obtained for masses Mp > 20 MJ —in the brown dwarf regime. In a recent numerical study, however, Masset & Ogilvie (2004) present evidence that the resolution attained by Papaloizou, Nelson, & Masset (2001) was probably
semi-major axis / AU
semi-major axis / AU
Figure 4. Left hand panel: the distribution of extrasolar planets discovered via radial velocity surveys in the a-Mp sin(j) plane. The dashed diagonal traces a line of equal ease of detectability—planets on circular orbits lying along lines parallel to this cause the same amplitude of stellar radial velocity variations. Right hand panel: the number of detected planets with Mp sin(j) > 1 Mj is plotted as a function of semi-major axis. The solid curve shows the predicted distribution from a pure migration model by Armitage et al. (2002). The theoretical curve is unaltered from the 2002 version except for an arbitrary normalization.
inadequate to determine the sign of eccentricity evolution for Jupiter mass planets, which remains uncertain. Further numerical work is needed.
4.4. Comparison with statistics of extrasolar planetary systems The estimated time scale for migration of a giant planet from 5 AU to the hot Jupiter region is of the order of a Myr. This time scale is short enough—compared to the lifetime of typical protoplanetary disks—to make migration a plausible origin for hot Jupiters, while not being so short as to make large-scale migration inevitable (the latter would raise obvious concerns as to why there is no evidence for substantial migration of Jupiter itself). Having passed this initial test, it is then of interest to try and compare quantitative predictions of migration, most obviously the expected distributions of planets in mass and orbital radius, with observations. Models that attempt this include those by Trilling et al. (1998), Armitage et al. (2002), Trilling, Lunine, & Benz (2002), Ida & Lin (2004a) and Ida & Lin (2004b). Accurate knowledge of the biases and selection function of the radial velocity surveys is essential if such exercises are to be meaningful, making analyses such as those of Cumming, Marcy, & Butler (1999) and Marcy et al. (2005) extremely valuable.
Figure 4 shows how the distribution of known extrasolar planets with semi-major axis compares to the pure migration model of Armitage et al. (2002). In this model, we assumed that giant planets form and gain most of their mass at an orbital radius (specifically 5 AU) beyond where most extrasolar planets are currently being detected. Once formed, planets migrate inward via Type II migration and are either (a) swallowed by the star, or (b) left stranded at some intermediate radius by the dispersal of the protoplane-tary disk. We assumed that disk dispersal happens as a consequence of photoevaporation (Johnstone, Hollenbach, & Bally 1998), and that the probability of planet formation per unit time is constant over the (relatively short) window during which a massive planet can form at 5 AU and survive without being consumed by the star. Although clearly oversimplified, it is interesting that this model continues to reproduce the orbital dis-
—, I I I I I I i I I I I I r o - da/dt = 1.0 X 10~7 (a/AU)1/2 (Mpl/Me) AU/yr
Figure 5. The minimum metallicity required to form a gas giant planet as a function of orbital radius, from simplified core accretion calculations by Rice & Armitage (2005). Host metallicity, expressed by the parameter /aust, is proportional to 10tFe/Hl. The models assume that dispersion in disk metallicity dominates over dispersion in either disk gas mass or disk lifetime in controlling the probability of planet formation. Type I migration of the core prior to accretion of the envelope is included, using several different prescriptions indicated by the different symbols, though none of the Type I rates are as large as the baseline rate of Tanaka, Takeuchi, & Ward (2002).
tribution of known planets out to radii of a few AU, once selection effects have been taken into account (here, by simply ignoring low-mass planets that are detectable only at small orbital radii). Moreover, it predicts that substantial outward migration ought to occur in disks where strong mass loss prompts an outwardly directed radial velocity in the giant planet forming region (Veras & Armitage 2004). Planets at these large radii are potentially detectable today via their effect on debris disks (Kuchner & Holman 2003).
Additional clues to the role of migration in forming the observed population of extrasolar planets may be possible by combining large planet samples with knowledge of the host stars' metallicity. It is now clear that the frequency of detected planets increases rapidly with host metallicity, and that the measured metallicity reflects primarily the primordial composition of the star-forming gas rather than subsequent pollution of the convective envelope (Santos, Israelian, & Mayor 2004; Fischer & Valenti 2005; and references therein). The existence of such a correlation is not in itself surprising, given that the time scale for core accretion decreases quickly with increasing surface density of planetesimals. However, the sharpness of the rise in planet frequency over a fairly narrow range of stellar [Fe/H] is striking, since it suggests that metallicity, rather than variations in initial gas disk mass or gas disk lifetime, may well be the single most important parameter determining the probability of giant planet formation around a particular star.
Motivated by these observations, we have investigated the use of the critical or threshold metallicity for giant planet formation as a discriminant of different planet formation models (Rice & Armitage 2005). Using simplified core accretion models similar to those of Ida & Lin (2004a), we have calculated the radial dependence of the threshold metallic-ity under the assumption that disks around different stars have similar initial gas masses and lifetimes (this could be replaced with the much weaker assumption that the gas mass and disk lifetime are not correlated with the metallicity). A sample result is shown in Figure 5. By definition, planets that just manage to form as the gas disk is being lost suffer little or no Type II migration, so delineating the threshold metallicity curve ob-servationally can yield information on planet formation that is independent of Type II migration uncertainties. We find that the most important influence on the shape of the threshold metallicity curve is probably Type I core migration prior to accretion of the gas envelope. When this is included, we derive a monotonically rising, minimum-metallicity curve beyond about 2 AU. In the absence of significant core migration, the threshold metallicity is flat beyond the snow line (with a weak dependence on the surface density profile of planetesimals), and the location of the snow line may be preserved in the observed distribution of planets in the orbital radius/metallicity plane.
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