Small Orbits: The Migration Imperative

Remarkably, inward (and outward) migration of protoplanets in disks had been predicted prior to the detection of the extrasolar planets, with explicit predictions by Ward (1981), Lin and Papaloizou (1986), Ward & Hourigan (1989), and Artymowicz (1993). Lin (1986) wrote, ``Once the nebula is tidally truncated, gas cannot stream from the outer region of the nebula inside the orbit of Jupiter.'' This implies that, ``Eventually, the inner region of the nebula transfers angular momentum to Jupiter at a slower rate than does Jupiter to the outer region, so that Jupiter loses angular momentum and spirals toward the protosun.'' These predictions of inward migration, a decade before discovery of 51 Peg seem clairvoyant, if not absurd, as Jupiter offers little evidence of migration.

There appear to be two possible modes of angular momentum loss. Small giant planet cores of a few Earth-masses, which cannot clear a gap, lose angular momentum via interactions with the Lindblad resonances induced in the disk (Artymowicz 1993, Ward 1997). The time scale for orbital decay in a nominal disk appears to be 300,000 yr. For a planet of 1 Mjup, Lindblad coupling may cause it to spiral into the star within less than a million years. As the planet grows by accretion up to 0.1 Mjup, the time scale may shorten below 300,000 yr, much less than the T Tauri disk ages of 3 Myr. If correct, orbital decay plays a significant role, even prior to full formation of a jupiter.

The second migration mode occurs when the protoplanet acquires a mass above 0.1 Mjup which will clear a gap in the disk. Torques between the planet and the inner and outer gap edges force the planet (and gap) to flow inward along with the viscous disk material toward the star.

Cartoon of Migration of a Giant Planet

One recent calculation of this migration for a nominal protoplanetary disk has been carreid out by Trilling et al. Astrophys. J. June 10, 1998. The time for material at 5 AU to drain into a typical T Tauri disk appears to be ~ 1 Myr, from measured values of the mass accretion rate and the disk mass. The best analyses of accretion rates come from analysis of the UV and blue excesses of T Tauri stars (Valenti et al. 1993,A.J.,106,2024; Hartmann et al. 1998, Astrophys. J. 495, 385). The measured values of mass accretion rates for a large sample of T Tauri stars in Taurus and Chamaeleon from Hartmann et al. (1997) are dM/dt = 3x10^-8 Msun/yr for typical T Tauri stars (Valenti et al. 1993, Hartmann et al. 1997). The typical total disk mass out to 100 AU is 0.03 Msun . But the disk mass within 5 AU must be less than 1/3 of 0.03 Msun, for any expected density profile (falling off with radius more gently than r^-1.9).

Thus, a parcel of material at 5 AU will drain into the star in about 1 Myr. Disk material at 5 AU hits the star in 1 Myr, assuming that there is no more than 10 Mjup within that radius in a typical T Tauri disk. The lifetime of a parcel at 5 AU can be extended to the full age of a typical T Tauri star, 3 Myr, only if there is 100 Mjup within 5 AU. Such a large disk mass implies, in turn, an efficiency for planet building of ~1%, in disagreement with the supposed efficient feeding of material onto a giant protoplanet. A giant planet, entrained in its cleared gap, must follow this draining gas inward (Ward, 1997, Lin et al. 1995). Indeed, even for parcels of material starting at 100 AU, the viscous time scale is 3 Myr, given the observed accretion rate. Matter initially at 100 AU would hit the central star in 3 Myr. At 5 AU, the material must hit the star much sooner.

Both modes of migration imply that jupiters initially at 5 AU may not survive for even 1/10 the lifetime of the disk, 3 Myr. The second mode of migration is conceivably avoidable, if the midplane is characterized by very low viscosity, and hence a longer viscous time scale. This might be the case if the viscosity results from an ionized, magnetic gas component, as in the Balbus-Hawley effect (cf Hawley et al. 1996). In that case, only the partially ionized material located above and below the midplane is actually accreting onto the star. The resulting ``dead zone'' in the midplane could allow jupiters to incubate longer than 10$^6$ yr. However, the first mode of angular momentum loss would still not be avoided. Thus, the inward migration of proto-jupiters seems difficult to circumvent theoretically (Lin 1986, Ward 1997). The survival of Jupiter itself deserves reexamination.

One possible solution to the migration problem is to acknowledge its imperative. Jupiters may form, and spiral into the star within 300,000 yr. Successive jupiters may form, each meeting the same fate (Lin, 1996). Finally, the viscous time scale is matched by the remaining lifetime of the disk, at which point the last jupiter survives, left at some intermediate point in its travel. Such a migration imperative implies, speculatively, two predictions.

First, there may be some maximum planet mass, such that the time scale for draining that mass into the star is slightly longer than the remaining lifetime of the disk. Such planets would barely survive. More massive planets may well form, constructed from more massive disks via the standard paradigm of planet formation. However, viscous or Lindblad migration in those massive disks may simply destroy those massive giant planets. Massive planets may form in young, massive disks, such as around HL Tau. But they may not survive against the draining flow into the star.

Second, inward migration implies that giant planets will be found at a range of orbital distances, 0 -- 20 AU from the host star. Their final distribution may not be uniform, as the migration speed, dr/dt, depends on the surface density profile of the disk, for a given dM/dt. The deposited jupiters might be statistically located with a probability distribution that is proportional to dtau, which in turn is proportional to the local surface mass density, \sigma (r). If sigma(r) increases inward for typical T Tauri disks, then jupiters may be found preferentially close to the star.

This idea has been explored further by Trilling et al. (Astophys. J. June 10, 1998). The inward migration may be stopped at about 0.05 AU either by tidal interactions with the spin of the star or by the clearing of the inner disk by the stellar magnetosphere (Lin et al. 1996). Trilling et al. (1998) consider the possibility that mass transfer can halt the migration, though the mechanism is difficult to implement. Another parking mechanism derives from the possibility that a number of planets orbit at different orbital distances, each in resonance with each other. Each planet delivers angular momentum to the slower-moving planet outside it, and receives angular momentum from the planet orbiting inside it. In this way, the planetary orbits remain stable, as the innermost planet receives angular momentum tidally from the spinning T Tauri star.

Halting the inward migration requires careful computation of tidal effects between planet and star (Rasio, 1996, Lubow et al. 1996), including special attention paid by L. Bildsten to dissipation in both the star and the planet. Tidal interaction is sufficiently strong to transfer spin angular momentum from the star to the orbit of the planet. The survival of these close jupiters against evaporation or UV stripping of the envelope has also been considered by Lin et al. Tidal circularization could have caused the low eccentricity for the three closest extra-solar planets, but tides could not circularize the orbits of 55 rho Cnc and rho CrB, nor of 47 UMa . If those three migrated inward, they must have stopped some other way. Perhaps, as described above, jupiters are constantly migrating inwards during the lifetime of disk, and occasionally the disk clears with the jupiter at intermediary distances, from 0.1 - 5 AU. Such migration might have been the case for Jupiter in the Solar System as well.