Ge/Ay133 What effects do 1-10 MEarth cores have on the surrounding disk? Today = Gaps Wednesday = Migration (included here) Ge/Ay133

Disks can be unstable globally: Toomre’s criterion Q ≡ kc/(pGS) < 1 (axisymmetric perturbations) k = epicyclic frequency

Disks can be unstable globally: AB Aur k = epicyclic frequency k2 = r-3 d/dr[(r2W)2] In a Keplerian disk, where W2 = GM/r3 , k2 = W2 2" S. Corder et al. 2005, OVRO

Local resonances can propagate globally!

Linblad resonance equations:

Balance torques from tides and viscous response, or… Inner/outer tidal torque, f≈0.2 (const.) Torque from the viscous disk. Balance torques from tides and viscous response, or…

Look at time to open a gap as compared to the viscous response time scale of the disk gas. Find: Planet mass needed to open gap:

How big can gaps grow? To clear the inner disk,

Local resonances can propagate globally! Gas accretion can drive global structures in the disk.

Local resonances can propagate globally! Sufficiently large planets can create gaps, but gas accretion does continue. Can these structures assist in the formation of additional Jovian planets?

Ge/Ay133 As 1-10 MEarth cores grow and interact with the disk, what forces are involved? Ge/Ay133

The is a radial force on the planet Migration. Inner/outer tidal torque, f≈0.2 (const.) If the inner & outer torque are not balanced… The is a radial force on the planet Migration.

If you do a LINEAR analysis in a LAMINAR disk, three types of migration mechanisms emerge: Type I – “Low mass” cores w/o an induced gap. Type II – “High mass” core with an induced gap. Type III – Runaway migration in high mass disks (really needs a non-linear analysis) Type I: Ruden review: Ward, Icarus 126, 261(1997).

More on non-linear effects in a bit…

A low mass gas disk is needed to avoid driving the cores into the central star… Numerically:

More on non-linear effects in a bit…

Type II: When a gap opens, the force balance changes. The growing planet is now tied to the disk transport timescale(s). For a laminar disk: Ruden review: Recall Type I migration has Ward, Icarus 126, 261(1997).

Type II versus Type I: Type II is slower, but in a linear analysis the migration rate can still be very fast! Type III: With a massive disk, “runaway migration” can occur: Very sensitive to the mass surface density profile (can go out!).

What can we think about that might slow down migration rates? Idea #1: Turbulent disks & stochastic migration.

Idea #2: Do not use linearity assumptions! Non-linearities in the gas flow around an accreting protoplanet should scale as: Where q is the secondary to primary mass ratio and h=H/r, the disk scale height/radius. In this scenario, deviations from linearity should follow: Masset, D’Angelo & Kley (2006, ApJ, in press)

2-D and 3-D Disk simulations reveal significant non-linearity: Tests with disk viscosity. This can dramatically alter the outcome of migration. With sufficiently shallow mass surface density profiles the direction of migration can even be changed. (With a sufficiently shallow profile the migration is outward.)

Another interesting facet: Resonant systems migrate more slowly! From Armitage. Why? Couple of reasons, including wider gap opening and exchange of momentum between the interacting planets & disk.

Comparison of migration predictions to observables? The plot at left is a prediction that uses a simple disk evolutionary model with gas dissipation (more next time) and Type I+Type II migration. The histogram data at right are from the extrasolar planet sample in Fischer & Valenti (2005) that is cut at 30 m/s for completeness. It also ignores the “hot Jupiter pile up.” The green line is the simple disk evolution model+migration.