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What effects do 1-10 MEarth cores
have on the surrounding disk?
Today = Gaps
Friday = 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 = W 2
2"
S. Corder et al. 2005, OVRO
Local resonances can
propagate globally!
Linblad
resonance
equations:
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?
As 1-10 MEarth cores grow and
interact with the disk, what forces
are involved?
Ge/Ay133
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:
Ruden review:
When a gap opens,
the force balance
changes. The growing
planet is now tied to
the disk transport
timescale(s). For a
laminar disk:
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:
Linear 
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.)
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 km/s for
completeness. It also ignores the “hot
Jupiter pile up.” The green line is the
simple disk evolution model+migration.