AST3020.

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AST3020. Lecture 07 Migration type I,
II, III Talk by Sherry on migration Ups And and
the need for disk-planet interaction during the
formation of multiplanetary systems Torques and
migration type I Numerical calculations of gap
opening and type II situation, as well as fast
migration Test problem and difference between
codes Last Mohican scenario and the viability of
Earths

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Orbital radii masses of the extrasolar planets
(picture from 2003)
Radial migration
Hot jupiters
These planets were found via Doppler
spectroscopy of the hosts starlight. Precision
of measurement 3 m/s
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Marcy and Butler (2003)
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2005
2003
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m sin i vs. a
Blurry knowledge of exoplanets in 2006
Zones of avoidance?
multiple
single



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m sin i vs. a
Zones of avoidance? Migration?
mass
Pile-up
Distance



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Eccentricity of exoplanets vs. a and m sini
a e ? m
a e ? m
m, a, e somewhat correlated a e ?

m
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Eccentricity of exoplanets vs. a and m sini
a e ? m
a e ? m
m, a, e somewhat correlated a e ?

m
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• Upsilon Andromedae

And the question of planet-planet vs. disk planet
interaction
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The case of Upsilon And examined Stable or
unstable? Resonant? How, why?...
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Upsilon Andromedaes two outer giant planets
have STRONG interactions
Inner solar system (same scale)
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Definition of logitude of pericenter (periapsis)
a.k.a. misalignment angle
.
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Classical celestial mechanics
In the secular pertubation theory, semi-major
axes (energies) are constant (as a result of
averaging over time). Eccentricities and orbit
misalignment vary, such as to conserve the
angular momentum and energy of the system. We
will show sets of thin theoretical curves for
(e2, dw). There are corresponding (e3, dw)
curves, as well. Thick lines are numerically
computed full N-body trajectories.
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0.8 Gyr integration of 2 planetary orbits with
7th-8th order Runge-Kutta method
Initial conditions not those observed!
eccentricity
Orbit alignment angle
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Upsilon And The case of very good alignment of
periapses orbital elements practically
unchanged for 2.18 Gyr
unchanged
unchanged
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N-body (planet-planet) or disk-planet
interaction? Conclusions from modeling Ups
And 1. Secular perturbation theory and numerical
calculations spanning 2 Gyr in agreement. 2. The
apsidal resonance (co-evolution) is
expected and observed to be strong, and
stabilizes the system of two nearby, massive
planets 3. There are no mean motion resonances 4.
The present state lasted since formation
period 5. Eccentricities in inverse relation to
masses, contrary to normal N-body trend tendency
for equipartition. Alternative a lost most
massive planet - very unlikely 6. Origin still
studied, Lin et al. Developed first
models involving time-dependent axisymmetric disk
potential
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Diversity of exoplanetary systems likely a result
of cores? disk-planet
interaction a m e (only
medium) yes planet-planet interaction a
m? e
yes star-planet interaction a m
e? yes disk breakup
(fragmentation into GGP) a m e?
Metallicity no
X
X
X
X
X
X
X
X
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resonances and waves in disks, orbital
evolutionmigration type I - embedded planets

• Disk-planet interaction

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SPH (Smoothed Particle Hydrodynamics) Jupiter in
a solar nebula (z/r0.02) launches waves at LRs.
The two views are (left) Cartesian, and (right)
polar coordinates.
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Inner and Outer Lindblad resonances in an SPH
disk with a jupiter
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Laboratory of disk-satellite interaction
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A gap-opening body in a disk Saturn rings,
Keeler gap region (width 35 km) This new 7-km
satellite of Saturn was announced 11 May 2005.
To Saturn
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Migration Type I embedded in fluid
Migration Type II more in the open (gap)
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Illustration of nominal positions of Lindblad
resonances (obtained by WKB approximation. The
nominal positions coincide with the mean motion
resonances of the type m(m-1) in celestial
mechanics, which doesnt include pressure.)
Nominal radii converge toward the planets
semi-major axis at high azimuthal numbers m,
causing problems with torque calculation
(infinities!).
On the other hand, the pressure-shifted positions
are the effective LR positions, shown by the
green arrows. They yield finite total LR torque.
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Wave excitation at Lindblad resonances (roughly
speaking, places in disk in mean motion
resonance, or commensurability of periods, with
the perturbing planet) is the basis of the
calculation of torques (and energy transfer)
between the perturber and the disk. Finding
precise locations of LRs is thus a prerequisite
for computing the orbital evolution of a
satellite or planet interacting with a disk.
LR locations can be found by setting radial
wave number k_r 0 in dispersion relation of
small-amplitude, m-armed, waves in a disk.
Wave vector has radial component k_r and
azimuthal component k_theta m/r This
location corresponds to a boundary between the
wavy and the evanescent regions of a disk. Radial
wavelength, 2pi/k_r, becomes formally infinite
at LR.
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One-sided and differential torques, type I
migration
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Migration Type I, II
Underlying fig. from Protostars and Planets IV
(2000)

Time-scale (years)
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gap opening thermal criterionviscous
criterionmigration type II - non-embedded
planets

• Disk-planet interaction

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The diffusion equation for disk surface density
at work additional torque to to planet added.
Type II migration inside the gap. Speed
viscous speed (timescale t_dyn Re)
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This case illustrates the fact that outer parts
of a disk spread OUT, carrying the planet with
it. In any case, migration type II is very slow,
since the viscous time scale is 1 Myr or a
significant fraction thereof.
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Eccentricity evolution

• Disk-planet interaction

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Eccentricity in type-I situation is always
strongly damped.
Eccentricity pumping
--gt m(z/r)
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Migration Type I embedded in fluid
Migration Type III partially open (gap)
Migration Type II in the open (gap)
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Migration Type I, II, and III
Underlying fig. from Protostars and Planets IV
(2000) cf. Protostars and Planets V (2006)
this talk for type III data

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type III
Time-scale (years)
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Disk-planet interactionNumerics
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ANTARES/FIREANT Stockholm Observatory 20 cpu
(Athlons) mini-supercomputer (upgraded in 2004
with 18 Opteron 248 CPUs inside SunFire V20z
workstations)
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AMRA
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AMRA
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MNRAS (2006)
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Code comparison project EU RTN, Stockholm
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FARGO
AMRA
Comparison of Jupiter in an inviscid disk after
t100P
FLASH-AG
FLASH-AP
FLASH-AP
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RH2D
NIRVANA-GD
Jupiter in an inviscid disk t100P
RODEO
PARA-SPH
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Surface density comparison
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jupiter
vortex
L4
Surface density
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Vortensity specific vorticity vorticity /
Sigma
De Val Borro et al (2006, MNRAS)
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Disk-planet interactionhow do
supergiant planets (10 M_Jup) form?
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Gas flows through despite the gap. This
result explains the possibility of
superplanets with mass 10 MJ Migration
explains hot jupiters.
Mass flows through the gap opened by a
jupiter-class exoplanet
gt Superplanets can form
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An example of modern Godunov (Riemann solver)
code PPM VH1-PA. Mass flows through a wide
and deep gap!
Surface density Log(surface density)
Binary star on circular orbit accreting from a
circumbinary disk through a gap.
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Pan opens Encke gap in A-ring of Saturn
Shepherding by Prometheus and Pandora
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Prometheus (Cassini view)
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1. Early dispersal of the primordial nebula gt
no material, no mobility 2. Late formation
(including Last Mohican scenario)
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What the permeability of gaps tells us about our
own Jupiter - Jupiter was potentially able to
grow to 5-10 mJ, if left accreting from a
standard solar nebula for 1 Myr - the most
likely reason why it didnt the nebula was
already disappearing and not enough mass was
available. What the permeability of gaps tell us
about exoplanets - some, but not too many, grew
in disks to become superplanets - most didnt,
and we cant invoke the perfect timing
argument. One way to uderstand the ubiquitous
small exo-giants is that we see the LAST MOHICANS
(survivors from an early epoch of planet
migration and demise inside the suns).
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Disk-planet interactionDirection
rate of fast migration
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Migration Type I embedded in fluid
Migration Type II more in the open (gap)
(1980s 90s)
(1980s)
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AMR PPM (Flash) simulation of a Jupiter in a
standard solar nebula. 5 levels/subgrids.
(Peplinski and Artymowicz 2004)
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Variable-resolution PPM (Piecewise Parabolic
Method) Artymowicz 1999 Jupiter-mass
planet, fixed orbit a1, e0. White oval
Roche lobe, radius r_L 0.07 Corotational
region out to x_CR 0.17 from the planet
disk
gap (CR region)
disk
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Outward migration type III of a
Jupiter Inviscid disk with an inner clearing
peak density of 3 x MMSN Variable-resolution, ad
aptive grid (following the planet). Lagrangian
PPM. Horizontal axis shows radius in the range
(0.5-5) a Full range of azimuths on the vertical
axis. Time in units of initial orbital period.
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Simulation of a Jupiter-class planet in a
constant surface density disk with soundspeed
0.05 times Keplerian speed. PPM Piecewise
Parabolic Method Artymowicz (2000), resolution
400 x 400
Although this is clearly a type-II situation (gap
opens), the migration rate is NOT that of the
standard type-II, which is the viscous
accretion speed of the nebula.
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Consider a one-sided disk (inner disk only). The
rapid inward migration is OPPOSITE to the
expectation based on shepherding (Lindblad
resonances).
Like in the well-known problem of sinking
satellites (small satellite galaxies merging
with the target disk galaxies), Corotational
torques cause rapid inward sinking. (Gas is
trasferred from orbits inside the perturber to
the outside. To conserve angular momentum,
satellite moves in.)
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Now consider the opposite case of an inner hole
in the disk. Unlike in the shepherding case, the
planet rapidly migrates outwards.
Here, the situation is an inward-outward
reflection of the sinking satellite problem.
Disk gas traveling on hairpin (half-horeseshoe)
orbits fills the inner void and moves the planet
out rapidly (type III outward migration).
Lindblad resonances produce spiral waves and try
to move the planet in, but lose with CR torques.
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Saturn-mass protoplanet in a solar nebula disk
(1.5 times the Minimum Nebula, PPM, Artymowicz
2003)
Azimuthal angle (0-360 deg)
Type III outward migration Condition for FAST
migration disk mass in CR region planet
mass. Notice a carrot-shaped bubble of vacuum
behind the planet. Consisting of material
trapped in librating orbits, it produces CR
torques smaller than the matrial in front of the
planet. The net CR torque powers fast migration.
1
2
3
radius
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AMR PPM (FLASH). Jupiter simulation by Peplinski
and Artymowicz (in prep.). Red color marks the
fluid initially surrounding the planets orbit.
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Variable-resolution PPM (Piecewise Parabolic
Method) 1. Gas surface density, accentuating
LR-born waves (surf) 2. Vortensity, showing gas
flow (rip-tide) 0.1 Jupiter mass planet in a
z/r0.05 gas nebula Horizontal tick mark 0.1
a Corotational region out to xCR 0.08 a
away from the planet
azimuth
0.8 1 1.2 1.4 radius
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What is more important Lindblad Resonances
(waves) or Corotation?
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Impulse approximation
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No migration of planet
Outward migration
Flow fields obtained from simplification of
Hills equations of motion. (Guiding center
trajectories.)
One result xCR 2.5 rL
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Guiding center trajectories in the Hill problem
Unit of length Hill sphere Unit of
da/dt Hill sphere radius per dynamical time
AnimationEduardo Delgado
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xCR
NO MIGRATION In this frame, comoving with the
planet, gas has no systematic radial velocity
V 0, r a semi-major axis of
orbit. Symmetric horseshoe orbits,
torque 0
0
r
a
Librating Corotational (CR) region
protoplanet
disk
Librating Hill sphere (Roche lobe) region
xCR
half-width of CR region, separatrix distance
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SLOW MIGRATION In this frame, comoving with
the planet, gas has a systematic radial velocity
V - da/dt -(planet
migr.speed) asymmetric horseshoe
orbits, torque da/dt
0
r
a
FAST MIGRATION CR flow on one side of the
planet, disk flow on the other Surface
densities in the CR region and the disk are, in
general, different. Tadpole orbits, maximum
torque
0
r
a
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M
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Migration type III, neglecting LRs viscous
disk flow
independent of planet mass,
e.g., in MMSN
at a 5 AU, the type-III
time-scale 48 Porb
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Peplinski and Artymowicz (MNRAS, 2006, in
prep.) AMR code FLASH adaptive multigrid,
PPM, Cartesian coordinates local resolution up to
0.0003 a 0.0015 AU 225000 km 3 Jupiter
radii NUMERICAL CONVERGENCE when
gas given higher temperature near the planet -
results not sensitive to gravitational softening
length - or resolution
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As theorized - no significant dependence on mass
4 jupiter masses
Radius (a)
1 jupiter mass
2
Disk gap Smooth initial disk
1
0 50 100 P
time
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As theorized - no significant dependence on mass
4 jupiter masses
Radius (a)
1 jupiter mass
2
Disk gap Smooth initial disk
1
0 50 100 P
time
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ALL TORQUES RESTORED (LRs, viscous)
Outward migr.
Inward migr.
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Mass deficit
Global migration reverses at the outer boundary
Migration rate
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How can there be ANY SURVIVORS of the rapid
type-III migration?!
Migration type III
Structure in the disk gradients of
density, edges, gaps, dead zones
Migration stops, planet grows/survives
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Unsolved problem of the Last Mohican scenario of
planet survival in the solar systemCan the
terrestial zone survive a passage of a giant
planet?

• N-body simulations, N1000 (Edgar Artymowicz
  2004)
• A quiet disk of sub-Earth mass bodies reacts to
  the rapid passage of a much larger protoplanet
  (migration speed input parameter).
• Results show increase of velocity
  dispersion/inclinations and limited reshuffling
  of material in the terrestrial zone.
• Migration type III too fast to trap bodies in
  mean-motion resonances and push them toward the
  star
• Evidence of the passage can be obliterated by gas
  drag on the time scale ltlt Myr ---gt passage of
  a pre-jupiter planet(s) not exluded dynamically.

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Edges or gradients in disks Magnetic cavitie
s around the star Dead zones
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Summary of type-III migration

• New type, sometimes extremely rapid (timescale
  lt 1000 years). CRs gtgt LRs
• Direction depends on prior history, not just on
  disk properties.
• Supersedes a much slower, standard type-II
  migration in disks more massive than planets
• Conjecture modifies or replaces type-I
  migration
• Very sensitive to disk density (or vortensity)
  gradients.
• Migration stops on disk features (rings, edges
  and/or substantial density gradients.) Such edges
  seem natural (dead zone boundaries,
  magnetospheric inner disk cavities,
  formation-caused radial disk structure)
• Offers possibility of survival of giant planets
  at intermediate distances (0.1 - 1 AU),
• ...and of terrestrial planets during the passage
  of a giant planet on its way to the star.
• If type I superseded by type III then these
  conclusions apply to cores as well, not only
  giant protoplanets.

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Migration type 0 type I type II
IIb type III N-body
Interaction Gas drag Radiation
press. Resonant excitation of waves (LR) Tidal
excitation of waves (LR) Corotational flows
(CR) Gravity
Timescale of migration from 1e2 yr to disk
lifetime (up to 1e7 yr) gt 1e4 yr gt 1e5 yr gt
1e2 - 1e3 yr gt 1e5 yr (?)