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 openingand type II situation, as well as fast migrationTest problem and difference between codes
Last Mohican scenario and the viability of Earths
Orbital radii + masses of the extrasolar planets (picture from 2003)
These planets were foundvia Doppler spectroscopyof the host’s starlight.
Precision of measurement:~3 m/s
Hot jupitersRadial migration
Marcy and Butler (2003)
~2003
2005
m sin i vs. a
Zones ofavoidance?
multiple
single
Blurry knowledgeof exoplanetsin 2006
m sin i vs. aZones ofavoidance?
Migration?
Distance
mass
Pile-up
Eccentricity of exoplanets vs. a and m sini
m, a, e somewhat correlated: a e ? m
a e ? m
a e ? m
Eccentricity of exoplanets vs. a and m sini
m, a, e somewhat correlated: a e ? m
a e ? m
a e ? m
Upsilon Andromedae
And the question of planet-planet vs. disk planet interaction
The case of Upsilon And examined: Stable or unstable? Resonant? How, why?...
Upsilon Andromedae’s two outer giant planets have STRONG interactions
Innersolarsystem(samescale)
.
1
2Definition of logitude of pericenter (periapsis) a.k.a.misalignment angle
In the secular pertubation theory, semi-major axes (energies) are constant (as a result of averaging over time).
Eccentricities and orbit misalignment vary, such asto 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.
Classical celestial mechanics
ecce
ntr
icit
y
Orbit alignment angle
0.8 Gyr integration of 2 planetary orbitswith 7th-8th order Runge-Kutta method
Initial conditionsnot those observed!
Upsilon And: The case of very good alignment of periapses: orbital elements practically unchanged for 2.18 Gyr
unchanged
unchanged
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 expectedand observed to be strong, and stabilizes the systemof two nearby, massive planets3. There are no mean motion resonances4. The present state lasted since formation period5. Eccentricities in inverse relation to masses, contrary to normal N-body trend tendency for equipartition.Alternative: a lost most massive planet - very unlikely6. Origin still studied, Lin et al. Developed first modelsinvolving time-dependent axisymmetric disk potential
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
XX X
X X
X
X
resonances and waves in disks, orbital evolution
migration type I - embedded planets
Disk-planet interaction
.
.
.
SPH (Smoothed Particle Hydrodynamics)Jupiter in a solar nebula (z/r=0.02) launches waves at LRs. The two views are (left) Cartesian, and (right) polar coordinates.
Inner and Outer Lindblad resonances in an SPH disk with a jupiter
Laboratory of disk-satellite interaction
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
Migration Type I :embedded in fluid
Migration Type II :more in the open (gap)
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 doesn’t include pressure.) Nominal radii converge toward the planet’s 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.
Wave excitation at Lindblad resonances (roughly speaking,places in disk in mean motion resonance, or commensurabilityof 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 isthus 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 numberk_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 andthe evanescent regions of a disk. Radial wavelength, 2*pi/k_r, becomes formally infinite at LR.
One-sided and differential torques, type I migration
Migration Type I, II
Tim
e-sc
a le
( y
e ars
)
Underlying fig. from: “Protostars and Planets IV (2000)”;
gap opening:
thermal criterionviscous criterion
migration type II - non-embedded planets
Disk-planet interaction
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)
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.
Eccentricity evolution
Disk-planet interaction
--> m(z/r)Ecc
entr
icit
y pu
mpi
ng
Eccentricity in type-Isituation is always strongly damped.
Migration Type I :embedded in fluid
Migration Type III partially open (gap)
Migration Type II :in the open (gap)
Migration Type I, II, and III
Tim
e-sc
a le
( y
e ars
)
Underlying fig. from: “Protostars and Planets IV (2000)”;cf. “Protostars and Planets V (2006)” & this talk for type III data
type III
?
Disk-planet interaction:
Numerics
ANTARES/FIREANT
Stockholm Observatory
20 cpu (Athlons)mini-supercomputer
(upgraded in 2004 with 18 Opteron 248CPUs inside SunFireV20z workstations)
AMRA
AMRA
MNRAS (2006)
Code comparison project: EU RTN, Stockholm
AMRA FARGO
FLASH-AG FLASH-AP
Comparison of Jupiter in an inviscid disk after t=100P
FLASH-AP
RH2DNIRVANA-GD
PARA-SPH
Jupiter in an inviscid disk t=100P
RODEO
Surface density comparison
jupitervortex
L4
Sur
face
den
sity
Vortensity = specific vorticity = vorticity / Sigma
De Val Borro et al(2006, MNRAS)
Disk-planet interaction:
how do supergiant planets (~10 M_Jup) form?
Mass flows through the gapopened by a jupiter-class exoplanet
==> Superplanets can form
Gas flows through& despite the gap.
This resultexplains the possibility of “superplanets” with mass ~10 MJ
Migration explains
hot jupiters.
Binary star on circular orbitaccreting from a circumbinary disk through a gap.
Surface density Log(surface density)
An example of modern Godunov (Riemann solver) code:PPM VH1-PA. Mass flows through a wide and deep gap!
Shepherding by
Prometheus and Pandora
Pan opens Encke gap in A-ring of Saturn
Prometheus (Cassini view)
1. Early dispersal of the primordial nebula ==> no material, no mobility2. Late formation (including Last Mohican scenario)
What the permeability of gaps tells us aboutour 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 didn’t: 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 didn’t, and we can’t invoke the perfect timing argument.One way to uderstand the ubiquitous small exo-giants is thatwe see the LAST MOHICANS (survivors from an earlyepoch of planet migration and demise inside the suns).
Disk-planet interaction:
Direction & rate of fast migration
Migration Type I :embedded in fluid
Migration Type II :more in the open (gap)
(1980s & 90s) (1980s)
Migration Type III partially open (gap)
(2003)
AMR PPM (Flash) simulation of a Jupiter in a standard solar nebula. 5 levels/subgrids.(P
epli
n sk i
and
Art
ymo w
icz
2 00 4
)
Variable-resolutionPPM (Piecewise Parabolic Method)[Artymowicz 1999]
Jupiter-mass planet,fixed orbit a=1, e=0.
White oval = Roche lobe, radius r_L= 0.07
Corotational region outto x_CR = 0.17 from the planet
disk
disk gap (CR region)
Outward migration type IIIof a Jupiter
Inviscid disk with an inner clearing & peak density of 3 x MMSN
Variable-resolution,adaptive grid (following the planet). Lagrangian PPM.
Horizontal axis showsradius in the range (0.5-5) a
Full range of azimuthson the vertical axis.
Time in units of initialorbital period.
Simulation of a Jupiter-class planet in a constant surface density disk with soundspeed = 0.05 times Keplerian speed.PPM = Piecewise Parabolic MethodArtymowicz (2000),resolution 400 x 400
Although this is clearly a type-II situation (gap opens), the migrationrate is NOT that of the standard type-II, which is the viscous accretionspeed of the nebula.
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.)
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.
Saturn-mass protoplanet in a solar nebula disk (1.5 times the Minimum Nebula,PPM, Artymowicz 2003)
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 trappedin librating orbits, it producesCR torques smaller than the matrial in front of the planet. The net CR torque powers fast migration.
radius1 2 3
Azimuthalangle (0-360 deg)
AMR PPM (FLASH). Jupiter simulation by Peplinski and Artymowicz (in prep.).Red color marks the fluid initially surrounding the planet’s orbit.
Variable-resolutionPPM (Piecewise Parabolic Method)
1. Gas surface density,accentuating LR-born waves (surf)2. Vortensity, showinggas flow (rip-tide)
0.1 Jupiter mass planetin a z/r=0.05 gas nebula
Horizontal tick mark = 0.1 a
Corotational region out
to xCR = 0.08 a away from the planet
0.8 1 1.2 1.4 radius
azim
uth
What is more important: Lindblad Resonances (waves) or Corotation?
Impulse approximation
No migration of planet Outward migration
Flow fields obtained from simplification of Hill’s equations of motion. (Guiding center trajectories.)
One result: xCR ~ 2.5 rL
Animation:Eduardo Delgado
Guiding center trajectories in the Hill problem
Unit of length = Hill sphere
Unit of da/dt = Hill sphere radiusper dynamical time
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
r
a
0
disk
Librating Corotational (CR) region
Librating Hill sphere (Roche lobe) region
xCR
protoplanet
xCR = half-width of CR region, separatrix distance
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
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
r
r
a
0
0
a
M
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M CR 324 /~
Migration type III, neglecting LRs & viscous disk flow
independentof planet mass, e.g., in MMSN at a= 5 AU, the type-III time-scale = 48 Porb
Peplinski and Artymowicz (MNRAS, 2006, in prep.)
AMR code FLASH adaptive multigrid, PPM,Cartesian coordinateslocal 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
time
Radius (a)
Dis
k g
ap
Sm
ooth
init
ial d
isk
4 jupiter masses1 jupiter mass
0 50 100 P
1
2
As theorized - no significant dependence on mass:
time
Radius (a)
Dis
k g
ap
Sm
ooth
init
ial d
isk
4 jupiter masses1 jupiter mass
0 50 100 P
1
2
As theorized - no significant dependence on mass:
Outward migr.
Inward migr.
ALL TORQUES RESTORED (LRs, viscous)
Massdeficit
Migration rate
Global migrationreverses at the outer boundary
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
Unsolved problem of the Last Mohican scenario of planet survival in the solar system:Can the terrestial zone survive a passage of a giant planet?
N-body simulations, N~1000 (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 << Myr ---> passage of a pre-
jupiter planet(s) not exluded dynamically.
Edges or gradients in disks:
Magneticcavities aroundthe star
Dead zones
Summary of type-III migration New type, sometimes extremely rapid (timescale < 1000
years). CRs >> 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.
Migration:
type 0
type I
type II & IIb
type III
N-body
Timescale of migration:
from ~1e2 yr to disk lifetime (up to 1e7 yr)
> 1e4 yr
> 1e5 yr
> 1e2 - 1e3 yr
> 1e5 yr (?)
Interaction:
Gas drag + Radiation press.
Resonant excitation of waves (LR)
Tidal excitation of waves (LR)
Corotational flows (CR)
Gravity