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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3

a aq

52.

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aq gapdsk

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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