Sculpting Circumstellar

Disks

NetherlandsApril 2007

Alice QuillenUniversity of Rochester

Motivations• Planet detection via disk/planet interaction –

Complimentary to radial velocity and transit detection methods

• Rosy future – ground and space platforms• Testable – via predictions for forthcoming observations.• New dynamical regimes and scenarios compared to

old solar system• Evolution of planets, planetesimals and disks

Collaborators: Peter Faber, Richard Edgar, Peggy Varniere, Jaehong Park, Allesandro Morbidelli , Alex Moore

All extrasolar planets discovered by radial velocity (blue dots), transit (red) and microlensing (yellow) to 31 August 2004. Also shows detection limits of forthcoming space- and ground-based instruments.

Discovery space for planet detections based on disk/planet interactions

Discovery Space

Dynamical Regimes for Circumstellar Disks with central clearings

• Young gas rich accretion disks – “transitional disks” e.g., CoKuTau/4.

Planet is massive enough to open a gap (spiral density waves).

Hydrodynamics is appropriate for modeling.

Dynamical Regimes– continued

2. Old dusty diffuse debris disks – dust collision timescale is very long; e.g., Zodiacal cloud.

Collisionless dynamics with radiation pressure, PR force, resonant trapping and removal of particle in corotation region

3. Intermediate opacity dusty disks – dust collision timescale is in regime 103-104 orbital periods;

e.g., Fomalhaut, AU Mic debris disks

This Talk:

• Planets in accretion disks – The transition disks

• Planets in Debris disks with clearings – Fomalhaut

• Embryos in Debris disks without clearings– AU Mic

• Number of giant planets in old systems

What mass objects are required to account for the observed clearings, what masses are ruled out?

Transition Disks

1-3 Myr old stars with disks with central clearings, silicate emission features,discovered in young cluster surveys

Challenges to explain:Accreting vs non Dust wallClearing timesStatisticsDust properties

4 AU

10 AU

CoKuTau/4D’Alessio

et al. 05

Wavelength μm

Models for Disks with Clearings

2. Planet formation, gap opening followed by clearing (Quillen, Varniere) -- more versatile than photo-ionization models but also more complex Problems: Failure to predict dust density contrast, 3D structurePredictions: Planet masses required to hold up disk edges, and clearing timescales, detectable edge structure

1. Photo-ionization models (Clarke, Alexander) Problems:

-- clearings around brown dwarfs, e.g., L316, Muzerolle et al. -- accreting systems such as DM Tau, D’Alessio et al. -- wide gaps such as GM Aur; Calvet et al.Predictions: Hole size with time and stellar UV luminosity

Minimum Gap Opening Planet In an Accretion Disk

Edgar et al. 07

radiation

accretion, optically

thick Gapless disks lack planets

0.48 0.8 0.42 0.08min * *q M M L

Minimum Gap Opening Planet Mass in an Accretion Disk

2*

Different mass stars

M M

=0.01

Planet trap?

Smaller planets can open gaps in self- shadowed disks

Prospects with

ALMAPV plot

5km/s for a planet at 10AU

Edgar’s simulations

Fomalhaut’s eccentric

ring• steep edge profile

hz/r ~ 0.013• eccentric

e=0.11• semi-major axis

a=133AU• collision

timescale =1000 orbits based on measured opacity at 24 microns

• age 200 Myr• orbital period

1000yr

Free and forced

eccentricity

radii give you eccentricity

If free eccentricity is zero then the object has the same eccentricity as the forced one

cose v

sine v

eforced

forcedv

freevefree

longitude of pericenterv

Pericenter glow model• Collisions cause orbits to be near closed ones. This implies

the free eccentricities in the ring are small.• The eccentricity of the ring is then the same as the forced

eccentricity

• We require the edge of the disk to be truncated by the planet

• We consider models where eccentricity of ring and ring edge are both caused by the planet. Contrast with precessing ring models.

23/ 213/ 2

( )

( )forced planetp

b ae e

b a

~ 1 ring forced planete e e

Disk dynamical boundaries• For spiral density waves to be driven into a disk

(work by Espresate and Lissauer)

Collision time must be shorter than libration time

Spiral density waves are not efficiently driven by a planet into Fomalhaut’s disk

A different dynamical boundary is required• We consider accounting for the disk edge with the

chaotic zone near corotation where there is a large change in dynamics

• We require the removal timescale in the zone to exceed the collisional timescale.

Chaotic zone boundary and removal within

What mass planet will clear out objects inside the chaos zone fast enough that collisions will not fill it in?

Mp > Neptune

Saturn size

Neptune size

removal

N ND

a a t

colli

sion

less

life

time

Chaotic zone boundaries for particles with zero free eccentricity

Hamiltonian at a first order mean motion resonance

2 1/ 2 1/ 2

1/ 2 1/ 20 1

5/ 2 1 5/ 2 23/ 2 3/ 2

5/ 4 5/ 40 31 1 27

( ; , ) cos( )

cos( ) cos( )

4 2

2 2

With secular terms only there is

p p

p p

H a b c d

g g

c b d b

g f g f

-

21/ 2 1/ 23/ 2

13/ 2

a fixed point at

, that is the 0 orbitp p free

be

b

corotationregular

resonance

secular terms2

Poincare variables

~ ,

only depends on

e

a

Dynamics at low free eccentricity

Expand about the fixed point (the zero free eccentricity orbit)

For particle eccentricity equal to the forced eccentricity and low free eccentricity, the corotation resonance cancels recover the 2/7 law, chaotic zone same width

2

1/ 2 1/ 2 1/ 20 0 1

( ; , )

cos( ) ( ) cos( )f p p

H I a b cI

g I g g

goes to zero near the planet

same as for zero eccentricity planet

Dynamics at low free eccentricity is similar to that at low eccentricity near a planet in a circular orbit

No difference in chaotic zone width, particle lifetimes, disk edge velocity dispersion low e compared to low efreeplanet mass

wid

th o

f ch

ao

tic z

one

different eccentricity points

2/ 71.5

3/ 7~eu

Velocity dispersion in the disk edgeand an upper limit on Planet mass

• Distance to disk edge set by width of chaos zone

• Last resonance that doesn’t overlap the corotation zone affects velocity dispersion in the disk edge

• Mp < Saturn

2/ 7~ 1.5da

cleared out by perturbations

from the planet

Mp > Neptune

nearly closed orbits due to

collisions

eccentricity of ring equal to that of the

planet

Assume that the edge of the ring is the boundary of the

chaotic zone. Planet can’t be too massive otherwise the

edge of the ring would thicken Mp < Saturn

First Predictions for a planet just interior to Fomalhaut’s eccentric ring

• Neptune < Mp < Saturn• Semi-major axis 120 AU (16’’ from star)• Eccentricity ep=0.1, same as ring• Longitude of periastron same as the ring

The Role of Collisions• Dominik & Decin 03 and Wyatt 05

emphasized that for most debris disks the collision timescale is shorter than the PR drag timescale

• Collision timescale related to observables

1~ where is normal optical depth

The number of collisions per orbit ~ 18

2~ where is fraction stellar light

re-emitted in infrared

col n n

c n

n IR IR

t

N

rf f

dr

The numerical problem

• Between collisions particle is only under the force of gravity (and possibly radiation pressure, PR force, etc)

• Collision timescale is many orbits for the regime of debris disks 100-10000 orbits.

Numerical approaches

• Particles receive velocity perturbations at random times and with random sizes independent of particle distribution (Espresante & Lissauer)

• Particles receive velocity perturbations but dependent on particle distribution (Melita & Woolfson 98)

• Collisions are computed when two particles approach each other (Charnoz et al. 01)

• Collisions are computed when two particles are in the same grid cell – only elastic collisions considered (Lithwick & Chiang 06)

Our Numerical ApproachPerturbations independent of particle distribution:

• Espresate set the vr to zero during collisions. Energy damped to circular orbits, angular momentum conservation. However diffusion is not possible.

• We adopt

• Diffusion allowed but angular momentum is not conserved!• Particles approaching the planet and are too far away are

removed and regenerated • Most computation time spent resolving disk edge

0rv

v v v

Parameters of 2D simulations

collision rate, collisions per particle per orbit

- related to optical depth

tangential velocity perturbation size

- related to disk thickness

planet mass ratio

- unknown th

c

v

N

at we would like to

constrain from observations

Morphology of collisional disks

near planets• Featureless for low mass

planets, high collision rates and velocity dispersions

• Particles removed at resonances in cold, diffuse disks near massive planets

5 210 , 10 , 0.02cN e

4 310 , 10 , 0.01cN e

angle

radi

us

radi

us

Profile shapes

410

510

610

chaotic zone boundary 1.5 μ2/7

Rescaled by distance to chaotic zone boundary

Chaotic zone probably has a role in setting a length scale but does not completely determine the profile shape

Diffusive approximations

22

2

( ) where ~ ~

Consider various models for removal of particles by the planet

( ) 1 ( )

( ) 1 ( ) is an Airy function

( ) ( ) is

dvc

removal col Kplanet

lr

r

N Nf r uD D N

r r t t n v

f r N r e

f r r N r

f r e N r

1/ 2 2 / 7 1/ 2 1

a modified Bessel function

All have exponential solutions near the planet

with inverse scale length

~ and unknown function remove c dv removel t N t

Density decrement

• Log of ratio of density near planet to that outside chaotic zone edge

• Scales with powers of simulation parameters as expected from exponential model

10 10 6

10 102

Reasonable well fit with the function

log 0.12 0.23log10

0.1log 0.45log10 0.01

c dvN

Unfortunately this does not predict a nice form for tremove

Using the numerical measured fitTo truncate a disk a planet must have mass above

(here related to observables)

310 105 10

log 6 0.43log

/ 1.95

0.07

n

Ku v

Observables can lead to planet

mass estimates, motivation for better imaging leading to better estimates for the disk opacity and thickness

N c=10

-3

α=0.001

Log

Pla

net

ma

ssLog Velocity dispersion

N c=10-2

Application to Fomalhaut

• Upper mass limit confirmed by lack of resonance clumps

• Lower mass extended lower unless the velocity dispersion at the disk edge set by planet

• Velocity dispersion close to threshold for collisions to be destructive

Log Velocity dispersion

Log

Pla

net

mas

s

Quillen 2006, MNRAS, 372, L14 Quillen & Faber 2006, MNRAS, 373, 1245 Quillen 2007, astro-ph/0701304

Constraints on Planetary Embryos in Debris Disks

AU Mic JHKLFitzgerald, Kalas, & Graham

•Thickness tells us the velocity dispersion in dust

•This effects efficiency of collisional cascade resulting in dust production

•Thickness from gravitational stirring by massive bodies in the disk

h/r<0.02

The size distribution and collision cascade

Figure from Wyatt & Dent 2002

set by age of system scaling from dust opacity

constrained by gravitational stirring

observed

The top of the cascade

1

3

2

1

11 3 2 3

, *

Scaling from the dust:

ln( )

ln

( )

(multiply by )

As ~

2

Set and solve for

q

dd

q

dd

col

q

col col dd D

col age

d N aN a N

d a a

aa

a

a

t

a ut t

a Q

t t a

Gravitational stirring

2

22*

s*

4

In sheer dominated regime

1~ where mass density ratio

mass ratio

Solve: ( )

s s ss

s

d i

dt M ri

m

M

i t t

Comparing size distribution at top of collision cascade to that

required by gravitational stirring

>10o

bjec

ts

gravitation stirring

top of cascade

Hill sphere limit

size distribution might be flatter than 3.5 – more mass in high end runaway growth?

> 10

obje

cts

Comparison between 3 disks

with resolved vertical structure

107yr

107yr108yr

Debris Disk Clearing

• Spitzer spectroscopic observations show that dusty disks are consistent with one temperature, hence empty within a particular radius

• Assume that dust and planetesimals must be removed via orbital instability caused by planets

Disk Clearing by PlanetsL

og1

0 ti

me

(yr)

Faber & Quillen 07

μ=10

-7

μ=10

-3

Simple relationship between spacing, clearing time and planet mass

Invert this to find the spacing, using age of star to set the stability time.

Stable planetary system and unstable planetesimal ones.

How many planets?

• Between dust radius and ice line ~ 4 Neptune’s required

• Spacing and number is not very sensitive to the assumed planet mass

• It is possible to have a lot more stable mass in planets in the system if they are more massive

Summary

• Quantitative ties between observations, mass, eccentricity and semi-major axis of planets residing in disks

• In gapless disks planets can be ruled out – but we find preliminary evidence for embryos and runaway growth

• The total mass in planets in most systems is likely to be high, at least a Jupiter mass

• Better understanding of collisional regime• More numerical and theoretical work inspired by these

preliminary crude numerical studies• Exciting future in theory, numerics and observations