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