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Sculpting
Circumstellar
Disks
Alice Quillen
University of Rochester
Feb 2008
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
Discovery
Space
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
Dynamical Regimes for Circumstellar
Disks with central clearings
•
Young gas rich Myr old
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 particles 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, planetary migration, rapid
change is planetary architecture, planetary
growth
This Talk:
What mass objects are required to
account for the observed clearings,
what masses are ruled out?
• 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
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
e sin v
radii give you
eccentricity
efree
v
free
eforced
v
forced
e cosv
If free eccentricity is
zero then the object has
the same eccentricity as
the forced one
v longitude of pericenter
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
e forced
b3/2 2 ( )
 1
e planet
b3/ 2 ( )
a

ap
• We require the edge of the disk to be truncated by
the planet 
 ~ 1  ering  e forced  e planet
• We consider models where eccentricity of ring and
ring edge are both caused by the planet. Contrast
with precessing ring models.
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
  N 
N
D



and removal within
a  a  t
collisionless lifetime
removal
Neptune
size
Saturn
size
What mass
planet will
clear out
objects inside
the chaos
zone fast
enough that
collisions will
not fill it in?
Mp > Neptune
Chaotic zone boundaries for particles
with zero free eccentricity
Hamiltonian at a first order mean motion resonance
H (  ; ,  )  a 2  b  c  d 1/ 21/p 2 cos( -  p )
 g 0 1/ 2 cos(   )  g11/p 2 cos(   p )
Poincare variables
 ~ e2 ,
 only depends on a
c

regular
resonance
 b
5/ 2 1
3/ 2
4
g 0    2 5/ 4 f 31
d

corotation
 5/ 2b3/2 2
2
g1    2 5/ 4 f 27
With secular terms only there is a fixed point at
  p, 
1/ 2
secular terms
b3/2 2 1/ 2
 1  p that is the e free  0 orbit
b3/ 2
Dynamics at low free eccentricity
Expand about the fixed point (the zero free
eccentricity orbit)
H (  ; I ,  )  a 2  b  cI
same as for zero  g I 1/ 2 cos(
0
eccentricity planet
goes to zero near
the planet
  )  ( g 0 1/f 2  g11/p 2 ) cos(   p )
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
Dynamics at low free eccentricity is
similar to that at low eccentricity near
a planet in a circular orbit
width of
chaotic zone
different eccentricity
points
1.5 2/ 7
planet mass
No difference in
chaotic zone width,
particle lifetimes, disk
edge velocity
dispersion low e
compared to low efree
Velocity dispersion in the disk edge
and an upper limit on Planet mass
• Distance to disk edge
set by width of chaos
zone
2/ 7
da ~ 1.5
• Last resonance that
doesn’t overlap the
corotation zone
affects velocity
dispersion in the disk
edge
• Mp < Saturn
ue ~  3/ 7
cleared out by
perturbations from
the planet
Mp > Neptune
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
nearly closed
orbits due to
collisions
eccentricity of
ring equal to that
of the planet
First Predictions for a planet just
interior to Fomalhaut’s eccentric ring
• Neptune < Mp < Saturn
• Semi-major axis 120 AU (16’’ from star)
location predicted using chaotic zone as
boundary
• 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
tcol ~  n  
1
where  n is normal optical depth
The number of collisions per orbit N c ~ 18 n
2r
n ~
f IR where f IR is fraction stellar light
dr
re-emitted in infrared
The numerical problem
• Between collisions particle is only under
the force of gravity (and < 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)
A Simple Numerical Approach
Perturbations independent of particle distribution:
• Espresante set the vr to zero during collisions. Energy
damped to circular orbits, angular momentum
conservation. However diffusion is not possible.
• We adopt
vr  0
v  v  v
• 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
Parameters of 2D simulations
N c collision rate, collisions per particle per orbit
- related to optical depth
 v tangential velocity perturbation size
- related to disk thickness

planet mass ratio
- unknown that we would like to
constrain from observations
radius
Morphology of
collisional disks
near planets
  105 , N c  102 , e  0.02
  104 , N c  103 , e  0.01
radius
• Featureless for low mass
planets, high collision
rates and velocity
dispersions
• Particles removed at
resonances in cold,
diffuse disks near massive
planets
angle
Profile shapes
chaotic zone
boundary 1.5 μ2/7
  105
  104
  106
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
  N  Nf (r )
D

r  r  tremoval
planet
  dv 
u2
where D ~
~
N

c
tcol n 2
v
 K 
2
Consider various models for removal of particles by the planet
f (r )  1  N (r )  elr
f (r )  1  r  N (r ) is an Airy function
f (r )  e  r  N (r ) is a modified Bessel function
All have exponential solutions near the planet
with inverse scale length
1/ 2
l ~ tremove
 2 / 7 N c1/ 2 dv1 and unknown function tremove
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
Reasonable well fit with the function
  
log10   0.12  0.23log10  6 
 10 
 Nc 
  dv 
 0.1log10  2   0.45log10 

 10 
 0.01 
Unfortunately this does not predict a
simple form for tremove
decrement for different
planet masses as a
function of dispersion
To truncate a disk a
planet must have
mass above
 n 
log10   6  0.43log10 
3 
 5  10 
 u / vK 
 1.95 

 0.07 
(here related to
observables)
Log Planet mass
Using the numerical
measured fit
Log Velocity dispersion
Observables can lead to planet
mass estimates, motivation for
better imaging leading to better
estimates for the disk opacity
and thickness
• Upper mass limit
confirmed by lack of
resonance clumps
• Lower mass limit
extended unless the
velocity dispersion at the
disk edge set by planet
• Velocity dispersion close
to threshold for collisions
to be destructive
Quillen 2006, MNRAS, 372, L14
Quillen & Faber 2006, MNRAS, 373, 1245
Quillen 2007, MNRAS
Log Planet mass
Application to Fomalhaut
Log Velocity dispersion
Constraints on Planetary
Embryos in
Debris Disks
AU Mic JHKL
Fitzgerald, Kalas, & Graham
h/r<0.02
• Thickness tells us the velocity dispersion in dust
• This effects efficiency of collisional cascade resulting in dust
production
• Thickness increased by gravitational stirring by massive bodies in
the disk
The size distribution and
collision cascade
observed
Figure from Wyatt
& Dent 2002
set by age of
system scaling
from dust opacity
constrained by
gravitational
stirring
Scaling from the dust:
1 q
 a 
d ln N
 N (a)  N d  
d ln a
 ad 
3 q
 a 
 (a)   d  
 ad 
(multiply by a 2 )
As tcol ~  
The top of the
cascade
1
1 3
1 q
3
 a   u 
tcol  tcol ,d    * 
 ad   2QD 
Set tcol  tage and solve for a
2
related to observables,
however exponents not
precisely known
Gravitational stirring
In sheer dominated regime
2
 
1 d i
~ s2 s
 dt
i
where  s 
s 
Solve: i (t )  t 4
s
mass density ratio
M *r 2
ms
M*
mass ratio
Comparing size distribution at top
of collision cascade to that
required by gravitational stirring
size distribution
might be flatter
than 3.5 – more
mass in high end
 runaway
growth?
top of
cascade
Comparison
between 3 disks
with resolved
vertical structure
108yr
107yr
107yr
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 Planets
Log10 time(yr)
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.
Faber & Quillen 07
How many planets?
• Between dust radius and ice line ~ 4 Neptune’s
required
• ~a Jupiter mass in planets is required to explain
clearings in all debris disk systems
• 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
• Would be interesting to extend to relaxing and
scattering planetary systems….
Transition Disks
CoKuTau/4
D’Alessio
et al. 05
4 AU
10 AU
Wavelength μm
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 wall
Clearing times
Statistics
Dust properties
Transition disks
• 5-15% of disks in clusters 1-5Myr old as
found from surveys (e.g., Muzerolle et al.)
• 50% of them are still accreting at low rates
• 50% have hot fainter, optically thin inner
dust disks
• mm observations in 2 cases have resolved
clearings (Wilner’s collaboration)
SED modeling
• Dust edge in most cases dominated by small
amorphous grains
• Density contrasts clearing/edge in dust and gas are
likely large
• Light reradiated in wall sets dust wall thickness.
Consistent with temperature predicted via radiative
transfer
• A denser disk edge  increased flux at longer
wavelengths
• However in some cases large outer disk mass is
required from far infrared fluxes
• Examples of disks without massive outer disks
(CoKuTau4) and with (DMTau)
Models for Disks with Clearings
1. Photo-ionization models (Clarke, Alexander)
Problems:
-- clearings around brown dwarfs, e.g., L316, Muzerolle et al.
-- dense transition disks
Predictions: Hole size with time and stellar UV luminosity,
clearing when disk accretion drops below a wind outflow rate
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
Predictions: Planet masses required to
hold up disk edges, and clearing
timescales, detectable edge structure
Differentiating between models
from Najita et al 07, + are
transition disks
Photo-ionization
alone cannot
explain all
objects, because
some have high
disk masses
Najita et al 07
and Alexander et
al. 07
density slice
azimuthal
angle
Vertical
motions in
the disk
edge
vz in units of
Mach number
(radius)
Minimum Gap Opening Planet
In an Accretion Disk
accretion,
optically
thick
Gapless
disks lack
planets
qmin  M 0.48 0.8 M *0.42 L*0.08
Edgar et al. 07
Minimum Gap Opening Planet Mass
in an Accretion Disk
Different mass stars
M  M *2
 =0.01
Planet
trap?
Smaller
planets can
open gaps
in selfshadowed
disks
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