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Gravitationally Unstable Accretion
Disks
Roman Rafikov
(Princeton)
Gravitational Instability
Outline
• Evidence for the gravitationally unstable disks
• Gravitoturbulence vs fragmentation
• Properties of gravitoturbulent disks
• Constraints on fragmentation
• Applications
- Planet formation
- Star Formation in the Galactic Center
Gravitational Instability
Gravitational Instability (GI)
When a disk patch with size L starts
collapsing it has the following
contributions to energy.
To collapse need
Egr  Eth
Eth ~ L2cs2
 
2 2
G L
Egr ~
 G 2 L3
L
2
2
Erot ~ L L  2 L4
Egr  Erot
AND
G 2 L3  2 L4
G 2 L3  L2cs2
cs2
G
L 2
G

Thus, gravitational instability requires
cs 
1
G
Gravitational Instability
Gravitational Instability (GI)
Q 1
    2G | k |  k c
2
Get
2
2 2
s
 0
2
Q 1
2
Dispersion relation for density
waves in disk
2
and instability when
cs
Q
1
G
Toomre Q parameter
Q 1
k
Gravitational Instability
Greaves, Richards, Rice & Muxlow (2008)
Gravitational Instability
Observational Evidence
Planets: HD 8799
(Marois 2009)
• 3 young giant planets in almost
circular orbits around A star
• Masses around 10 M_J
• Star is 1.5 M_Sun, 40 pc away,
age 30-160 Myrs
• Projected separations between
24 AU (innermost) and 70 AU
(outer)
• Keplerian motion detected
• Probably the most compelling
case of a pristine system
Gravitational Instability
Stellar Disks in the Galactic Center
Genzel et al 2003
• Galactic Center contains
a supermassive black hole
(SMBH) with a mass of
M SMBH  3106 M sun
• Black hole’s gravity
dominates within roughly 1
pc from the center
• Inner 0.5 pc contain more
than 80 young bright O and
B stars
• Some arranged in disklike geometry (Genzel et al
2003)
Gravitational Instability
Stellar Rings
• Contain no more than M d  10 4 M sun in stars (Nayakshin et al 2005) –
otherwise rings would preccess excessively in the neighbor’s fiels
• extend from 0.05 pc to 0.5 pc (Paumard et al 2006)
• have small geometric thickness, <h/r> ~ 0.14
Levin & Beloborodov 2003
Stars in disks are
• very young, with ages of about 6
Myrs
• very massive, typically tens of
solar masses.
• lifetimes are less than 100 Myrs
• likely formed by gravitational
instability
Gravitational Instability
Hubble
Galactic disks
Kennicutt
Gravitational Instability
Fragmentation vs
Gravitoturbulence
Gravitational Instability
Disk fragmentation
Gammie (2001) showed that for fragmentation to set in one needs
tc  50
No fragmentation
tc  3
Gammie ‘01
tc  2
Fragmentation
2D hydro
1
When tc ~  fragments lose thermal support at the same rate at
which they collapse. Isothermal gas effectively has tc  0.
Gravitational Instability
3D simulations confirm this general picture
(tc  3  5).
Rice et al 2003
tcool  5
1
tcool  3
1
Gravitational Instability
Gravitoturbulent disks
Gravitational Instability
Gravitoturbulent disks

GM * 
2
F ~ 3 M  M
r
• Dissipated energy is radiated locally

• Angular momentum conservation
• By definition
cs2
 

M ~ 
&
Prescription for the angular momentum
transfer by gravitoturbulence
Fragmentation happens when
 1 !
tcool
cs2

F
1
~
tcool
Gravitational Instability
External Irradiation.
Toomre Q
Q
Rafikov 2009
1

T  Tirr
r
- parameter

1

T  Tirr
r
  3 103 , Tirr  30K
Gravitational Instability
External Irradiation.
Toomre Q
Q
Rafikov 2009
1

T  Tirr
r
- parameter

1

T  Tirr
r
  3 103 , Tirr  30K
Gravitational Instability
External Irradiation.
Toomre Q
Q
Rafikov 2009
fragmentation
1
r

- parameter

1

fragmentation
r
  3 103 , Tirr  30K
Gravitational Instability
External Irradiation.
• Disk can remain gravitationally unstable in the
presence of external irradiation
• Irradiation suppresses fragmentation
• Fragmentation is possible only at high mass
accretion rates
• In cold disks with dust opacity fragmentation is
possible in the earliest phases of disk formation, far
from the star (> 100 AU)
• At very low accretion rates disks remain viscous
everywhere
Gravitational Instability
Fragmenting disks
Gravitational Instability
Disk cooling.
Most unstable (to fragmentation) situation corresponds
to the shortest cooling time
4
Eth
c
c
k  1
tc 


   6
Fcool Fcool T
    cs
2
s
Fcool  T
2
s
4
4
T
cs2
k
Requirement that fragmentation takes place and planets
may be born then implies
4
  k  6
3  tc     cS
 
or

c 
3
6
s
k
 

4
Gravitational Instability
Fragmentation
+
GI
Instability requires
High T needed for short cooling:
  k 
 
c 
3   
4
cs
Q
1
G
6
s
Fragmentation condition
then sets a lower limit on
   k 
 

 3   
tc  3
cs :
This sets an upper limit
on c s :
4 1/ 6



 cs
   k 
 

 3   
cs 
+
4 1/ 6



 cs 
G

G

Gravitational Instability
Thermodynamical constraints
cs
GI
planet
formation

Constraint on
follows:
fragmentation
Rafikov 2005
   k 
 

 3   
naturally
4 1/ 6




G


As a result,
giant planet formation by GI requires
 
k
 

6 
 3 (G )   
7
  k
 
T 
 3G   
2
4 1/ 5



 21/ 10
 3 105 g cm  2 a AU
3/ 2 2 / 5



6 / 5
 2200 K a AU
( ~ 100 MMSN)
~T 
Sun
!
!!!
Gravitational Instability
Rafikov (2007)
With realistic opacities
f 1
Alexander et al 2005
find that planet
formation still requires
extreme properties of
protoplanetary disks!
( Cf. Boss 2004 )
Prad
MMSN
Prad
f 1
Gravitational Instability
Numerical results: grid based
Boss 2003
Boley et al 2006
Boss (2003) sees
fragmentation and
formation of bound objects
BUT
Boley (2006) do not
observe fragmentation
Gravitational Instability
Numerical results: SPH
Mayer et al 2007
Disks fragment in
simulations of Mayer et al
(2007)
Stamatellos et al 2008
BUT
They don’t in simulations
of Stamatellos &
Whitworth (2008)
Gravitational Instability
Numerical results: summary
Can’t draw any robust conclusions!
• Results depend on which method is used and which
group gets them
• No convergence between different groups
• Need to be EXTREMELY CAREFUL regarding resolution
and radiative transfer treatment (Nelson 2006)
Need numerical comparison projects !!!
Gravitational Instability
Conclusions
• Gravitational instability is important for accretion disks is
a variety of settings, from protoplanetary to galactic
• Gravitational instability results in two outcomes
depending on the cooling time: gravitoturbulence or
fragmentation
• Properties of gravitoturbulent disks can be derived
analytically
• Planet formation by gravitational instability requires
extreme properties of protoplanetary disks, but is feasible
beyond 100 AU from the star
• Star formation around SMBH in the Galactic Center is
natural at distances of 0.1 pc