The Formation and Long-term Evolution of Circumstellar
Disks
Shantanu Basu
The University of Western Ontario, Canada
&
Eduard Vorobyov (ICA, St. Mary’s U., Canada)
Isaac Newton Institute DDP Program seminarSeptember 24, 2009
The Envelope-Disk Connection
FU OriFUor’s are YSO’s with significant circumstellar material.
Calvet, Hartmann, & Strom (1999)C. Briceno
Typical disk accretion:
FU Ori: yr10
yr10104
78
sun
sun
M
M
Empirical Inference of YSO Accretion History
Observed frequency of FU Ori eruptions (last 50 years) is several times greater than the low-mass star formation rate within 1 kpc It is thought that all YSO’s undergo multiple eruptions.
Hartmann (1998), based on Kenyon et al. (1990)
New evidence from Spitzer (very low luminosity objects – VeLLO’s; Enoch et al. 2008) also reveals need for episodic accretion
Gravitational instability of a gaseous disk • The stability properties of gas disks are often expressed in terms of
the Toomre Q-parameter (Toomre 1964)
• If Q > 2 the disk is stable (but still may have low-amplitude non-axisymmetric density perturbations).
• If 1 < Q < 2 the disk is unstable and can develop observationally meaningful non-axisymmetric structure.
• If Q < 1 the disk is vigorously unstable and can fragment into self-gravitating clumps.
Fragment formation ultimately depends also upon cooling and heating rates (Gammie 2001; also, Lodato, Rice, Durisen, Pickett, Boss, and others) and/or upon mass accretion onto the disk (Vorobyov, Basu)
G
cQ s
Global Core Disk Formation/Accretion Simulations
• Our model is global, nonaxisymmetric, and includes disk self-gravity. Outer boundary at ~ 104 AU, i.e. prestellar core.•Integrate vertically (in z-direction) through cloud. Solve time-dependent equations for profiles in (r,) directions. IC’s from self-similar core collapse calculations. • With nonuniform mesh, can study large dynamic range of spatial scales, ~ 104 AU down to several AU • Allows efficient calculation of long-term evolution even with very small time stepping due to nonuniform mesh. Can study disk accretion for ~ 106 yr rather than ~103 yr (for 3D)• Can run a very large number of simulations – for statistics and parameter study • Last two still not possible for 3D simulations
We Employ the Thin-Disk Approximation Vorobyov & Basu (2006):
What’s not included in this model (for now)
• Magnetic braking
• Ambipolar diffusion, Ohmic dissipation, Hall term
• Model for inner disk (~ 5 AU) inside central sink cell
• Magnetorotational instability (can’t occur in thin-disk model)
• Stellar irradiation effects on disk
• Radiative transfer in disk - we use P= P(), barotropicrelation
• Photoevaporation of outer disk
Schematic from Armitage, Livio, & Pringle (2001)
2 2
1
vertically integrated gas pressure
( ),
1( ) P ,
P c c ,
p p
pp p p p p
s s crcr
p r
p
t
t
u r u
r rr
u
uu u
u
2 2
( , )( , )
2 cos( )
i j i i ji j
i j i i i i j j
r r rr G
r r r r
Basic Equations
2D convolution theorem (Binney & Tremaine, Galactic Dynamics) very useful to model isolated objects
Core initial conditions
0 0
2 20
22
00
0
,
2 1 1 .
r
r r
r r
r r
Pick r0 , so that core is mildly gravitationally unstable initially.
These profiles represent best analytic fits (Basu 1997) to axisymmetric models of magnetically supercritical core collapse (e.g. Basu & Mouschovias 1994).
3 30
4
-1 -1 14 -10
6.4 10 pc 1.3 10 AU
0.05 pc 10 AU
1.5 km s pc 4.9 10 rad s
out
r
r
Basic qualitative results are independent of details of initial profiles.
All scale as r -1 at large radii.
Self-consistent formation of the protostellar disk and envelope-induced evolution
Mass infall rateonto the protostar
Evolution of the protostellar disk
Early (Burst Mode) Disk Evolution
Time (yr)
0 10000 20000 30000 40000 50000
Mas
s ac
cret
ion
rate
(M8
yr-1
)
0
2e-4
4e-4
6e-4
8e-4
Too
mre
par
amet
er (
Q)
0
1
2
3
4
5
Mass accretion bursts and the Q-parameter
Black line - mass accretion rate onto the central sink; Red line – the Q-parameter
The disk is strongly gravitationally unstable when the bursts occur
Smooth mode Burst mode
G
cQ s
3sc
G
Vorobyov & Basu (2006)
Accretion history of young protostars
FU Ori outburst
envelope accretion
disk accretion
VeLLO’s?
Vorobyov & Basu (2007)
Burst mode Residual accretion , self-regulated mode
Disk mass stays well below central mass
Gravitational instability and clump formation can occur in low-mass protostellar disks.
M = 1 M, rout = 0.05 pc, = 0.275%
disk formation at 10 AU
Azimuthally Averaged Spatial Profiles – Into the Late Phase
T
Q
G
cQ s
Vorobyov & Basu (2007)
Accretion and instability help to self-regulate disks to a near-uniform Q distribution
.2/3 r
Sharp edge!
Self-regulation
Keplerian
Disk weakly nonisothermal
Nonaxisymmetry is essential for this result.
1.5r
1.5r
Slope of MMSN
Two Modes of Disk Accretion
-250 -150 -50 50 150 250
Radial distance (AU)
-250
-150
-50
50
150
250
Rad
ial
dis
tan
ce (
AU
)
678910111213
-200 -100 0 100 200
Radial distance (AU)
-200
-100
0
100
200
Rad
ial
dis
tan
ce (
AU
)
678910111213
Late self-regulated mode
Gravitational torque driven accretion, Q ~ 1, not GI
Diffuse spiral structure
Early burst mode
Episodic vigorous gravitational instability (GI). Distinct spiral modes
Clumps form and accreted inward
Binary formation may occur here
VB06
The Swing Amplifier – why fluctuations persist in self-regulated mode
Leading spiral waves can be unwound into trailing spiral waves. During the process, a transient instability feeds energy into the spiral mode.
For the process to work continuously, need a feedback loop, i.e. fresh sources of leading waves in the system. Where from??
Toomre (1981), based on work by Zang and Toomre. Also, Goldreich & Lynden-Bell (1965).
Compile accretion rates for various initial core masses
Vorobyov & Basu (2008)
Solid circles: time-average (class II phase, 0.5 to 3 Myr) values from models with differing initial mass. Bars represent variations from mean during same time period.
All other symbols: data from Muzerolle et al. (2005) and Natta et al. (2006).
Blue line – best fit to simulation averages. Black line – best fit to all data points. Red lines – best fits to low and higher mass regimes of data.
Blue line: 1.7*M M
Some key results
• Can fit mean observed T Tauri star (TTS) accretion rates using a model of gravitational torque driven accretion
• Model also produces near-Keplerian rotation and r -3/2 surface density profile in disk
• However, disk masses and disk-to-star mass ratios are a factor ~10 greater than observational estimates for TTSs and BDs (Andrews & Williams 2005; Scholz et al. 2006)
Observed disk masses underestimated?
• Grain growth in disks already significant. Standard opacity requires grain growth to 1 mm at ~100 AU, but what if they grow further? Larger grains would lead to higher disk mass estimates (Andrews & Williams 2007; Hartmann et al. 2006)
• Upper envelope of TTS accretion rate dM/dt ~ 10-7 Msun/yr implies Mdisk ~ dM/dt x 1 Myr ~ 0.1 Msun
• MMSN contains ~ 0.01 Msun material, barely enough to make Jupiter. Extrasolar systems with M sin i up to several Jupiter masses imply Mdisk >> 0.01 Msun
• Chondrule formation models (Desch & Connolly 2002; Boss & Durisen 2005) require a high density and Mdisk ~ 0.1 Msun
2 2
1
viscous stress tensor
kinematic viscosity
vertically integrated gas pressure
( ),
1( ) P ,
12 ,
3
,
P c c ,
p p
pp p p p p p
s
s s crcr
p r
p
t
t
c Z
u r u
r rr
u
uu u Π
Π u u e
u
2 2
( , )( , )
2 cos( )
i j i i ji j
i j i i i i j j
r r rr G
r r r r
unit tensor
Basic Equations with Viscosity
Vorobyov & Basu (2009, MNRAS, 393, 822)
The Additional Effect of viscosity
Radial distance (AU)
Red lines, = 0. Solid black lines, = 0.01.
Distances on horizontal axis in AU.
Density drops and disk is larger in viscous disk. Self-regulated disk structure is lost, and it is clearly gravitationally stable.
An effective alpha for models
eff3 sM c H at inner sink
Vorobyov & Basu (2009)
Can viscous approach model gravitational instability/torques?
In Burst mode: No! Global (mostly m=1) mode dominates
Burst mode
Self-regulated mode many higher order modes dominate
Vorobyov & Basu (2009)
Viscous approach may be useful for self-regulated mode
In self-regulated mode, many high order spirals, lots of mode-mode interaction a local approximation more suitable, e.g., Lin & Pringle (1987,1990), Lodato & Rice (2004), Vorobyov (2010).
Lodato & Rice 2004 – a self-regulated disk, Q ~ 1
Vorobyov & Basu (2007) – evolution of a ring
Summary
• Circumstellar disks that form self-consistently enter an early burst mode of episodic vigorous gravitational instability formation of clumps FU Ori-type bursts. Very low accretion states may correspond to VeLLO’s.
• At late (~ Myr) stages, disks enter a self-regulated mode, have a sharp edge and maintain persistent nonaxisymmetric density fluctuations non-radial gravitational forces torques that drive accretion at rates comparable to that of TTSs
• Self-regulation of disk in late phase leads to Q ~ const. and to surface density profile ~ r -3/2 ; same slope as MMSN
• For models with ~ 0.5 Msun and above, can fit observed dM/dt vs. M* relation.
• Disk mass stays well below central mass, but factor ~ 10 larger than observational estimates. Observed disk masses systematically underestimated?
• Addition of -viscosity increasingly undermines all of the above effects, and dominates even for = 10-2 . Other parametrizations of viscosity (Q -dependent) may provide a reasonable approximation to the self-regulated mode.