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.