The formation of stars and planets

Day 3, Topic 2:

Viscous accretion disksContinued...

Lecture by: C.P. Dullemond

Non-stationary (spreading) disks

• So far we assumed an infinitely large disk• In reality: disk has certain size• As most matter moves inward, some matter must

absorb all the angular momentum• This leads to disk spreading: a small amount of

outer disk matter moves outward

Non-stationary (spreading) disksGiven a viscosity power-law function , one can solve the Shakura-Sunyaev equations analytically in a time-dependent manner. Without derivation, the resulting solution is:

€

ν ~ rχ

Lynden-Bell & Pringle (1974), Hartmann et al. (1998)

€

Σ=C

3πν 1ϖχθ−(5 / 2−χ ) /(2−χ ) exp −

ϖ 2−χ

θ

⎡

⎣ ⎢

⎤

⎦ ⎥

where we have defined

€

ν1 ≡ ν (r1)

€

ϖ ≡r /r1

€

θ ≡t / ts +1

with r1 a scaling radius and ts the viscous scaling time:

€

ts =1

3(2 − χ )2

r12

ν 1

Non-stationary (spreading) disks

Time steps of 2x105 year

Lynden-Bell & Pringle (1974), Hartmann et al. (1998)

Formation & viscous spreading of disk

Formation & viscous spreading of disk

Formation & viscous spreading of disk

Formation & viscous spreading of disk

From the rotating collapsing cloud model we know:

€

rcentrif ~ t 4

Initially the disk spreads faster than the centrifugal radius.

Later the centrifugal radius increases faster than disk spreading

Formation & viscous spreading of diskA numerical model

Formation & viscous spreading of diskA numerical model

Formation & viscous spreading of diskA numerical model

Formation & viscous spreading of diskA numerical model

Formation & viscous spreading of diskA numerical model

Formation & viscous spreading of disk

Hueso & Guillot (2005)

Disk dispersal

Haisch et al. 2001

It is known that disks vanish on a few Myr time scale.

But it is not yet established by which mechanism. Just viscous accretion is too slow.

- Photoevaporation? - Gas capture . by planet?

Photoevaporation of disks(Very brief)

Ionization of disk surface creates surface layer of hot gas. If this temperature exceeds escape velocity, then surface layer evaporates.

€

vesc ≈GM

r

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 2

Evaporation proceeds for radii beyond:

€

r ≥GM

csHII

2≡ rgr

Some special topics

‘Dead zone’MRI can only work if the disk is sufficiently ionized.

Cold outer disk (T<900K) is too cold to have MRI

Cosmic rays can ionize disk a tiny bit, sufficient to drive MRI

Cosmic rays penetrate only down to about 100 g/cm2.

full penetration of cosmic rays

partial penetration of cosmic rays

‘Dead zone’Hot enough to ionize gas

Only surface layer is ionized by cosmic rays

Tenuous enough for cosmic rays

Above dead zone: live zone of fixed Σ = 100 g/cm2. Only this layer has viscosity and can accrete.

Accumulation of mass in ‘dead zone’

€

vr = −3

2

ν

r

Remember:

€

ν ≡rχ

€

vr = −3

2rχ −1

Stationary continuity equation (for active layer only):

€

∂(ΔΣr vr)

∂r~ −ΔΣ

∂( rχ )

∂r≠ 0

For >0 we have mass loss from active layer (into dead zone)

Gravitational (in)stabilityIf disk surface density exceeds a certain limit, then disk becomes gravitationally unstable.

Toomre Q-parameter:

€

Q =hΩK

2

π GΣ

€

≈h

r

M*

Mdisk

For Q>2 the disk is stableFor Q<2 the disk is gravitationally unstable

Unstable disk: spiral waves, angular momentum transport, strong accretion!!

Gravitational (in)stability

Spiral waves act as `viscosity’

Rice & Armitage

Episodic accretion: FU Orionis outbursts1. Dead zone: accumulation of mass

2. When Q<2: gravitational instability

3. Strong accretion, heats up disk

4. MRI back to work, takes over the viscosity

5. Massive dead zone depleted

6. Temperature drops

7. Main accretion event ends

8. New dead zone builds up, another cycle

time (year)

Armitage et al. 2001

FU Orionis stars

McNeal’s Nebula: a new FU Ori?

Effect of an external companion

Augereau & Papaloizou (2004)

Observations of disks

Silhouette disks in Orion Nebula

Photoevaporation of disks: from outsideMany low mass stars with disks in Orion near Trapezium cluster of O-stars. Their disks are being photoevaporated.

Images of isolated disks: scattered light

C. Grady

HD100546

Images of isolated disks: scattered light

C. Grady

HD163296

Measuring the Keplerian rotation

CO, CN lines

HD163296: MWC 480:

Qi (PhD Thesis) 2001

Measuring the Keplerian rotation

Pietu, Guilloteau & Dutrey (2005)

AB Aurigae: nearly Kepler, but deviations

13CO 2-1

AB Aurigae: spiral arms and clumps

Pietu, Guilloteau & Dutrey (2005)

AB Aurigae: spiral arms and clumps

Fukagawa et al. 2004

Scatteredlight