Protostellar Disks: Accretion Processes

Sudhir Raskutti

Princeton University

November 6, 2012

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Protostellar Disks: Accretion Processes

Introduction

Modelling Protostellar Disks

Angular Momentum Transport

Accretion by the Magnetorotational Instability:

Ideal CaseResistive Case and Ionization StructureOther non-ideal e↵ectsNon-linearities

Accretion by Hydrodynamic Instabilities

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Protostellar Disks: Accretion Processes

Protostellar Disks

Masses around 0.01� 0.1M� and sizes around 10� 100AU

Thin Disks (Minimum Mass Solar Nebula):

⌃(r) ⇡ 1700�

r1 AU

��3/2gcm�2

hr = cs

⌦r ⇡ 0.03�

r1 AU

�1/4

Cool and Dusty: T (r) ⇡ 280�

r1 AU

��1/2K

Magnetic fields between 10�2 � 1G

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Protostellar Disks: Accretion Processes

Mass Accretion

Figure: (Kitamura, 2002)

Disks last for ⇠ 1� 10Myr

Must accrete or disperse diskmass in this time

Accretion rates⇠ 10�9 � 10�7

M�yr�1

Disk evolves, accretes massonto protostar by

Loss of mass and angularmomentum(photoevaporation, diskbraking, disk winds)

Angular MomentumTransport

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Protostellar Disks: Accretion Processes

Angular Momentum Transport

Local turbulence creates viscoscity ⌫ = ↵

c2s⌦ related to the local

stress

Wr� =

�vr�v� � BrB�

4⇡⇢

�

⇢

Wr� = ↵c

2s

This viscoscity drives disk evolution

@⌃

@t

=3

r

@

@r

pr

@

@r

�⌫⌃

pr

��

M = 6⇡r1/2@

@r

(2⌃⌫r1/2)

Accretion and di↵usion outwards if @(⌫⌃)@⌃ < 0

Most internal methods of accretion require sustained turbulence

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Protostellar Disks: Accretion Processes

MHD Turbulence

Can create turbulence by:

Self-gravity (Wendy)Hydrodynamic InstabilitiesMagnetorotational Instability (MRI)

J x B

J x B

B0+ δB

1

2v

v

rφzx

B0

1

2

In Ideal MHD@B@t = r⇥ [v ⇥B]

Di↵erential rotation createstension along field lines

Excites turbulence, drivessome mass inwards, angularmomentum outwards

Excited if ddr

�⌦2

�< 0

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Protostellar Disks: Accretion Processes

Complications

In reality, there are non-ideal e↵ects

@B

@t

= r⇥v ⇥B� ⌘r⇥B� J⇥B

ene+

(J⇥B)⇥B

c�⇢i⇢

�.

Magnetic field drifts due to di↵usion terms

Ohmic Di↵usion: ⌘ = c2

4⇡�c

Ambipolar Di↵usion: (J⇥B)⇥B

c�⇢i⇢

Hall Di↵usion: J⇥B

ene

Stronger coupling between magnetic field and fluid required forMRI

With Di↵usion, what regions of the disk accrete?

When are non-ideal e↵ects important and what e↵ect do theyhave?

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Protostellar Disks: Accretion Processes

Ohmic Di↵usion

Magnetic Di↵usion due to finite resistivity ⌘ = c2me�e⇢4⇡e2ne

Important for low ionization fraction since resistivity increaseswith neutral fraction

Suppresses MRI when resistive damping ⌧⌘ ⇠ �2

⌘ is shorter than

growth rate ⌧ ⇠ �vA

Equivalent to:

ReM =hvA

⌘

. 1

ne

n

= x ⇠ 5⇥ 10�13

✓h/r

0.05

◆�1 ✓vA

0.1cs

◆�1

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Protostellar Disks: Accretion Processes

Disk Ionization

Thermal ionization, at typical densities nH ⇠ 1013gcm�3,reaches x ⇠ 10�13 for T & 103K

Protoplanetary disks are much colder than most astrophysicaldisks, does not hold beyond r ⇠ 0.1AU

Stellar X-rays (1 AU, 5 keV)

Cosmic rays (unshielded)

Radioactive decay of 26Al

Likely surfacedensity at 1 AU

Figure: (Armitage, 2010)

Non-thermal sources ofionization dominate

Radioactive Decay

Cosmic Rays

Protostellar X-rays

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Protostellar Disks: Accretion Processes

Dead Zones

Leads to tiered structure of protostellar disks (Gammie, 1996)

Thermally ionized and MRI turbulent interiorNon-thermally ionized and MRI turbulent exteriorIntermediate region with thin active layer and mid-plane deadzone

dead zone

collisional ionization at T > 103 K (r < 1 AU),MRI turbulent

resistive quenchingof MRI, suppressedangular momentumtransport MRI-active

surface layer

non-thermal ionizationof full disk column

cosmicrays?

ambipolar diffusiondominates

X-rays

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Protostellar Disks: Accretion Processes

Layered Accretion

Accretion proceeds through the active layer onto the dead zone

Uneven accretion leads to gravitational instability and heating toabove ⇠ 103K

Mass accreted onto dead zone rapidly accreted onto protostar(Variable/Bursty Accretion)

Accretes su�cient mass onto the dead zone (Gammie, 1996)

M ⇡ 1.3⇥ 10�3⇣

↵

10�2

⌘2✓

⌃a

100gcm�2

◆3

0

✓�t

104yr

◆M�

But:

Results ignore Hall and Ambipolar Di↵usionVery sensitive to the exact opacity/recombination rate

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Protostellar Disks: Accretion Processes

Uncertainties

x determined by balance between ionization and recombination

Ionization (by X-rays) slightly uncertain

Gas-phase recombination well known (though sensitive to metalabundance)

Recombination onto grains dependent on both the fraction ofdust and the size of dust grains

nI,dust

nI,gas⇠ 20

✓fd

10�2

◆⇣x

10�12

⌘�1✓

T

100 K

◆✓a

1 µm

◆�1

Even if initial dust distribution is known, the rate ofsedimentation is unknown

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Protostellar Disks: Accretion Processes

Uncertainties: Dust Fraction

Gammie, 1996 results assume⌃a ⇡ 100gcm�2, which is onlytrue for small dust fractions Figure: (Sano, Miyama, 2000)

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Protostellar Disks: Accretion Processes

Uncertainties: Dust Size

Significant active layer onlyfor large dust sizes and smalldust fractions

This is true if disk hasevolved significantly

Figure: (Sano, Miyama, 2000)

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Protostellar Disks: Accretion Processes

Uncertainties: Disk Density

Changes in disk surface density have less impact, but stilluncertain

Figure: (Sano, Miyama, 2000)

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Protostellar Disks: Accretion Processes

Non-Ideal E↵ects: Ambipolar Di↵usion

Other e↵ects don’t destroy flux but let it drift wrt neutral fluid

@B

@t

= r⇥⇥(v + vB)⇥B� ⌘(r⇥B)k

⇤

vB = vP + vH

J x B

J

v

v + vB

vB

1

r

φ

z

B0

J x B

J

v

v + vB

2

vB

Ambipolar di↵usion (lowdensity, high x) dominateswhen the field is frozen to ions,with a drift due to neutral drag

vP =J⇥B

c�i⇢i⇢

�i =h�vii

mi +m

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Protostellar Disks: Accretion Processes

Non-Ideal E↵ects: Hall Di↵usion

Hall di↵usion dominates when the field frozen to electrons aloneand induces a drift due to the di↵erential ion-electron motion

vH = � J

ene

The Hall e↵ect depends on the field direction and can eitherreinforce or entirely suppress the MRI

1

2

J x B

J

J x B

J

vP

vH

v

v + vB

vB vP

vH

v

v + vB

vB

r

φ

zB0

JJ x B

vP

vH

v v + vB

vB

1

JJ x B

vP

vH

vv + v

B

vB 2

XB0

r

φ

z

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Protostellar Disks: Accretion Processes

Non-Ideal E↵ects in Protoplanetary Discs

Figure: (Sano, Stone, 2002)

Compare di↵usion terms:OI ⌘ 1

ReMAI ⌘ ⌦

�⇢iHI ⌘ X

2

Assume equilibrium:

�c = e2nemennh�vie

⌘ = c2

4⇡�c

X = ⌘⌦2v2

A

Typical protoplanetary disks are Hall/Ohm dominated in innerregions and Ambipolar-dominated beyond r ⇠ 20AU

Hall di↵usion may be very important for mass accretion

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Protostellar Disks: Accretion Processes

Non-Ideal E↵ects: Linear Regime

–2 0 2 40

1

2

3

4

s ηH Ω / vA2

0.100.3

0.4

0.5

0.6

0.7

(η +

ηA

) Ω /

v A2

0.7

0.5

0.4

0.3

1

0.2

0.1

∞0.75

0.2

–2 0 2 40

1

2

3

4

s ηH Ω / vA2

(η +

ηA

) Ω /

v A2

1.2

0.5 0.4 0.3

1

∞

0.2

1.5

0.6

0.7

0.8 0.9

2

0

Maximum growth rate (Black), wavenumber (Blue) and largest stable wavenumber (Right).

(Wardle, Salmeron 2012)

For B = sBz (s = ±1) under perturbations exp(⌫t� ikz)

Weakly coupled electron-ion-neutral plasma:

⌘A = B2

4⇡ �i ⇢⇢i

⌘H = cB4⇡ e ne

Pure ohmic and ambipolar di↵usion tend to decrease the growthrate and increase the maximum wavelength of perturbations

Hall Di↵usion increases the maximum growth rate

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Protostellar Disks: Accretion Processes

Non-Ideal E↵ects: Linear Regime

0

1

2

3

4

5

0.0

0.5

1.0

1.5

2.0

2.5

0

1

2

3

4

5

0.0

0.5

1.0

1.5

2.0

2.5

B (G)

0

0.25

0.5

0.75

10–3 10–2 10–1 100 101 1020

1

2

3

4

5

0.0

0.5

1.0

1.5

2.0

2.5

z / h

ohmic only

Bz > 0

Bz < 0

ν / Ω

z (1

012 c

m)

z / hz / h

z (1

012 c

m)

z (1

012 c

m)

Mdust / Mgas = 0

Figure: Maximum growth rate vs.height above mid-plane. (Wardle,Salmeron, 2012).

10–3 10–2 10–1 100 101 102

100

101

102

103

ohmohm+ amb

ohm+amb+hall (Bz > 0)

Mdust / Mgas = 10–4

activ

e co

lum

n (

g cm

–2 )

B (G)

ohm+amb+hall (Bz < 0)

a = 1µm

Hall di↵usion does stabilise(destabilise) the disk compared toohmic di↵usion alone

Up to 2 orders of magnitudedi↵erence in active layer columndensity

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Protostellar Disks: Accretion Processes

Non-Ideal E↵ects and Dust

E↵ect of Hall Di↵usionprobably dwarfed byuncertainty in dustfraction

No grains: Couplingcan probably bemaintained atmidplane

1% mass in grains(early evolution), nosignificant active layer

If grains remain small(turbulence), nosignificant active layer

10–3 10–2 10–1 100 101 102

100

101

102

103

B (G)

10–4

10–6

10–2

10–2

10–410–6

0

0

activ

e co

lum

n (

g cm

–2 )

Figure: Active layer size at 1 AU for positive andnegative magnetic fields and various dust mass fractionswith a = 1µm. (Wardle, Salmeron, 2012)

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Protostellar Disks: Accretion Processes

Non-Ideal E↵ects: Simulations

No guarantee that linear conditions will guarantee a steady stateof MHD turbulence and outwards transport of angularmomentum

Figure: Radial Velocity and Magnetic Field. (Sano,Stone, 2002)

Non-linear e↵ectscaptured in 2-fluidsimulations

Small initialperturbations in gaspressure ⇠ 10�6

Angular MomentumTransport as in idealMHD

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Protostellar Disks: Accretion Processes

Non-Ideal E↵ects: Simulations

Figure: (Sano, Stone, 2002)

Ohmic di↵usion condition remainsthe same

Hall di↵usion marginally changes thesaturation stress

Hall di↵usion has no e↵ect on thecritical Reynolds number

But don’t probe regime of halldomination XReM > 2 and ReM < 1

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Protostellar Disks: Accretion Processes

Hydrodynamic Instabilities

Angular Momentum Transport may be achieved with purehydrodynamic instabilities:

ConvectionPlanet-Driven EvolutionBaroclinic Instability

Likely to be subdominant to MHD turbulence

Can be important in dead zones

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Protostellar Disks: Accretion Processes

Baroclinic Instability

thermalization due to diffusion

buoyantsinking,roughlyadiabatic

buoyantrise, acceleration

thermalization

vortex

dTpert / dq�= 0

r

entropygradient

Radial entropy gradients and e�cient cooling produce vorticity

Particles moving inwards are cooler, drawn to lower orbits

E�cient thermal di↵usion heats them up along Keplerian orbits

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Protostellar Disks: Accretion Processes

Momentum Transport

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

0.0035

0.004

50 100 150 200

< !

>

t

Figure: (Lesur, 2010)

Vorticity if: Convectively unstable

N

2r = � 1

�⇢

dP

dr

d

drln

✓P

⇢

�

◆< 0

E�cient cooling

Significant initial perturbation(Subcritical)

2D simulations show growth of vorticity and weak angularmomentum transport

Not necessarily the same in 3D

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Protostellar Disks: Accretion Processes

Conclusions

MRI turbulence important in mass accretion for protoplanetarydisks

Leads to layered disk structure, with accretion through a thinactive layer

Still large uncertainties concerning:

Exact Modelling of Disk (MMSN)Dominant Ionization SourcesRecombination Rate and the large part played by dust grainsBehaviour of Hall Di↵usion in the non-linear regime

Baroclinic Instability possible source of additional accretion indead zones

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Protostellar Disks: Accretion Processes

References

1 Armitage, P. J. (2010, November 5). Dynamics of ProtoplanetaryDisks. arXiv.org. doi:10.1146/annurev-astro-081710-102521

2 Sano, T., Miyama, S. M., Umebayashi, T., and Nakano, T.(2000, May 23). Magnetorotational Instability in ProtoplanetaryDisks. II. Ionization State and Unstable Regions. arXiv.org.doi:10.1086/317075

3 Bai, X.-N., and Stone, J. M. (2011). E↵ect of AmbipolarDi↵usion on the Nonlinear Evolution of MagnetorotationalInstability in Weakly Ionized Disks. The Astrophysical Journal,736(2), 144. doi:10.1088/0004-637X/736/2/144

4 Sano, T., and Stone, J. M. (2002a, January 11). The E↵ect ofthe Hall Term on the Nonlinear Evolution of theMagnetorotational Instability: I. Local AxisymmetricSimulations. arXiv.org. doi:10.1086/339504

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References

1 Sano, T., and Stone, J. M. (2002b, May 22). The E↵ect of theHall Term on the Nonlinear Evolution of the MagnetorotationalInstability: II. Saturation Level and Critical Magnetic ReynoldsNumber. arXiv.org. doi:10.1086/342172

2 Wardle, M., and Salmeron, R. (2012). Hall di↵usion and themagnetorotational instability in protoplanetary discs. MonthlyNotices of the Royal Astronomical Society, 422(4), 27372755.doi:10.1111/j.1365-2966.2011.20022.x

3 Gammie, C. F. (1996). Layered Accretion in T Tauri Disks.Astrophysical Journal v.457, 457, 355. doi:10.1086/176735

4 Lesur, G., and Papaloizou, J. C. B. (2010). The subcriticalbaroclinic instability in local accretion disc models. Astronomyand Astrophysics, 513, 60. doi:10.1051/0004-6361/200913594

5 Kitamura, Y., Momose, M., Yokogawa, S., Kawabe, R., Tamura,M., and Ida, S. (2002). Investigation of the Physical Propertiesof Protoplanetary Disks around T Tauri Stars by a 1 ArcsecondImaging Survey: Evolution and Diversity of the Disks in TheirAccretion Stage. The Astrophysical Journal, 581(1), 357380.doi:10.1086/344223

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