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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