Theory of Protostellar Disk Formation

Zhi-Yun Li (University of Virginia)

Collaborators: Ruben Krasnopolsky (ASIAA)
Sienny Shang (ASIAA)
1

Focus on magnetic effects on disk formation• Investigated by many groupse.g., Tomisaka 1998; Krasnopolsky & Konigl 2002; Allen+2003; Galli+2006;
Price & Bate 2007; Mellon & Li 2008, 2009; Hennebelle & Fromang 2008;
Duffin & Pudritz 2009; Krasnopolsky+2010, 2011, 2012; Dapp & Basu
2010, Inutsuka+2010; Machida+2010, 2011, 2014; Li+2011, 2013, 2014;
Seifried+2011, 2012, 2013, 2015; Santos-Lima+2012, 2013; Dapp+2012;
Joos+2012, 2013; Myers+2013, Braiding & Wardle 2012a,b; Krumholz+2013; Tomida+2013, 2014; Padovani+2013, 2015; Fielding+2015;
Hopkins & Raives 2015; Tsukamoto+2015• By members or former members of NAOJ
to be reviewed by Machida later today?
2Outline
I. Difficulty with disk formation in ideal MHD limit
magnetic fields in dense cores; magnetic flux freezing; magnetic
split monopole; excessive magnetic braking

II. Can non-ideal MHD effects save disk in 2D?
ambipolar diffusion; Ohmic dissipation; Hall effectIII. Can 3D effects save the disk?
magnetic interchange instability in inner protostellar accretion flow

IV. How to form rotationally supported disks?
review proposed mechanisms and some speculations3

I. Difficulty with disk formation in the ideal MHD limit
magnetic fields prevalent in molecular clouds

Planck Collaboration 20144

Are magnetic fields dynamically important?

• Dynamical importance of the magnetic field measured by themass-to-flux ratio

=⇥M
⇥
⇥=
1
2 G1/2(Nakano & Nakamura 1978)⇥B⇥
12 G1/2(Can the magnetic field support a core against the gravity?)• Supercritical if λ > 1, subcritical if λ < 1 (larger λ, weaker B field)
5Magnetic Field Strength in Cores: Observations
Prestellar core L1544(Troland & Crutcher 2008)

0.05 pc
10,000AU(Shirley et al. 2000)

OH Zeeman survey of dark cloud cores using Arecibo telescope (Troland & Crutcher 2008)

mean mass-to-flux ratio λcore,los ≈ 4.8 ± 0.4 without geometric corrections
Geometric corrections reduce the ratio by a factor of 2-3, to λ ~ 2(magnetic/gravitational ~λ-2 ~ tens of percent >> rotational/gravitational ~ a few%)

Not strong enough to prevent collapse, but strong enough to affect disk formation

Magnetic core collapse in ideal MHD: split monopole
Meridian plane

λ~3Collapse of 1 M!core w/ uniform B, no Ω
Flux freezing during collapse (pinched B)
Flattened mass distribution (pseudodisk)(Galli & Shu 1993)

Split magnetic monopole near origin(making disk formation difficult)7

Magnetic braking catastrophe in ideal
MHD limit(Galli et al. 2006)• Flow dynamics near split monopole
Br ~ r-2, vr ~ vff ~ r-1/2, ρ ~ ρff ~ r-3/2• Magnetic energy density
B2 ~ r-4• Flow energy density
ρ v2 ~ r-5/2• Flow completely magneticallydominated as r " 0vφ " 0 as r " 0 (no RSD)

8

Magnetic Braking & Disk Formation
Slower rotation

Faster rotation• Differential twist of field lines
generates a toroidal magnetic field• Magnetic tension of twisted field
yields a braking torque on thefaster rotating, inner material• Braking rate ~ Bp x Bφ
• Stronger field, harder to form disk
9

Magnetic braking catastrophe in ideal MHD: 2D
numerical example with ZeusMP (Mellon & Li 2008)
Initial configuration
λ ~ 13, weak field

Meridian plane+10a0

Rotation speed

Infall on equator-10a

Central massspeed

Kepleriantime

Equatorial radius(see also, e.g., Allen et al. 2003, Machida et al. 2005, Banerjee &Pudritz 2006, Price & Bate 2007, Hennebelle & Fromang 2008, Duffin& Pudritz 2009, Seifried et al. 2012, Santos-Lima et al. 2012)

Disk formation suppressed for λ ~ a few in ideal
10 MHDaka “magnetic braking catastrophe”

II. Can non-ideal MHD effects savethe disk in 2D? (Li, Krasnopolsky & Shang 2011)
1020upAmbipolar
diffusion
Ambipolar

1018
10161022

Magnetic diffusivities (cm2s-1)

Magnetic diffusivities (cm2s-1)

1111111111111111111111111111111111111111111111111111111111111111111
11111111111
1022
Cover

Halleffect
Hall

1014
1012

Cover upup
Cover

Ambipolar diffusion
Ambipolar

1020
Hall
111018
Hall effect10161014
1012
Ohmic

Ohmic

1010

Ohmic
Ohmic

104

1010
106

108 1010
nH(cm-3)

1012

1014

Typical core density

Large grain size: 1 µm
AD dominates over most densities

104

106

108 1010
nH(cm-3)

1012

11014

Power-law size distribution
w/ small grains (MRN)(Mathis, Rumpl & Nordsieck 77)

11

Smaller grain, Hall more important

Ambipolar diffusion & magnetic flux redistribution(Li, Krasnopolsky & Shang 2011)• 2D (axisymmetric) collapse of
initially uniform, λ=2.9 non-rotating
core, with only AD (including MRNgrains), in spherical coordinate
system using ZEUS–TW

2D AD calculation

IMHD• Magnetic flux piles up outside star(high magnetic pressure, slowingdown collapse) (Li & McKee 1996, also

(cm/s)

cm/s

Ciolek & Konigl 1998, Krasnopolsky & Kronigl
2002, Tassis & Mouschovias 2007)

ionsAD shockions

Infall speed

Free-fall

Equatorial radius (cm)Meridian view
12Can ambipolar diffusion save therotationally supported disk (RSD)?
• No! No rotationally supported disk (see also Krasnopolsky & Konigl 2002)
Almost complete braking of rotation
-19.0

2•105

-18.0

-17.0

-16.0

-15.0

-14.0

pointmass= 1.1375878e+33
frame=600-13.0

1•1016

Meridian plane-180•11.0

Infall and rotation speed (cm/s)

Keplerian
Keplerian

1•105

5•1015Rotation
Rotation-1700

Infall1.0•10-17

1.0•10AD shock-5•1015

Free-fall
Free-fall-1•105

1.0•

Infall speedof neutrals-2•105
1014

1015
1016
Radius along equator (cm)Magnetically induced disk!

-1•10161017

0

10 -185.0•1015

1.0•1016

1.5•1016

2.0•1016 13(cm)Can ambipolar diffusion, Ohmic dissipation & Hall
effect save the rotationally supported disk (RSD)?
AD, Ohmic, Hall, B & Ω anti-parallel
11111111111
111Hall spin-up (Wardle & Ng 1999)

Rotation

Rotation
AD onlyAD, Ohmic, Hall, B & Ω parallel

infallunable to save disks, at least in 2D
14

III. Can 3D effects save the disk?
2D accretion flows expected to be unstable
2D AD calculation• Magnetically supported post-AD
shock region expected to beunstable to magnetic interchange

Meridian plane

instability (Li & McKee 1996, Krasnopolsky& Konigl 2002)(cm/s)

cm/s• What happens in 3D?

ionsions

AD shock

Infall speed

free-fall

Equatorial radius (cm)

15(cm)

III. Can 3D effects save the disk?
(Krasnopolsky, Li, Shang & Zhao 2012)
Non-rotating collapse in 3D with AD and localized Ohmic dissipation• 3D collapse of initially uniform, non-rotating dense core of λ=2.9 (late times)• Shown are density and Bz on equatorial plane, simulated with ZEUS-TWdensity

Field strength•Instability driven by magnetic flux decoupled from matter going into the central object(also B. Zhao+2011, with ENZO AMR MHD code, Seifried+2011 with FLASH, & Cunningham+2012,Joos+2012 with RAMSES)•Magnetically dominated circumstellar region, potentially making disk formation difficult16

III. Can 3D effects save the disk?
Rotating collapse in 3D with AD and localized Ohmic dissipation
Equatorial planedensity

Field strength• Rotationally supported disk does not appear to form in 3D during the protostellar accretion phase• Instability alleviated but not eliminated the obstacle to disk formation – strong circumstellar field
3D instabilities unable to save disks

IV. How to form rotationally supported disks (RSDs)?
(1) Simplest resolution: most dense cores weakly magnetized?
unlikely for the following reason
Consider a uniform core of 1 Msun and n(H2)=104cm-3, radius=2.17x1017cm
For a dimensionless mass-to-flux ratio of λ, the core field strength must be1/2

2⇥G M
B== 4.4µG2(⇥r )5⇥

4.4µG is less than median field strength inferred for atomic CNM of ~6µG!
(Heiles & Troland 2005; see also Troland’s unpublished results for B field of GMC envelope, ~10µG)

>>

core

be enough to make formation of large-scale rotationally
5 may
supported disk difficult during protostelllar accretion phase18

IV. How to form rotationally supported disks?
(2) Disk formation through misaligned rotation & magnetic axes?
• Possible motivation: CARMA observations (Hull et al. 2013)• Magnetic fields on ~1000 AU scale• Some B vectors aligned withoutflow/rotation axis; most not• Angle consistent with being random
19

Can magnetic field-rotation axis misalignment enable large-scale disk formation?
Joos, Hennebelle & Ciardi (2012)(also Price & Bate 2007; Hennebelle & Ciardi 2009)

Angle between magnetic field & rotation•• Disks form in weakly magnetizedcores of λ~17 for all angles,but not for λ~2, median value
inferred by Troland & Crutcher (2008)• For intermediate cases of λ~3-5,
large misalignment enables
disk formation
Dimensionless mass-to-flux ratio λ20

Illustration: Li et al 2013, λ~14 (weak B), density
map & velocity vectors on equatorial plane

Aligned caseOrthogonal case

400AU
21

Illustration: Li et al 2013, λ~14 (weak B), density
map & velocity vectors on a meridian plane

Orthogonal case

2000AU

Aligned case

No RSD

Outflow is a key difference! (see also Ciardi & Hennebelle 2010)

RSD22

IV. How to form rotationally supported disks (RSDs)?
(2) Disk formation through misaligned rotation & magnetic axes?
(Li, Krasnopolsky & Shang 2013)eff=13.7
λ~14,θ=00

4•1015A0=0o2•10150λ~7,
eff=6.85
θ=00

4•1015-2•1015-4•1015

B

2•10152•101500-2•1015-2•1015-4•1015-4•1015-2•1015

0

2•1015λ~14, θ=450

4•1015

4•1015

D-2•1015

02•1015λ~7, θ=450

4•1015E-4•101500-2•1015-2•1015-2•1015-4•1015-4•1015-4•1015

0=45o2•1015-4•1015-2•1015

02•1015

λ~14, θ=9004•1015

4•1015G

2•1015-2•1015

02•1015

λ~7, θ=9004•1015

H

00-2•1015-2•1015-2•1015-4•1015-4•1015-4•1015

4•1015F• No, for moderately strongfield case of λ~4 (B field two
where there is NO disk even inthe orthogonal case!

-4•10152•10152•1015times weaker than inferred value),-2•101502•1015λ~4, θ=9004•1015

2•101504•1015I

Dual requirement for large disks:
Rather weak B AND large misalignment

0=90o0-4•1015

4•1015-2•1015

λ~4, θ=450

4•10152•10150• Yes, for weak-field cases ofλ~14 and 7, where disk
formation is enabled by largetilt, in broad agreement with
Joos et al. (2012)

C-4•1015-4•10154•10152•1015λ~4,
eff=4.12
θ=00

4•1015-2•101502•1015

4•1015

Unlikely for the majority of cores23-4•1015-4•1015-2•101502•1015

4•1015-4•1015-2•101502•10154•1015

Rough estimate of the fraction of dense cores withmisalignment-enabled large-scale disks(Krumholz et al. 2013; Li et al. 2013)• Median λ~2 for dense cores (Troland & Crutcher 2008)• Half cores with λ>2, capable of disk formation for large
misalignment according to Joos et al. 2012
• If misalignment random (Hull et al. 2013), half cores θ>600
eff=13.7

4•1015A0=0o2•10150-2•1015-4•1015-2•1015

0

2•10152•101500-2•1015-2•1015

0=45o-2•1015-4•1015-2•1015

02•10154•1015E-2•1015

02•1015G

4•10152•101500-2•1015-2•1015

0=90o-2•1015-4•1015-2•1015

02•10154•1015H-2•1015

0

2•1015

4•101502•10154•1015disk fraction reduced by a factor of ~2F" disk fraction ~10%-4•1015-2•101502•10152•10152•101500-2•1015-2•10154•1015I

4•1015-4•1015-4•1015-2•1015-4•1015-4•10154•10152•1015-4•10152•1015

4•1015• However, if only λ>4 cores form disks (Li et al. 2013)4•1015-4•1015-4•1015" disk fraction ~ ½ x ½ ~ ¼

C-4•1015-4•10154•10152•1015eff=4.12

4•10152•1015

4•1015D

4•10150B-4•1015-4•10150

eff=6.85

4•1015! majority of cores may not produce large disks
deduring the protostellar accretion phase?

-4•1015-4•1015-2•1015

0

2•1015

4•1015-4•1015-2•1015

0

2•1015

4•1015

Comparison with observations: what fraction ofprotostars have large, 100AU-scale RSDs?
Direct evidence for large disks around???

• many evolved “Class II” YSOs (little envelope)

Courtesy of C.-F. Chiang• increasing number of “Class I” YSOs (someenvelope, transition from deeply embedded to revealed)• “Class 0” protostars?? (deeply embeddedwithin a massive envelope) – objects to comparewith simulations
Class IITW Hydrae

13COClass IC18OClass I object TMC1A observed withPdBI in 13CO and C18O (2-1)(Harsono et al. 2014)25

What fraction of protostars havelarge, 100AU-scale RSDs?

???

• Disks difficult to observed in deeply embedded objects; confused with
inner envelope• Continuum interferometer data often reveal compact emission indicating
significant mass concentration on small scales•Example: SMA data on NGC1333-IRAS2A(Jorgensen et al. 2005)"Compact component of 0.02 M! or more
•Unbiased CARMA survey of protostars inSerpens reveals that 6 out of 9 have compactemission (Enoch et al. 2011) ##"Compact component appears common• Maury et al. (2010) surveyed 5 “Class 0” objects w/ PdBI at 0.3-0.5”
resolution & found no evidence for large-scale (~100 AU) RSDs
• Kinematic information on small scales needed26Comparison with observations: what fraction of“Class 0” protostars have large, 100AU-scale
RSDs? (Sakai’s talk)• Using CARMA, Tobin et al. (2012) detected in L1527 a roughly Keplerian disk
13COL1527 in Taurus

Tobin et al. 2012

• L1527: a case for large rotationally
supported “Class 0” disk?
•ALMA observations…

27

What fraction of “Class 0” protostars have large,
100AU-scale RSDs? ALMA observations of L1527
Rotation curve along major axis

Courtesy N. Ohashi, C18O 2-1, velocity map(km/s)

Vφ~ r-0.5

Vφ~ r-1(AU)• Keplerian-like inside ~60 AU, case for rotationally supported “Class 0” disk strengthened• Unclear how common…two other good cases (VLA 1623, Murillo+13; HH212, Codella+14)• If rare, strong magnetic braking may be ok or even desirable
L1527 and others: perhaps rare cases of BOTH weak magnetic field AND large B-rotation misalignment?

• If common, key ingredient missing?

28

IV. How to form rotationally supported disks?
(3) turbulence-enabled large disk formation?
• Plausibility first demonstrated numerically by Santos-Lima et al. (2012)

Preliminary simulations• Confirmed and extended by Seifried et al. (2012, 2013), Myers et al. (2013),
Joos et al. (2013) & Li et al. (2014)29

Illustration: effects of increasing level of turbulence on disk formation (Li+2014)
Non-turbulent

M=0.0

Equatorial density map

M=0.1

M=0.3
Sonic turbulence

1000 AU

M=0.5

M=0.7

30

M=1.0

Turbulence-enabled disk formation:
reasons debated• Magnetic flux loss from central region due toturbulence-induced reconnection (Santos-Lima et al.
2012, 2013, motivated by Lazarian & Vishniac 1999)• Tangling of field lines and strong shear conducive to
disk formation (Seifried et al. 2012, 2013)• Turbulence induces magnetic flux loss & B-rotation
misalignment (Joos et al. 2013; Myers et al. 2013)• Enhanced magnetic flux loss via
pseudodisk warping (Li et al. 2014)! promising but more work needed31

IV. How to form rotationally supported disks?
(4) Ohmic dissipation-enabled small disk formation• Outflow a defining characteristic of “Class 0” objects (Andre et al. 1993)• Fast (~100 km/s) jets (e.g., HH211) most likely driven by disks on sub-AU scale• Small, AU-scale rotationally supported disks can form through Ohmic dissipation,
the dominant non-ideal MHD effect at highest densities
demonstrated semi-analytically by Dapp & Basu (2010) and Dapp et al. (2012)
numerically by Machida et al. (2010) and Tomida et al. (2013)(see also related work by Shu et al. 2006 and Krasnopolsky et al. 2010)

SiO (8-7), proper motion " ~170 km/s
(Lee et al. 2007)32

IV. How to form rotationally supported disks?
(4) Ohmic dissipation-enabled small disk formation• Example of small disk formation: radiative MHD simulations with nested grid(Tomida et al. 2013; see also Tomida et al. 2015; Tsukamoto et al. 2015; talk by Machida)
Equatorial plane, ρ, V

Equatorial plane, T•Include both first &second (stellar) core
•Challenging simulations,stopped ~ 1yr after 2nd core
Ideal MHD•No RSD in ideal MHD•Small (0.35AU) RSD with
Ohmic dissipation"How does the small disk
grow in time, in presenceof AD & Hall effect?
With Ohmic dissipation

33

IV. How to form rotationally supported disks?
(4) Non-ideal MHD-enabled disk formation• More likely b/c of reduction in cosmic ray ionization rate
on disk-forming scales (Padovani et al. 2013, 2015)(1) helicoidal motions " longer path, larger column of matter(2) magnetic mirroring " fewer CR particles deeper inStrong•

weakIonization rate may be
reduced to as low as~10-20 S-1 on 102AU scalein some cases•

strongReduction should be
conducive to diskformation & more work
warranted
34

IV. How to form rotationally supported disks?
(5) Late formation of large disks from envelope depletion?
Slower rotation
Faster rotation• IF the slowly rotating envelope isremoved, the braking may become
inefficient " rapid disk growth?
• By outflows? (Mellon & Li 2008)

11111(rapid disk growth during Class 0" I?)• By accretion? (Machida et al. 2010)35

Summary: difficulties with disk formation• Disk formation suppressed in ideal MHD for realisticallymagnetized dense cores under axisymmetry (2D)• Microscopic non-ideal effects (AD, Ohmic, Hall)
do not appear to enable large disk formation in 2D• Protostellar accretion flows unstable to magnetic
interchange instability driven by flux redistribution! Disk formation not as simple asgenerally believed(basic problem: magnetic flux concentration by accretion)

Summary: Possible Resolutions• Weak core magnetization probably not consistent withavailable data
• Misalignment weakens magnetic braking
but may not enable large disk formationin majority of dense cores

1111

1111

• Turbulence is found to facilitate
disk formation, but details debated• Ohmic dissipation enables small disks, but growth uncertain• Outflow may weaken magnetic braking
through envelope stripping, yet to be quantified! Problem of disk formation not yet resolved definitively