A resolution of the magnetic braking catastrophe during the

second collapse

cc2yso UWO,May 17, 2010 – Wolf Dapp

Wolf B. Dapp & Shantanu Basu

Protostellar disksw

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Magnetic flux and angular momentum problem

• the resolution of those two problems are interlinked (preceding talks by Galli, Li)

• cloud cores have

cc2yso UWO,May 17, 2010 – Wolf Dapp

ideal MHD

Magnetic braking

• coupling of disk’s magnetic field with external field

• torsional Alfvén waves transfer angular momentum from disk to low-density external medium

Ambipolar diffusion

• ions gyrate around magnetic field lines

• neutrals effectively ‘feel’ the magnetic field through collisions

• they drift only slowly past the ions

• dominant flux loss mechanism in the regime n < ~1010 cm-3

(c) 2006 Pearson Education, Inc., publishing as Addison Wesley

Ohmic dissipation

• if charged particles are not well-coupled to the magnetic field, collisions can knock them off, and flux is dissipated

• dominant flux loss mechanism between ~1012 < n < 1015 cm-3 (Nakano et al. 2002, Kunz & Mouschovias 2010)

• Introduction

• Method and Initial State

• Results

• Future work

• Summary

Out

line

cc2yso UWO,May 17, 2010 – Wolf Dapp

• a disk forms under the right conditions

common approach

cc2yso UWO,May 17, 2010 – Wolf Dapp

AU-sized sink cell

resolution down to

stellar sizes

our approach

• AU-sized sink cells, only first core resolved

• no disk formation found

Method• axisymmetric, rotating, thin disk

• logarithmic, adaptive grid, N = 1024, rmin = 0.02 , resolving the 2nd core

• ambipolar diffusion, ohmic dissipation, magnetic braking, and force-free external B

• barotropic pressure-density relation

• disk is hydrostatic in z-direction, incl point mass/disk gravity, magnetic pinching, thermal and external pressure

cc2yso UWO,May 17, 2010 – Wolf Dapp

Magnetic braking and ohmic dissipation

cc2yso UWO,May 17, 2010 – Wolf Dapp

from steady-state Alfvén wave propagation

(Basu & Mouschovias 1994)

resistivity,Machida et al. (2007), Nakano et al. (2002)

ionization fraction

Barotropic pressure-density relation

Masunaga & Inutsuka (2000)

eff = 1.1

= 7/5

collapsing dense core

“first core”

cc2yso UWO,May 17, 2010 – Wolf Dapp

second collapse

Dissociation of H2

@4.5 eV

secondcoreIonization of HI

@13.6 eV

Initial state

• central number density• column density• rotation rate

• external number density• vertical magnetic field• mass-to-flux ratio• Temperature

nc = 4.4 x 106 cm-3

c = 0.23 g cm-2

edge = 0.3 km s-1 pc-1

= 10-14 s-1

next = 103 cm-3

Bz = 200 G

0 = 2

T = 10 K

cc2yso UWO,May 17, 2010 – Wolf Dapp

• Introduction

• Method and Initial State

• Results

• Future work

• Summary

Out

line

cc2yso UWO,May 17, 2010 – Wolf Dapp

Results: Density profile

cc2yso UWO,May 17, 2010 – Wolf Dapp

magnetic wall

added centrifg support under flux freezing

first core

second core Dapp & Basu (2010)

ohmic dissipation

flux-freezing

expansion wave, r -1/2

prestellar infall

profile, r -1

Results: Magnetic Field

cc2yso UWO,May 17, 2010 – Wolf Dapp

magnetic wall

Dapp & Basu (2010)

}3 orders of magnitude difference

Results: Mass-to-flux ratio

cc2yso UWO,May 17, 2010 – Wolf Dapp

Dapp & Basu (2010)

Results: Angular velocity

cc2yso UWO,May 17, 2010 – Wolf Dapp

Dapp & Basu (2010)

magnetic braking

catastrophe

expansion wave, r -2

Disk formation!

cc2yso UWO,May 17, 2010 – Wolf Dapp

• introduce sink cell (a few ) after 2nd core formsDapp & Basu (2010)

centrifugal balance

• centrifugal balance is achieved

Disk formation!

• infall velocity plummets

cc2yso UWO,May 17, 2010 – Wolf Dapp

Dapp & Basu (2010)

Disk formation!

• disk fragments into ring

cc2yso UWO,May 17, 2010 – Wolf Dapp

classical Toomre

instability

Dapp & Basu (2010)

Fut

ure

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k

• very fast runs, allows for large parameter searches

• Add non-axisymmetry or effective viscosity to stabilize disk / long-term disk evolution

cc2yso UWO,May 17, 2010 – Wolf Dapp

• we resolve the 2nd core• despite magnetic braking, a disk does form

at a very early age, very close to the 2nd core• we can differentiate between prestellar and

centrifugal disks• we resolve and identify features like

– expansion waves in – magnetic wall(s)

• Ohmic dissipation – removes flux efficiently within 1st core,– effectively shuts off magnetic braking, – increases m-t-f ratio by ~103

cc2yso UWO,May 17, 2010 – Wolf Dapp

Sum

mar

y

The

End

cc2yso UWO,May 17, 2010 – Wolf Dapp

Thin-disk test

• thin-disk model is justified within the 1st core, and in the prestellar profile outside

• it’s not applicable within the 2nd core, as expected

cc2yso UWO,May 17, 2010 – Wolf Dapp

Z = r

Initial profile

• collapse profile with and

• angular velocity goes as column density

cc2yso UWO,May 17, 2010 – Wolf Dapp

Dapp & Basu (2010)

Expansion wave effects

• gravitational field just outside the central stellar core instead of as further out

• free-fall profile outside of star, – infall velocity – steady-state mass accretion

• angular velocity now– angular momentum

cc2yso UWO,May 17, 2010 – Wolf Dapp

Mass-to-flux-ratio in the ISM

• observations consistent with = 1

• assembled from ionized subcritical HI gas

• problems with higher :– accumulation length

~1 kpc for = 1– accumulation speed

10 km/s ↔ 10 pc/Myr– collapse as soon as > 1

• large scale fields ordered

• Emag ~ Egrav

cc2yso UWO,May 17, 2010 – Wolf Dapp

Basu (2005)

Alves et al. (2008)