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
wor
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)