A resolution of the magnetic braking catastrophe during the second collapse cc2yso UWO, May 17, 2010 – Wolf Dapp Wolf B. Dapp & Shantanu Basu

Protostellar disks

Protostellar disks

Angular momentum and magnetic flux problem Mellon & Li (2008): magnetic braking; disk requires Mellon & Li (2009): add ambipolar diffusion; disk still requires and weak ionization cc2yso UWO, May 17, 2010 – Wolf Dapp

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 < ~10 10 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 ~10 12 < n < cm -3 (Nakano et al. 2002, Kunz & Mouschovias 2010)

Introduction Method and Initial State Results Future work Summary Outline 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,  r min = 0.02, resolving the 2 nd 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 H eV secondc ore Ionization of eV

Initial state central number density column density rotation rate external number density vertical magnetic field mass-to-flux ratio Temperature n c = 4.4 x 10 6 cm -3  c = 0.23 g cm -2  edge = 0.3 km s -1 pc -1 = s -1 n ext = 10 3 cm -3 B z = 200  G  0 = 2 T = 10 K cc2yso UWO, May 17, 2010 – Wolf Dapp

Introduction Method and Initial State Results Future work Summary Outline 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 classical Toomre instability introduction of sink cell (a few in size) after 2 nd core formation centrifugal balance is achieved, and disk fragments into ring Dapp & Basu (2010) centrifugal balance disk fragments

Disk formation! cc2yso UWO, May 17, 2010 – Wolf Dapp introduce sink cell (a few ) after 2 nd core forms Dapp & 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)

Disk formation! cc2yso UWO, May 17, 2010 – Wolf Dapp Dapp & Basu (2010)

Future work 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 2 nd core despite magnetic braking, a disk does form at a very early age, very close to the 2 nd 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 1 st core, –effectively shuts off magnetic braking, –increases m-t-f ratio by ~10 3 cc2yso UWO, May 17, 2010 – Wolf Dapp Summary

The End cc2yso UWO, May 17, 2010 – Wolf Dapp

initial equilibrium profile free-fall profile first core second core free-fall profiles magnetic subkeplerian structures magnetic sub- Keplerian structures! ∝ r -2 ∝ r -1 effects of central object Dapp & Basu (in prep.) ∝ r -2 ∝ r -1

Thin-disk test thin-disk model is justified within the 1 st core, and in the prestellar profile outside it’s not applicable within the 2 nd 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 E mag ~ E grav cc2yso UWO, May 17, 2010 – Wolf Dapp Basu (2005) Alves et al. (2008)