The Growth of the MRI in Protoplanetary Disks

The Growth of the MRI in Protoplanetary Disks Mark Wardle Macquarie University Sydney, Australia Protoplanetary disks Magnetic diffusion MRI with diffusion Ionisation equilibrium Live and Dead zones

Minimum-mass solar nebula (Weidenschilling 1977; Hayashi 1981)

Kitamura et al 2002 Ap. J

Protostellar disks are poorly conducting • high density implies low conductivity – recombinations relatively rapid – drag on charged particles • deeper layers shielded from ionising radiation for r < 5 AU – x-ray attenuation column ~10 g/cm 2 – cosmic ray attenuation column ~100 g/cm 2 – “dead zone” near midplane (Gammie 1996)

Magnetic diffusion regimes fully ionized partially ionized Ideal MHD ions and electrons tied to field ions, electrons and neutrals tied to field Ambipolar – neutrals decoupled ions and electrons decoupled ions, electrons and neutrals decoupled Hall Ohmic

• If the only charged species are ions and electrons, • Three distinct diffusion regimes: log B – Ohmic (resistive) ambipolar – Hall – Ambipolar hall ohmic log n

Wardle 2007

Magnetorotational instability

Resistivity calculations • minimum mass solar nebula – assume isothermal in z-direction • ionisation by cosmic rays and/or x-rays from central star • simple reaction scheme following Nishi, Nakano & Umebayashi (1993) – H+, H 3+, He+, C+, molecular (M+) and metal ions (M+), e-, and charged grains – extended to allow high grain charge (T larger than in molecular clouds) • adopt model for grains – results for “no grains” or 0. 1 mm grains presented here • evaluate resistivity components – when can the field couple to the shear in the disc? – which form of diffusion is dominant?

x-ray ionisation rate cosmic rays Igea & Glassgold 1999

Reaction scheme

Abundances: 1 AU, no grains e M+ m+ C+ He+ H 3+ log n / n. H H+ log z(s-1) z/h Wardle 2007

log (cm 2 s-1) Resistivities: 1 AU, no grains poor coupling ( = h cs) Ambipolar Hall Ohmic 1 G 0. 1 G z/h Wardle 2007

Wardle 2007

MRI growth rate (Ω)

MRI growth rate (Ω) no Hall diffusion

Salmeron & Wardle 2005

Salmeron & Wardle 2005

Abundances: 1 AU, 0. 1 mm grains m+ log n / n. H C+ He+ M 0 + H 3+ e H+ log z(s-1) 1 2 3 -4 -11 -3 -12 -2 -13 -14 z/h Wardle 2007

Wardle 2007

MRI growth rate (Ω)

Wardle 2007

MRI growth rate (Ω)

MRI growth rate (Ω)

MRI growth rate (Ω)

MRI growth rate (Ω) no Hall diffusion

Salmeron & Wardle 2008

Notes • No grains: coupling can be maintained even at the midplane at 1 AU – Hall diffusion dominates – active ≈ 1700 g cm– 2 • Grains increase magnetic diffusion – 1 AU: 0. 1 µm active ≈ 2 g cm– 2 3 µm active ≈ 80 g cm– 2 – 5 AU: 1 µm active ≈ total • No cosmic rays? – in absence of grains, X-rays active ≈ 150 g cm– 2 at 1 AU – with grains, X-rays dominate ionization of active layer in any case • Increase disk mass? – active ≈ unchanged • Small grains? – aaaargh

Dead zones vs live zones • critical for disk evolution – vertical and radial transport – grain evolution – planet migration • external ionizing sources – cosmic rays (maybe) dominate at midplane – x-rays dominate at surface – stellar energetic particles? • poisoning of magnetic coupling by grains – dead zone boundary: grains just able to soak up most electrons – key grain parameter: Integral a n(a) da – e. g, 10– 3 cf standard 0. 1 µm grains keeps midplane alive at 1 AU in MMSN