Mark Wardle Raquel Salmeron (ANU) Macquarie University BP .

Dead zones in protoplanetary discsMark WardleMacquarie UniversityRaquel Salmeron (ANU)BP Pandey (Macquarie) Protoplanetary discsMagnetic fieldsDead zonesMRI and diffusionModification to dead zones

Magnetic fields Magnetic fields are present in PPDs– measured B in molecular clouds B 10 mG in PPDsZeeman, submm polarization Weak magnetic fields create turbulence– subequipartition B and rotational shear MHD turbulent torquemagnetorotational instability (MRI)– accretion, heating, stirring (chemistry, dust grains)disc evolutionobservational signaturesplanet formationplanet migration (!) Accretion rates require B 0.1 – 1G at 1AU

Magnetorotational instability (MRI)! magnetic field couples different radii in disc" tension transfers angular momentum outwards" kh 1 required to fit in disc, i.e. vA/cs 1! resulting turbulence transports angular momentum outwards!

Flux freezing breaks down in PPDs high density and low ionisation– drag on charged particles deeper layers shielded from ionising radiation for r 5 AU– x-ray attenuation column 10 g/cm2– cosmic ray attenuation column 100 g/cm2– “dead zone” near midplane (Gammie 1996)

MRI with !dead zone!Turner & Sano 2008!

Magnetic diffusionIdeal MHDelectrons, ions and neutralstied to magnetic fieldAmbipolar diffusionneutrals decoupledHall diffusionions and neutrals decoupledOhmic diffusionelectrons, ions and neutralsdecoupled

AmbipolarHall

Wardle 2007

Mark Wardle & Raquel Salmeron!!01stableηP Ω / vA22Ik0kc!!0!"ohmic ambipolardiffusionIIk0k!!"kIIIIIIkideal MHDIII0–2–1012s ηH Ω / vA2gure 5. Schematic dependence of the behaviour of the MRI on the Hall and Pedersen (ohmic ambipolar) diffusivities, ηH and ηinitial magnetic field sB ẑ in a Keplerian disc, where s 1 is the sign of Bz . The field is assumed to be weak so that stratifin be neglected. Physical values of the Pedersen conductivity ηP are non-negative, and there are no unstable modes for Hall diffus22H Ω/vA 2. The unstable region sηH Ω/vA 2 is subdivided according to the dependence of the growth rate on wave numhich is sketched in the insets labelled I–III. In region I (outside the red locus given by eq. 32), wave numbers less than a cutoffstable with the maximum growth rate ν0 attained at k0 (see eqs 33, 34, and 35). Between theWardlered and &blueloci (regionII) alSameron2012

–50 Z –41Charged particle abundances–90 Z –81–22–6Mdust / Mgas 10–4m –14C HeH3H log n / nHlog n / nH–10 M e–70 Z –61–60 Z –51–80 Z –71–18–50 Z –41–22–6Mdust / Mgas 0M , e–14–18m C Helog ζH (s–1)H3H –220log n / nHlog n / nH–101 23z/h45ratios of 10 2 , 10 4 and 0 are presented in Figure 11. Thediffusionandisthemagnetorotatiionisation rate Hallas a functionof depthplottedin the lowerpanel – interstellar cosmic rays are the dominant source bemalow 2 scale heights, above this ionisation by stellar x-rays–6raydominates. In the no-graincase, electrons and metal ionsMdust / Mgas 10–2umare the dominant charged species because the metal ionsdephave the smallest recombination rate coefficient. The top –10panel shows the effect of adding a populationmIgeof 1 µm-radius1 Z 10C grains withtotal mass 1% of the gas mass:Hegrains acquire rayaZ 0 Hattcharge viasticking of Melectronsand ions fromthe gas phase. 11 Z 20 3Htercharge is determined by thee–14Above z/h 2.5 the grainZ –61andcompetitive rates of sticking of ions –70and electrons,with the–60 Z –51effecoulomb repulsion of electrons by negatively-charged grains–80 Z –71offsetting their greater thermal velocity compared to ions–18–50This Z leads–41 to a gaus(Spitzer 1941; Draine & Sutin 1987).ratsian grain charge distribution with mean charge (in unitsion–90 Z –81of e) Zg 4akT /e2 67 andstandard deviationpan–22 Zg 1/2 8. Most recombinations still occur in the gaslow–6phase. The abundance of metal ions and electrons is reduceddomover thegrain-freecasefor z/h 5 where the charge storedMdust/ Mgas 10–4areon grains becomes comparable to the electron abundance.havFor z/h 2–2.5, the abundances of ions and electrons have–10panC declined to the point that the majority of electronsstick to Hegramgrain surfaces before they can recombineHin3 the gas phase,chaand most neutralisations occur when ionsH stick to negativelyAb–14charged grains (Nakano & Umebayashi 1980). Closer to the comMmid-plane,the ionisation fraction is so low that most grainsecouare lightly charged, so that coulomb–70attraction Z –61or repulsionoffs–60 Z –51Then ions and–18of ions and electrons by grains is negligible.(Sp–80 encounter,Z –71electrons stick to any grain that theywith resiacombinations occurring on grain surfaces.middle panel–50 ZThe –41ofof Fig. 11 shows what happens if 99% of the grains are re–22 moved (e.g. by settling to the mid-plane). The capacity ofpha–6the grain population to soak up electrons from the gas phaseoveis reducedand the height below which grainsM a /hundredfold,M 0

0.01052.0! / Ω40.751.530.51.00.50.2502z / hz (10 12 cm)Mdust / Mgas 0Bz 010.002.55Bz 02.041.531.020.510.002.55ohmic onlyz / hz / hz (10 12 cm)2.5z / hass ratios 0 and 10 2 are displayed in Figs 12 and 13 re2.55ectively.0First consider the case in which grains Barez absentig. 12).2.0The bottom panel shows the fastest growth rate4ubject to kh 1) as a function of magnetic field strengthd height if Hall and ambipolar diffusion are neglected and1.5 diffusion is included. In this case the departure3ly ohmic2om ideal MHD is measured by ηΩ/vA(the inverse of thesasser 1.0number; see section 6.1), which strongly decreases2th height as the fractional ionisation and Alfvén speedth increase strongly. For the ionisation profile plotted in0.512g. 11 it turns out that ηΩ/vAis small near the surfaceo that ideal MHD holds) and large near the mid-plane (so0.0 damping is severe). As a result, near the sur0at ohmicce the largest achievable growth rate is close to the ideal2.55lue 0.75 Ω, but declines rapidly below theheightwhereohmiconly2Ω/vA 1. The height of the transition between these4gimes 2.0declines with increasing field strength, simply be2 2use ηΩ/vA B . A second consideration is that thenge of 1.5MRI-unstable wave numbers is bounded above by3(see eq 33), and this must be larger than 1/h if any unstae modes are to exist with kh 1. It is this criterion that2ovides 1.0the upper and lower envelopes to the unstable re2on in Figs. 12 and 13. Near the surface where ηΩ/vA 1,is approximately3Ω/vA , and as vA rapidly increases0.51th height the unstable range shifts to wave numbers with 1 and it is no longer possible to find unstable modes0.00at fit withina scale –2height. Thisoccurs0 when themagnetic–3–1121010101010essure roughlyexceedsthe10gas pressure.Bycontrast,close 2B (G)the mid-plane where ηΩ/vA 1, kc 3vA /η and theave numbers are pushed out of the relevant range by thez (10 12 cm)Mark Wardle & Raquel SalmeronMaxgrowth rate: no dust0z (10 12 cm)60.5Wardle & Sameron 2012

0.51Hall diffusion and the0magnetorotationalinstability in protoplanetary discs–217Max growth rate: Mdust / Mgas 1005Bz 02.0z (10 12 cm)z (10 12 cm)Mdust / Mgas 10–2! / Ω40.751.530.51.00.50.2502z / hz / h10.002.55Bz 02.041.531.020.510.002.55z / hz / hz (10 12 cm)2.5z (10 12 cm)0.0MRI while extending the unstable wavelengths to shorteravelengths(see Fig. 7). As a result, super-equipartition2.55elds are unstable near the mid-plane. We emphasisethatBz 0respective of the magnitudes of the other diffusivities Hall2.042ffusion is completely stabilising when ηH Ω/vA 2. Thisresponsible for the sharp cutoff at the lower boundary in1.5 panel of Fig. 12.3he middleWhen a full complement of 1 µm dust grains are present.e. dust-to-gasmass ratio 10 2 ; charged species as in lower1.02anel of Fig. 11), the diffusivities are greatly increased nearhe mid-plane because electrons are locked up by grains and0.5immobile. Fig. 13 displays the same trends as1 innderedhe zero-grain case are apparent, but the MRI-unstable reon is 0.0now restricted to the upper layers of the disc, and0all cases the bulk of the disc is stable to the MRI. While2.55he differencesbetween the three panels might appear lessohmic onlyvere in this case, the strong density stratificationmeanshat there2.0 are orders of magnitude differences in the column4ensity of the MRI-unstable region.Fig. 14 shows the magnetically-active column density1.53s a function of field strength for different assumptions rearding the diffusivity when 99% of grains are assumed toave settled.The active column density varies by 1-2 orders1.02magnitude depending on the accuracy of the treatmentmagnetic diffusion. First, neglect Hall diffusion and con0.51der either ohmic diffusion alone or ohmic and ambipolarffusion (i.e. Pedersen diffusion) operating in concert (red0.0 curves). Ohmic diffusion dominates ambipolar0ng-dashed–3–2–101210101010ffusion exceptfor10strong magneticfields and10low densities,o there is little difference betweenB (G) these two cases exceptr magnetic fields in excess of 1 G where the additionalohmic only2.0Wardle & Sameron 20124

Column density of active layer!Wardle & Salmeron, MNRAS submitted!Wardle & Sameron 2012

Column density of active layer!Wardle & Salmeron, MNRAS submitted!Wardle & Sameron 2012

Summary & Discussion Ionisation levels determine extent of magnetically turbulent regions inprotoplanetary discs– dead zones: 0.3 – 3 AU from central star– topped by magnetically active layers Hall diffusion modifies thickness of active layer by an order ofmagnitude– dead zones: Bz 0 vs Bz 0 ; depends on magnitude of B– can sustain accretion rates in these layers across radial extent of dead zone– may drive “undead” zones– more general geometries destabilise disk (Pandey & Wardle submitted) MHD simulations?– linear analysis such as used here appears to be a good predictor of deadzone extent– no MHD simulations with strong hall diffusion (yet)– PS: simulations with non net flux are unrealistic

BP Pandey (Macquarie) Protoplanetary discs Magnetic fields Dead zones MRI and diffusion Modification to dead zones. Magnetic fields Magnetic fields are present in PPDs – measured B in molecular clouds B 10 mG in PPDs Zeeman, s